Main:z from

Percentage Accurate: 91.6% → 99.3%

Time: 28.3s

Alternatives: 25

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
   (- (sqrt (+ z 1.0)) (sqrt z)))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 91.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
   (- (sqrt (+ z 1.0)) (sqrt z)))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ t_2 := \sqrt{y + 1}\\ t_3 := \sqrt{1 + z}\\ t_4 := t\_3 - \sqrt{z}\\ t_5 := \sqrt{1 + x}\\ t_6 := t\_4 + \left(\left(t\_5 - \sqrt{x}\right) + \left(t\_2 - \sqrt{y}\right)\right)\\ \mathbf{if}\;t\_6 \leq 0.99:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + t\_5}\right) + t\_4\right) + t\_1\\ \mathbf{elif}\;t\_6 \leq 1.999998:\\ \;\;\;\;t\_1 + \left(\left(t\_5 + \mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{y} + t\_2}\right)\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(\left(\left(1 + t\_5\right) + \mathsf{fma}\left(y, 0.5, \frac{1}{\sqrt{z} + t\_3}\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_2 (sqrt (+ y 1.0)))
        (t_3 (sqrt (+ 1.0 z)))
        (t_4 (- t_3 (sqrt z)))
        (t_5 (sqrt (+ 1.0 x)))
        (t_6 (+ t_4 (+ (- t_5 (sqrt x)) (- t_2 (sqrt y))))))
   (if (<= t_6 0.99)
     (+ (+ (fma 0.5 (sqrt (/ 1.0 y)) (/ 1.0 (+ (sqrt x) t_5))) t_4) t_1)
     (if (<= t_6 1.999998)
       (+
        t_1
        (-
         (+ t_5 (fma 0.5 (sqrt (/ 1.0 z)) (/ 1.0 (+ (sqrt y) t_2))))
         (sqrt x)))
       (+
        t_1
        (-
         (+ (+ 1.0 t_5) (fma y 0.5 (/ 1.0 (+ (sqrt z) t_3))))
         (+ (sqrt y) (sqrt x))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + t)) - sqrt(t);
	double t_2 = sqrt((y + 1.0));
	double t_3 = sqrt((1.0 + z));
	double t_4 = t_3 - sqrt(z);
	double t_5 = sqrt((1.0 + x));
	double t_6 = t_4 + ((t_5 - sqrt(x)) + (t_2 - sqrt(y)));
	double tmp;
	if (t_6 <= 0.99) {
		tmp = (fma(0.5, sqrt((1.0 / y)), (1.0 / (sqrt(x) + t_5))) + t_4) + t_1;
	} else if (t_6 <= 1.999998) {
		tmp = t_1 + ((t_5 + fma(0.5, sqrt((1.0 / z)), (1.0 / (sqrt(y) + t_2)))) - sqrt(x));
	} else {
		tmp = t_1 + (((1.0 + t_5) + fma(y, 0.5, (1.0 / (sqrt(z) + t_3)))) - (sqrt(y) + sqrt(x)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_2 = sqrt(Float64(y + 1.0))
	t_3 = sqrt(Float64(1.0 + z))
	t_4 = Float64(t_3 - sqrt(z))
	t_5 = sqrt(Float64(1.0 + x))
	t_6 = Float64(t_4 + Float64(Float64(t_5 - sqrt(x)) + Float64(t_2 - sqrt(y))))
	tmp = 0.0
	if (t_6 <= 0.99)
		tmp = Float64(Float64(fma(0.5, sqrt(Float64(1.0 / y)), Float64(1.0 / Float64(sqrt(x) + t_5))) + t_4) + t_1);
	elseif (t_6 <= 1.999998)
		tmp = Float64(t_1 + Float64(Float64(t_5 + fma(0.5, sqrt(Float64(1.0 / z)), Float64(1.0 / Float64(sqrt(y) + t_2)))) - sqrt(x)));
	else
		tmp = Float64(t_1 + Float64(Float64(Float64(1.0 + t_5) + fma(y, 0.5, Float64(1.0 / Float64(sqrt(z) + t_3)))) - Float64(sqrt(y) + sqrt(x))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$4 + N[(N[(t$95$5 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 0.99], N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$6, 1.999998], N[(t$95$1 + N[(N[(t$95$5 + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(N[(1.0 + t$95$5), $MachinePrecision] + N[(y * 0.5 + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.98999999999999999

   1. Initial program 55.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - \color{blue}{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right) - x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - x}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\color{blue}{\sqrt{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 55.4

          \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \color{blue}{\sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 55.4%

      \[\leadsto \left(\left(\color{blue}{\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \frac{1}{\color{blue}{\sqrt{x}} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + \color{blue}{\sqrt{1 + x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 73.7

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + \sqrt{\color{blue}{1 + x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Simplified 73.7%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.98999999999999999 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.99999799999999994

   1. Initial program 95.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 95.7

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \color{blue}{\sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 95.7%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + x}} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + x}} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \color{blue}{\frac{1}{\sqrt{y} + \sqrt{1 + y}}}\right)\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \frac{1}{\color{blue}{\sqrt{y} + \sqrt{1 + y}}}\right)\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \frac{1}{\color{blue}{\sqrt{y}} + \sqrt{1 + y}}\right)\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{y} + \color{blue}{\sqrt{1 + y}}}\right)\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{y} + \sqrt{\color{blue}{1 + y}}}\right)\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. sqrt-lowering-sqrt.f64 44.0

          \[\leadsto \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Simplified 44.0%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.99999799999999994 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 98.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z + 1}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 98.3

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \color{blue}{\sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 98.3%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot y + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot y + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Simplified 60.4%

      \[\leadsto \color{blue}{\left(\left(\left(1 + \sqrt{1 + x}\right) + \mathsf{fma}\left(y, 0.5, \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) \leq 0.99:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) \leq 1.999998:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right)\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(1 + \sqrt{1 + x}\right) + \mathsf{fma}\left(y, 0.5, \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{1 + x}\\ t_4 := \left(t\_2 - \sqrt{z}\right) + \left(\left(t\_3 - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right)\\ t_5 := \sqrt{\frac{1}{z}}\\ \mathbf{if}\;t\_4 \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + t\_5, t\_3\right) - \sqrt{x}\\ \mathbf{elif}\;t\_4 \leq 2.0001:\\ \;\;\;\;1 + \left(\mathsf{fma}\left(0.5, t\_5, t\_1\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{elif}\;t\_4 \leq 2.99999997:\\ \;\;\;\;\left(\left(\left(1 + t\_1\right) + \left(t\_2 - \sqrt{x}\right)\right) - \sqrt{z}\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(3 - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (sqrt (+ 1.0 x)))
        (t_4 (+ (- t_2 (sqrt z)) (+ (- t_3 (sqrt x)) (- t_1 (sqrt y)))))
        (t_5 (sqrt (/ 1.0 z))))
   (if (<= t_4 1.00005)
     (- (fma 0.5 (+ (sqrt (/ 1.0 y)) t_5) t_3) (sqrt x))
     (if (<= t_4 2.0001)
       (+ 1.0 (- (fma 0.5 t_5 t_1) (+ (sqrt y) (sqrt x))))
       (if (<= t_4 2.99999997)
         (- (- (+ (+ 1.0 t_1) (- t_2 (sqrt x))) (sqrt z)) (sqrt y))
         (+
          (- (sqrt (+ 1.0 t)) (sqrt t))
          (- (- 3.0 (sqrt x)) (+ (sqrt y) (sqrt z)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((1.0 + z));
	double t_3 = sqrt((1.0 + x));
	double t_4 = (t_2 - sqrt(z)) + ((t_3 - sqrt(x)) + (t_1 - sqrt(y)));
	double t_5 = sqrt((1.0 / z));
	double tmp;
	if (t_4 <= 1.00005) {
		tmp = fma(0.5, (sqrt((1.0 / y)) + t_5), t_3) - sqrt(x);
	} else if (t_4 <= 2.0001) {
		tmp = 1.0 + (fma(0.5, t_5, t_1) - (sqrt(y) + sqrt(x)));
	} else if (t_4 <= 2.99999997) {
		tmp = (((1.0 + t_1) + (t_2 - sqrt(x))) - sqrt(z)) - sqrt(y);
	} else {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + ((3.0 - sqrt(x)) - (sqrt(y) + sqrt(z)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = sqrt(Float64(1.0 + x))
	t_4 = Float64(Float64(t_2 - sqrt(z)) + Float64(Float64(t_3 - sqrt(x)) + Float64(t_1 - sqrt(y))))
	t_5 = sqrt(Float64(1.0 / z))
	tmp = 0.0
	if (t_4 <= 1.00005)
		tmp = Float64(fma(0.5, Float64(sqrt(Float64(1.0 / y)) + t_5), t_3) - sqrt(x));
	elseif (t_4 <= 2.0001)
		tmp = Float64(1.0 + Float64(fma(0.5, t_5, t_1) - Float64(sqrt(y) + sqrt(x))));
	elseif (t_4 <= 2.99999997)
		tmp = Float64(Float64(Float64(Float64(1.0 + t_1) + Float64(t_2 - sqrt(x))) - sqrt(z)) - sqrt(y));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(3.0 - sqrt(x)) - Float64(sqrt(y) + sqrt(z))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 1.00005], N[(N[(0.5 * N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$5), $MachinePrecision] + t$95$3), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.0001], N[(1.0 + N[(N[(0.5 * t$95$5 + t$95$1), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.99999997], N[(N[(N[(N[(1.0 + t$95$1), $MachinePrecision] + N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.00005000000000011

   1. Initial program 82.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + t} + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \sqrt{1 + t}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{t}\right)}\right) + \sqrt{1 + t} \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \sqrt{t}\right)} + \sqrt{1 + t} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   5. Simplified 37.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      13. sqrt-lowering-sqrt.f64 38.9

          \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   8. Simplified 38.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   10. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right)} - \sqrt{x} \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       9. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x} \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x} \]

       11. sqrt-lowering-sqrt.f64 16.4

           \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}} \]

   11. Simplified 16.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}} \]

   if 1.00005000000000011 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.00010000000000021

   1. Initial program 97.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 16.9

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 16.9%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) \]

      13. sqrt-lowering-sqrt.f64 6.4

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

   8. Simplified 6.4%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   10. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

       3. --lowering--.f64 N/A

          \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

       4. +-commutative N/A

          \[\leadsto 1 + \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto 1 + \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto 1 + \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto 1 + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto 1 + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto 1 + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       10. +-commutative N/A

           \[\leadsto 1 + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto 1 + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

       12. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto 1 + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) \]

       13. sqrt-lowering-sqrt.f64 27.7

           \[\leadsto 1 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{y} + \color{blue}{\sqrt{x}}\right)\right) \]

   11. Simplified 27.7%

       \[\leadsto \color{blue}{1 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   if 2.00010000000000021 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.9999999700000002

   1. Initial program 97.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 53.7

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 53.7%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) \]

      13. sqrt-lowering-sqrt.f64 44.3

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

   8. Simplified 44.3%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   9. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \sqrt{x}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} \]

      3. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \sqrt{x}\right) - \sqrt{z}\right) - \sqrt{y}} \]

      4. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \sqrt{x}\right) - \sqrt{z}\right) - \sqrt{y}} \]

   10. Applied egg-rr 44.3%

       \[\leadsto \color{blue}{\left(\left(\left(1 + \sqrt{1 + y}\right) + \left(\sqrt{1 + z} - \sqrt{x}\right)\right) - \sqrt{z}\right) - \sqrt{y}} \]

   if 2.9999999700000002 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 99.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate-+r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \sqrt{1 + y}\right)} - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(1 + \sqrt{1 + x}\right)} + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(1 + \color{blue}{\sqrt{1 + x}}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{\color{blue}{1 + x}}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \color{blue}{\left(\sqrt{1 + y} - \sqrt{x}\right)}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\color{blue}{\sqrt{1 + y}} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \color{blue}{\sqrt{x}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. sqrt-lowering-sqrt.f64 99.5

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 99.5%

      \[\leadsto \color{blue}{\left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + 2\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + 2\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + y}} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + y}} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 99.5

          \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 99.5%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(3 - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. associate--r+ N/A

          \[\leadsto \color{blue}{\left(\left(3 - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(3 - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \left(\color{blue}{\left(3 - \sqrt{x}\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(3 - \color{blue}{\sqrt{x}}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. +-commutative N/A

          \[\leadsto \left(\left(3 - \sqrt{x}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(3 - \sqrt{x}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(3 - \sqrt{x}\right) - \left(\color{blue}{\sqrt{z}} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       8. sqrt-lowering-sqrt.f64 99.5

          \[\leadsto \left(\left(3 - \sqrt{x}\right) - \left(\sqrt{z} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 99.5%

       \[\leadsto \color{blue}{\left(\left(3 - \sqrt{x}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) \leq 2.0001:\\ \;\;\;\;1 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) \leq 2.99999997:\\ \;\;\;\;\left(\left(\left(1 + \sqrt{y + 1}\right) + \left(\sqrt{1 + z} - \sqrt{x}\right)\right) - \sqrt{z}\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(3 - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ t_4 := \sqrt{1 + x}\\ t_5 := \left(t\_2 - \sqrt{z}\right) + \left(\left(t\_4 - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right)\\ \mathbf{if}\;t\_5 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;t\_3 + 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)\\ \mathbf{elif}\;t\_5 \leq 1.99:\\ \;\;\;\;t\_3 + \left(t\_4 + \left(\frac{1}{\sqrt{y} + t\_1} - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \left(\left(\left(1 + t\_4\right) + \mathsf{fma}\left(y, 0.5, \frac{1}{\sqrt{z} + t\_2}\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_4 (sqrt (+ 1.0 x)))
        (t_5 (+ (- t_2 (sqrt z)) (+ (- t_4 (sqrt x)) (- t_1 (sqrt y))))))
   (if (<= t_5 2e-5)
     (+ t_3 (* 0.5 (+ (sqrt (/ 1.0 x)) (+ (sqrt (/ 1.0 y)) (sqrt (/ 1.0 z))))))
     (if (<= t_5 1.99)
       (+ t_3 (+ t_4 (- (/ 1.0 (+ (sqrt y) t_1)) (sqrt x))))
       (+
        t_3
        (-
         (+ (+ 1.0 t_4) (fma y 0.5 (/ 1.0 (+ (sqrt z) t_2))))
         (+ (sqrt y) (sqrt x))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((1.0 + z));
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double t_4 = sqrt((1.0 + x));
	double t_5 = (t_2 - sqrt(z)) + ((t_4 - sqrt(x)) + (t_1 - sqrt(y)));
	double tmp;
	if (t_5 <= 2e-5) {
		tmp = t_3 + (0.5 * (sqrt((1.0 / x)) + (sqrt((1.0 / y)) + sqrt((1.0 / z)))));
	} else if (t_5 <= 1.99) {
		tmp = t_3 + (t_4 + ((1.0 / (sqrt(y) + t_1)) - sqrt(x)));
	} else {
		tmp = t_3 + (((1.0 + t_4) + fma(y, 0.5, (1.0 / (sqrt(z) + t_2)))) - (sqrt(y) + sqrt(x)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_4 = sqrt(Float64(1.0 + x))
	t_5 = Float64(Float64(t_2 - sqrt(z)) + Float64(Float64(t_4 - sqrt(x)) + Float64(t_1 - sqrt(y))))
	tmp = 0.0
	if (t_5 <= 2e-5)
		tmp = Float64(t_3 + Float64(0.5 * Float64(sqrt(Float64(1.0 / x)) + Float64(sqrt(Float64(1.0 / y)) + sqrt(Float64(1.0 / z))))));
	elseif (t_5 <= 1.99)
		tmp = Float64(t_3 + Float64(t_4 + Float64(Float64(1.0 / Float64(sqrt(y) + t_1)) - sqrt(x))));
	else
		tmp = Float64(t_3 + Float64(Float64(Float64(1.0 + t_4) + fma(y, 0.5, Float64(1.0 / Float64(sqrt(z) + t_2)))) - Float64(sqrt(y) + sqrt(x))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 2e-5], N[(t$95$3 + N[(0.5 * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 1.99], N[(t$95$3 + N[(t$95$4 + N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(N[(N[(1.0 + t$95$4), $MachinePrecision] + N[(y * 0.5 + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.00000000000000016e-5

   1. Initial program 49.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + t} + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \sqrt{1 + t}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{t}\right)}\right) + \sqrt{1 + t} \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \sqrt{t}\right)} + \sqrt{1 + t} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   5. Simplified 23.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      13. sqrt-lowering-sqrt.f64 45.1

          \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   8. Simplified 45.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \color{blue}{\left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       9. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       10. /-lowering-/.f64 88.3

           \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   11. Simplified 88.3%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   if 2.00000000000000016e-5 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.98999999999999999

   1. Initial program 95.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 95.7

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \color{blue}{\sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 95.7%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} - \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} - \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + x}} + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + x}} + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} - \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\color{blue}{\frac{1}{\sqrt{y} + \sqrt{1 + y}}} - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\frac{1}{\color{blue}{\sqrt{y} + \sqrt{1 + y}}} - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\frac{1}{\color{blue}{\sqrt{y}} + \sqrt{1 + y}} - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\frac{1}{\sqrt{y} + \color{blue}{\sqrt{1 + y}}} - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + x} + \left(\frac{1}{\sqrt{y} + \sqrt{\color{blue}{1 + y}}} - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. sqrt-lowering-sqrt.f64 40.2

          \[\leadsto \left(\sqrt{1 + x} + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} - \color{blue}{\sqrt{x}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Simplified 40.2%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} - \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.98999999999999999 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 98.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z + 1}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 98.2

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \color{blue}{\sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 98.2%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot y + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot y + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Simplified 61.5%

      \[\leadsto \color{blue}{\left(\left(\left(1 + \sqrt{1 + x}\right) + \mathsf{fma}\left(y, 0.5, \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) \leq 1.99:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + x} + \left(\frac{1}{\sqrt{y} + \sqrt{y + 1}} - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(1 + \sqrt{1 + x}\right) + \mathsf{fma}\left(y, 0.5, \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ t_2 := \sqrt{1 + x} - \sqrt{x}\\ t_3 := t\_2 + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ t_4 := \sqrt{1 + z} - \sqrt{z}\\ t_5 := t\_4 + t\_3\\ t_6 := \sqrt{\frac{1}{z}}\\ \mathbf{if}\;t\_5 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;t\_1 + 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + t\_6\right)\right)\\ \mathbf{elif}\;t\_5 \leq 2.0001:\\ \;\;\;\;t\_1 + \left(t\_3 + 0.5 \cdot t\_6\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(t\_4 + \left(t\_2 + \mathsf{fma}\left(y, 0.5, 1 - \sqrt{y}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_2 (- (sqrt (+ 1.0 x)) (sqrt x)))
        (t_3 (+ t_2 (- (sqrt (+ y 1.0)) (sqrt y))))
        (t_4 (- (sqrt (+ 1.0 z)) (sqrt z)))
        (t_5 (+ t_4 t_3))
        (t_6 (sqrt (/ 1.0 z))))
   (if (<= t_5 2e-5)
     (+ t_1 (* 0.5 (+ (sqrt (/ 1.0 x)) (+ (sqrt (/ 1.0 y)) t_6))))
     (if (<= t_5 2.0001)
       (+ t_1 (+ t_3 (* 0.5 t_6)))
       (+ t_1 (+ t_4 (+ t_2 (fma y 0.5 (- 1.0 (sqrt y))))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + t)) - sqrt(t);
	double t_2 = sqrt((1.0 + x)) - sqrt(x);
	double t_3 = t_2 + (sqrt((y + 1.0)) - sqrt(y));
	double t_4 = sqrt((1.0 + z)) - sqrt(z);
	double t_5 = t_4 + t_3;
	double t_6 = sqrt((1.0 / z));
	double tmp;
	if (t_5 <= 2e-5) {
		tmp = t_1 + (0.5 * (sqrt((1.0 / x)) + (sqrt((1.0 / y)) + t_6)));
	} else if (t_5 <= 2.0001) {
		tmp = t_1 + (t_3 + (0.5 * t_6));
	} else {
		tmp = t_1 + (t_4 + (t_2 + fma(y, 0.5, (1.0 - sqrt(y)))));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_2 = Float64(sqrt(Float64(1.0 + x)) - sqrt(x))
	t_3 = Float64(t_2 + Float64(sqrt(Float64(y + 1.0)) - sqrt(y)))
	t_4 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	t_5 = Float64(t_4 + t_3)
	t_6 = sqrt(Float64(1.0 / z))
	tmp = 0.0
	if (t_5 <= 2e-5)
		tmp = Float64(t_1 + Float64(0.5 * Float64(sqrt(Float64(1.0 / x)) + Float64(sqrt(Float64(1.0 / y)) + t_6))));
	elseif (t_5 <= 2.0001)
		tmp = Float64(t_1 + Float64(t_3 + Float64(0.5 * t_6)));
	else
		tmp = Float64(t_1 + Float64(t_4 + Float64(t_2 + fma(y, 0.5, Float64(1.0 - sqrt(y))))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 + t$95$3), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$5, 2e-5], N[(t$95$1 + N[(0.5 * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.0001], N[(t$95$1 + N[(t$95$3 + N[(0.5 * t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(t$95$4 + N[(t$95$2 + N[(y * 0.5 + N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.00000000000000016e-5

   1. Initial program 49.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + t} + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \sqrt{1 + t}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{t}\right)}\right) + \sqrt{1 + t} \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \sqrt{t}\right)} + \sqrt{1 + t} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   5. Simplified 23.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      13. sqrt-lowering-sqrt.f64 45.1

          \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   8. Simplified 45.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \color{blue}{\left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       9. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       10. /-lowering-/.f64 88.3

           \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   11. Simplified 88.3%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   if 2.00000000000000016e-5 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.00010000000000021

   1. Initial program 96.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. /-lowering-/.f64 56.6

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + 0.5 \cdot \sqrt{\color{blue}{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 56.6%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 2.00010000000000021 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 98.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(1 + \frac{1}{2} \cdot y\right) - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\color{blue}{\left(\frac{1}{2} \cdot y + 1\right)} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\frac{1}{2} \cdot y + \left(1 - \sqrt{y}\right)\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\color{blue}{y \cdot \frac{1}{2}} + \left(1 - \sqrt{y}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\mathsf{fma}\left(y, \frac{1}{2}, 1 - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \mathsf{fma}\left(y, \frac{1}{2}, \color{blue}{1 - \sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 98.8

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \mathsf{fma}\left(y, 0.5, 1 - \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 98.8%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\mathsf{fma}\left(y, 0.5, 1 - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) \leq 2.0001:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \mathsf{fma}\left(y, 0.5, 1 - \sqrt{y}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := t\_1 - \sqrt{y}\\ t_3 := \sqrt{1 + z} - \sqrt{z}\\ t_4 := \sqrt{1 + t} - \sqrt{t}\\ t_5 := \sqrt{1 + x}\\ t_6 := t\_3 + \left(\left(t\_5 - \sqrt{x}\right) + t\_2\right)\\ \mathbf{if}\;t\_6 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;t\_4 + 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)\\ \mathbf{elif}\;t\_6 \leq 2:\\ \;\;\;\;t\_4 + \left(t\_5 + \left(\frac{1}{\sqrt{y} + t\_1} - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4 + \left(t\_3 + \left(t\_2 + \left(1 - \sqrt{x}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0)))
        (t_2 (- t_1 (sqrt y)))
        (t_3 (- (sqrt (+ 1.0 z)) (sqrt z)))
        (t_4 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_5 (sqrt (+ 1.0 x)))
        (t_6 (+ t_3 (+ (- t_5 (sqrt x)) t_2))))
   (if (<= t_6 2e-5)
     (+ t_4 (* 0.5 (+ (sqrt (/ 1.0 x)) (+ (sqrt (/ 1.0 y)) (sqrt (/ 1.0 z))))))
     (if (<= t_6 2.0)
       (+ t_4 (+ t_5 (- (/ 1.0 (+ (sqrt y) t_1)) (sqrt x))))
       (+ t_4 (+ t_3 (+ t_2 (- 1.0 (sqrt x)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = t_1 - sqrt(y);
	double t_3 = sqrt((1.0 + z)) - sqrt(z);
	double t_4 = sqrt((1.0 + t)) - sqrt(t);
	double t_5 = sqrt((1.0 + x));
	double t_6 = t_3 + ((t_5 - sqrt(x)) + t_2);
	double tmp;
	if (t_6 <= 2e-5) {
		tmp = t_4 + (0.5 * (sqrt((1.0 / x)) + (sqrt((1.0 / y)) + sqrt((1.0 / z)))));
	} else if (t_6 <= 2.0) {
		tmp = t_4 + (t_5 + ((1.0 / (sqrt(y) + t_1)) - sqrt(x)));
	} else {
		tmp = t_4 + (t_3 + (t_2 + (1.0 - sqrt(x))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0))
    t_2 = t_1 - sqrt(y)
    t_3 = sqrt((1.0d0 + z)) - sqrt(z)
    t_4 = sqrt((1.0d0 + t)) - sqrt(t)
    t_5 = sqrt((1.0d0 + x))
    t_6 = t_3 + ((t_5 - sqrt(x)) + t_2)
    if (t_6 <= 2d-5) then
        tmp = t_4 + (0.5d0 * (sqrt((1.0d0 / x)) + (sqrt((1.0d0 / y)) + sqrt((1.0d0 / z)))))
    else if (t_6 <= 2.0d0) then
        tmp = t_4 + (t_5 + ((1.0d0 / (sqrt(y) + t_1)) - sqrt(x)))
    else
        tmp = t_4 + (t_3 + (t_2 + (1.0d0 - sqrt(x))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0));
	double t_2 = t_1 - Math.sqrt(y);
	double t_3 = Math.sqrt((1.0 + z)) - Math.sqrt(z);
	double t_4 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double t_5 = Math.sqrt((1.0 + x));
	double t_6 = t_3 + ((t_5 - Math.sqrt(x)) + t_2);
	double tmp;
	if (t_6 <= 2e-5) {
		tmp = t_4 + (0.5 * (Math.sqrt((1.0 / x)) + (Math.sqrt((1.0 / y)) + Math.sqrt((1.0 / z)))));
	} else if (t_6 <= 2.0) {
		tmp = t_4 + (t_5 + ((1.0 / (Math.sqrt(y) + t_1)) - Math.sqrt(x)));
	} else {
		tmp = t_4 + (t_3 + (t_2 + (1.0 - Math.sqrt(x))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0))
	t_2 = t_1 - math.sqrt(y)
	t_3 = math.sqrt((1.0 + z)) - math.sqrt(z)
	t_4 = math.sqrt((1.0 + t)) - math.sqrt(t)
	t_5 = math.sqrt((1.0 + x))
	t_6 = t_3 + ((t_5 - math.sqrt(x)) + t_2)
	tmp = 0
	if t_6 <= 2e-5:
		tmp = t_4 + (0.5 * (math.sqrt((1.0 / x)) + (math.sqrt((1.0 / y)) + math.sqrt((1.0 / z)))))
	elif t_6 <= 2.0:
		tmp = t_4 + (t_5 + ((1.0 / (math.sqrt(y) + t_1)) - math.sqrt(x)))
	else:
		tmp = t_4 + (t_3 + (t_2 + (1.0 - math.sqrt(x))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = Float64(t_1 - sqrt(y))
	t_3 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	t_4 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_5 = sqrt(Float64(1.0 + x))
	t_6 = Float64(t_3 + Float64(Float64(t_5 - sqrt(x)) + t_2))
	tmp = 0.0
	if (t_6 <= 2e-5)
		tmp = Float64(t_4 + Float64(0.5 * Float64(sqrt(Float64(1.0 / x)) + Float64(sqrt(Float64(1.0 / y)) + sqrt(Float64(1.0 / z))))));
	elseif (t_6 <= 2.0)
		tmp = Float64(t_4 + Float64(t_5 + Float64(Float64(1.0 / Float64(sqrt(y) + t_1)) - sqrt(x))));
	else
		tmp = Float64(t_4 + Float64(t_3 + Float64(t_2 + Float64(1.0 - sqrt(x)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((y + 1.0));
	t_2 = t_1 - sqrt(y);
	t_3 = sqrt((1.0 + z)) - sqrt(z);
	t_4 = sqrt((1.0 + t)) - sqrt(t);
	t_5 = sqrt((1.0 + x));
	t_6 = t_3 + ((t_5 - sqrt(x)) + t_2);
	tmp = 0.0;
	if (t_6 <= 2e-5)
		tmp = t_4 + (0.5 * (sqrt((1.0 / x)) + (sqrt((1.0 / y)) + sqrt((1.0 / z)))));
	elseif (t_6 <= 2.0)
		tmp = t_4 + (t_5 + ((1.0 / (sqrt(y) + t_1)) - sqrt(x)));
	else
		tmp = t_4 + (t_3 + (t_2 + (1.0 - sqrt(x))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$3 + N[(N[(t$95$5 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 2e-5], N[(t$95$4 + N[(0.5 * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.0], N[(t$95$4 + N[(t$95$5 + N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$4 + N[(t$95$3 + N[(t$95$2 + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.00000000000000016e-5

   1. Initial program 49.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + t} + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \sqrt{1 + t}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{t}\right)}\right) + \sqrt{1 + t} \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \sqrt{t}\right)} + \sqrt{1 + t} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   5. Simplified 23.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      13. sqrt-lowering-sqrt.f64 45.1

          \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   8. Simplified 45.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \color{blue}{\left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       9. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       10. /-lowering-/.f64 88.3

           \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   11. Simplified 88.3%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   if 2.00000000000000016e-5 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2

   1. Initial program 96.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 96.9

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \color{blue}{\sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 96.9%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} - \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} - \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + x}} + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + x}} + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} - \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\color{blue}{\frac{1}{\sqrt{y} + \sqrt{1 + y}}} - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\frac{1}{\color{blue}{\sqrt{y} + \sqrt{1 + y}}} - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\frac{1}{\color{blue}{\sqrt{y}} + \sqrt{1 + y}} - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\frac{1}{\sqrt{y} + \color{blue}{\sqrt{1 + y}}} - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + x} + \left(\frac{1}{\sqrt{y} + \sqrt{\color{blue}{1 + y}}} - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. sqrt-lowering-sqrt.f64 45.2

          \[\leadsto \left(\sqrt{1 + x} + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} - \color{blue}{\sqrt{x}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Simplified 45.2%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} - \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 2 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 97.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 90.5

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 90.5%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) \leq 2:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + x} + \left(\frac{1}{\sqrt{y} + \sqrt{y + 1}} - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(1 - \sqrt{x}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{1 + z}\\ t_3 := t\_2 - \sqrt{z}\\ t_4 := \sqrt{1 + t} - \sqrt{t}\\ t_5 := \sqrt{1 + x}\\ t_6 := t\_3 + \left(\left(t\_5 - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right)\\ \mathbf{if}\;t\_6 \leq 0.9999999999998505:\\ \;\;\;\;t\_4 + \left(\frac{1}{\sqrt{x} + t\_5} + t\_3\right)\\ \mathbf{elif}\;t\_6 \leq 2.99999997:\\ \;\;\;\;t\_5 + \left(\left(t\_1 + \frac{1}{\sqrt{z} + t\_2}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4 + \left(\left(3 - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (- t_2 (sqrt z)))
        (t_4 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_5 (sqrt (+ 1.0 x)))
        (t_6 (+ t_3 (+ (- t_5 (sqrt x)) (- t_1 (sqrt y))))))
   (if (<= t_6 0.9999999999998505)
     (+ t_4 (+ (/ 1.0 (+ (sqrt x) t_5)) t_3))
     (if (<= t_6 2.99999997)
       (+ t_5 (- (+ t_1 (/ 1.0 (+ (sqrt z) t_2))) (+ (sqrt y) (sqrt x))))
       (+ t_4 (- (- 3.0 (sqrt x)) (+ (sqrt y) (sqrt z))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((1.0 + z));
	double t_3 = t_2 - sqrt(z);
	double t_4 = sqrt((1.0 + t)) - sqrt(t);
	double t_5 = sqrt((1.0 + x));
	double t_6 = t_3 + ((t_5 - sqrt(x)) + (t_1 - sqrt(y)));
	double tmp;
	if (t_6 <= 0.9999999999998505) {
		tmp = t_4 + ((1.0 / (sqrt(x) + t_5)) + t_3);
	} else if (t_6 <= 2.99999997) {
		tmp = t_5 + ((t_1 + (1.0 / (sqrt(z) + t_2))) - (sqrt(y) + sqrt(x)));
	} else {
		tmp = t_4 + ((3.0 - sqrt(x)) - (sqrt(y) + sqrt(z)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0))
    t_2 = sqrt((1.0d0 + z))
    t_3 = t_2 - sqrt(z)
    t_4 = sqrt((1.0d0 + t)) - sqrt(t)
    t_5 = sqrt((1.0d0 + x))
    t_6 = t_3 + ((t_5 - sqrt(x)) + (t_1 - sqrt(y)))
    if (t_6 <= 0.9999999999998505d0) then
        tmp = t_4 + ((1.0d0 / (sqrt(x) + t_5)) + t_3)
    else if (t_6 <= 2.99999997d0) then
        tmp = t_5 + ((t_1 + (1.0d0 / (sqrt(z) + t_2))) - (sqrt(y) + sqrt(x)))
    else
        tmp = t_4 + ((3.0d0 - sqrt(x)) - (sqrt(y) + sqrt(z)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0));
	double t_2 = Math.sqrt((1.0 + z));
	double t_3 = t_2 - Math.sqrt(z);
	double t_4 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double t_5 = Math.sqrt((1.0 + x));
	double t_6 = t_3 + ((t_5 - Math.sqrt(x)) + (t_1 - Math.sqrt(y)));
	double tmp;
	if (t_6 <= 0.9999999999998505) {
		tmp = t_4 + ((1.0 / (Math.sqrt(x) + t_5)) + t_3);
	} else if (t_6 <= 2.99999997) {
		tmp = t_5 + ((t_1 + (1.0 / (Math.sqrt(z) + t_2))) - (Math.sqrt(y) + Math.sqrt(x)));
	} else {
		tmp = t_4 + ((3.0 - Math.sqrt(x)) - (Math.sqrt(y) + Math.sqrt(z)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0))
	t_2 = math.sqrt((1.0 + z))
	t_3 = t_2 - math.sqrt(z)
	t_4 = math.sqrt((1.0 + t)) - math.sqrt(t)
	t_5 = math.sqrt((1.0 + x))
	t_6 = t_3 + ((t_5 - math.sqrt(x)) + (t_1 - math.sqrt(y)))
	tmp = 0
	if t_6 <= 0.9999999999998505:
		tmp = t_4 + ((1.0 / (math.sqrt(x) + t_5)) + t_3)
	elif t_6 <= 2.99999997:
		tmp = t_5 + ((t_1 + (1.0 / (math.sqrt(z) + t_2))) - (math.sqrt(y) + math.sqrt(x)))
	else:
		tmp = t_4 + ((3.0 - math.sqrt(x)) - (math.sqrt(y) + math.sqrt(z)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = Float64(t_2 - sqrt(z))
	t_4 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_5 = sqrt(Float64(1.0 + x))
	t_6 = Float64(t_3 + Float64(Float64(t_5 - sqrt(x)) + Float64(t_1 - sqrt(y))))
	tmp = 0.0
	if (t_6 <= 0.9999999999998505)
		tmp = Float64(t_4 + Float64(Float64(1.0 / Float64(sqrt(x) + t_5)) + t_3));
	elseif (t_6 <= 2.99999997)
		tmp = Float64(t_5 + Float64(Float64(t_1 + Float64(1.0 / Float64(sqrt(z) + t_2))) - Float64(sqrt(y) + sqrt(x))));
	else
		tmp = Float64(t_4 + Float64(Float64(3.0 - sqrt(x)) - Float64(sqrt(y) + sqrt(z))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((y + 1.0));
	t_2 = sqrt((1.0 + z));
	t_3 = t_2 - sqrt(z);
	t_4 = sqrt((1.0 + t)) - sqrt(t);
	t_5 = sqrt((1.0 + x));
	t_6 = t_3 + ((t_5 - sqrt(x)) + (t_1 - sqrt(y)));
	tmp = 0.0;
	if (t_6 <= 0.9999999999998505)
		tmp = t_4 + ((1.0 / (sqrt(x) + t_5)) + t_3);
	elseif (t_6 <= 2.99999997)
		tmp = t_5 + ((t_1 + (1.0 / (sqrt(z) + t_2))) - (sqrt(y) + sqrt(x)));
	else
		tmp = t_4 + ((3.0 - sqrt(x)) - (sqrt(y) + sqrt(z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$3 + N[(N[(t$95$5 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 0.9999999999998505], N[(t$95$4 + N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.99999997], N[(t$95$5 + N[(N[(t$95$1 + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$4 + N[(N[(3.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.999999999999850453

   1. Initial program 58.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - \color{blue}{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right) - x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - x}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\color{blue}{\sqrt{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 58.3

          \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \color{blue}{\sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 58.3%

      \[\leadsto \left(\left(\color{blue}{\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{x}} + \sqrt{1 + x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{x} + \color{blue}{\sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 61.0

         \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{\color{blue}{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Simplified 61.0%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.999999999999850453 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.9999999700000002

   1. Initial program 96.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z + 1}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 96.9

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \color{blue}{\sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 96.9%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\left(\color{blue}{\sqrt{1 + y}} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{\color{blue}{1 + y}} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \color{blue}{\frac{1}{\sqrt{z} + \sqrt{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\color{blue}{\sqrt{z} + \sqrt{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\color{blue}{\sqrt{z}} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \color{blue}{\sqrt{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) \]

      16. sqrt-lowering-sqrt.f64 29.7

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   7. Simplified 29.7%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   if 2.9999999700000002 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 99.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate-+r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \sqrt{1 + y}\right)} - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(1 + \sqrt{1 + x}\right)} + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(1 + \color{blue}{\sqrt{1 + x}}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{\color{blue}{1 + x}}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \color{blue}{\left(\sqrt{1 + y} - \sqrt{x}\right)}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\color{blue}{\sqrt{1 + y}} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \color{blue}{\sqrt{x}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. sqrt-lowering-sqrt.f64 99.5

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 99.5%

      \[\leadsto \color{blue}{\left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + 2\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + 2\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + y}} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + y}} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 99.5

          \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 99.5%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(3 - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. associate--r+ N/A

          \[\leadsto \color{blue}{\left(\left(3 - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(3 - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \left(\color{blue}{\left(3 - \sqrt{x}\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(3 - \color{blue}{\sqrt{x}}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. +-commutative N/A

          \[\leadsto \left(\left(3 - \sqrt{x}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(3 - \sqrt{x}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(3 - \sqrt{x}\right) - \left(\color{blue}{\sqrt{z}} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       8. sqrt-lowering-sqrt.f64 99.5

          \[\leadsto \left(\left(3 - \sqrt{x}\right) - \left(\sqrt{z} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 99.5%

       \[\leadsto \color{blue}{\left(\left(3 - \sqrt{x}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) \leq 0.9999999999998505:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) \leq 2.99999997:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(\sqrt{y + 1} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(3 - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{1 + x}\\ t_4 := \left(t\_2 - \sqrt{z}\right) + \left(\left(t\_3 - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right)\\ \mathbf{if}\;t\_4 \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, t\_3\right) - \sqrt{x}\\ \mathbf{elif}\;t\_4 \leq 2.99999997:\\ \;\;\;\;t\_3 + \left(\left(t\_1 + \frac{1}{\sqrt{z} + t\_2}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(3 - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (sqrt (+ 1.0 x)))
        (t_4 (+ (- t_2 (sqrt z)) (+ (- t_3 (sqrt x)) (- t_1 (sqrt y))))))
   (if (<= t_4 1.00005)
     (- (fma 0.5 (+ (sqrt (/ 1.0 y)) (sqrt (/ 1.0 z))) t_3) (sqrt x))
     (if (<= t_4 2.99999997)
       (+ t_3 (- (+ t_1 (/ 1.0 (+ (sqrt z) t_2))) (+ (sqrt y) (sqrt x))))
       (+
        (- (sqrt (+ 1.0 t)) (sqrt t))
        (- (- 3.0 (sqrt x)) (+ (sqrt y) (sqrt z))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((1.0 + z));
	double t_3 = sqrt((1.0 + x));
	double t_4 = (t_2 - sqrt(z)) + ((t_3 - sqrt(x)) + (t_1 - sqrt(y)));
	double tmp;
	if (t_4 <= 1.00005) {
		tmp = fma(0.5, (sqrt((1.0 / y)) + sqrt((1.0 / z))), t_3) - sqrt(x);
	} else if (t_4 <= 2.99999997) {
		tmp = t_3 + ((t_1 + (1.0 / (sqrt(z) + t_2))) - (sqrt(y) + sqrt(x)));
	} else {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + ((3.0 - sqrt(x)) - (sqrt(y) + sqrt(z)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = sqrt(Float64(1.0 + x))
	t_4 = Float64(Float64(t_2 - sqrt(z)) + Float64(Float64(t_3 - sqrt(x)) + Float64(t_1 - sqrt(y))))
	tmp = 0.0
	if (t_4 <= 1.00005)
		tmp = Float64(fma(0.5, Float64(sqrt(Float64(1.0 / y)) + sqrt(Float64(1.0 / z))), t_3) - sqrt(x));
	elseif (t_4 <= 2.99999997)
		tmp = Float64(t_3 + Float64(Float64(t_1 + Float64(1.0 / Float64(sqrt(z) + t_2))) - Float64(sqrt(y) + sqrt(x))));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(3.0 - sqrt(x)) - Float64(sqrt(y) + sqrt(z))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 1.00005], N[(N[(0.5 * N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.99999997], N[(t$95$3 + N[(N[(t$95$1 + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.00005000000000011

   1. Initial program 82.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + t} + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \sqrt{1 + t}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{t}\right)}\right) + \sqrt{1 + t} \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \sqrt{t}\right)} + \sqrt{1 + t} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   5. Simplified 37.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      13. sqrt-lowering-sqrt.f64 38.9

          \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   8. Simplified 38.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   10. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right)} - \sqrt{x} \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       9. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x} \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x} \]

       11. sqrt-lowering-sqrt.f64 16.4

           \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}} \]

   11. Simplified 16.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}} \]

   if 1.00005000000000011 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.9999999700000002

   1. Initial program 97.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z + 1}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 97.4

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \color{blue}{\sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 97.4%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\left(\color{blue}{\sqrt{1 + y}} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{\color{blue}{1 + y}} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \color{blue}{\frac{1}{\sqrt{z} + \sqrt{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\color{blue}{\sqrt{z} + \sqrt{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\color{blue}{\sqrt{z}} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \color{blue}{\sqrt{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) \]

      16. sqrt-lowering-sqrt.f64 35.8

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   7. Simplified 35.8%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   if 2.9999999700000002 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 99.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate-+r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \sqrt{1 + y}\right)} - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(1 + \sqrt{1 + x}\right)} + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(1 + \color{blue}{\sqrt{1 + x}}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{\color{blue}{1 + x}}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \color{blue}{\left(\sqrt{1 + y} - \sqrt{x}\right)}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\color{blue}{\sqrt{1 + y}} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \color{blue}{\sqrt{x}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. sqrt-lowering-sqrt.f64 99.5

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 99.5%

      \[\leadsto \color{blue}{\left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + 2\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + 2\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + y}} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + y}} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 99.5

          \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 99.5%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(3 - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. associate--r+ N/A

          \[\leadsto \color{blue}{\left(\left(3 - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(3 - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \left(\color{blue}{\left(3 - \sqrt{x}\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(3 - \color{blue}{\sqrt{x}}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. +-commutative N/A

          \[\leadsto \left(\left(3 - \sqrt{x}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(3 - \sqrt{x}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(3 - \sqrt{x}\right) - \left(\color{blue}{\sqrt{z}} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       8. sqrt-lowering-sqrt.f64 99.5

          \[\leadsto \left(\left(3 - \sqrt{x}\right) - \left(\sqrt{z} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 99.5%

       \[\leadsto \color{blue}{\left(\left(3 - \sqrt{x}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) \leq 2.99999997:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(\sqrt{y + 1} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(3 - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ t_2 := \sqrt{y + 1}\\ t_3 := \sqrt{1 + z} - \sqrt{z}\\ t_4 := \sqrt{1 + x}\\ \mathbf{if}\;t\_3 + \left(\left(t\_4 - \sqrt{x}\right) + \left(t\_2 - \sqrt{y}\right)\right) \leq 1:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + t\_4}\right) + t\_3\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(t\_3 + \left(\frac{\left(y + 1\right) - y}{\sqrt{y} + t\_2} + \left(1 - \sqrt{x}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_2 (sqrt (+ y 1.0)))
        (t_3 (- (sqrt (+ 1.0 z)) (sqrt z)))
        (t_4 (sqrt (+ 1.0 x))))
   (if (<= (+ t_3 (+ (- t_4 (sqrt x)) (- t_2 (sqrt y)))) 1.0)
     (+ (+ (fma 0.5 (sqrt (/ 1.0 y)) (/ 1.0 (+ (sqrt x) t_4))) t_3) t_1)
     (+
      t_1
      (+ t_3 (+ (/ (- (+ y 1.0) y) (+ (sqrt y) t_2)) (- 1.0 (sqrt x))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + t)) - sqrt(t);
	double t_2 = sqrt((y + 1.0));
	double t_3 = sqrt((1.0 + z)) - sqrt(z);
	double t_4 = sqrt((1.0 + x));
	double tmp;
	if ((t_3 + ((t_4 - sqrt(x)) + (t_2 - sqrt(y)))) <= 1.0) {
		tmp = (fma(0.5, sqrt((1.0 / y)), (1.0 / (sqrt(x) + t_4))) + t_3) + t_1;
	} else {
		tmp = t_1 + (t_3 + ((((y + 1.0) - y) / (sqrt(y) + t_2)) + (1.0 - sqrt(x))));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_2 = sqrt(Float64(y + 1.0))
	t_3 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	t_4 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(t_3 + Float64(Float64(t_4 - sqrt(x)) + Float64(t_2 - sqrt(y)))) <= 1.0)
		tmp = Float64(Float64(fma(0.5, sqrt(Float64(1.0 / y)), Float64(1.0 / Float64(sqrt(x) + t_4))) + t_3) + t_1);
	else
		tmp = Float64(t_1 + Float64(t_3 + Float64(Float64(Float64(Float64(y + 1.0) - y) / Float64(sqrt(y) + t_2)) + Float64(1.0 - sqrt(x)))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$3 + N[(N[(t$95$4 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$1), $MachinePrecision], N[(t$95$1 + N[(t$95$3 + N[(N[(N[(N[(y + 1.0), $MachinePrecision] - y), $MachinePrecision] / N[(N[Sqrt[y], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1

   1. Initial program 82.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - \color{blue}{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right) - x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - x}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\color{blue}{\sqrt{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 82.7

          \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \color{blue}{\sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 82.7%

      \[\leadsto \left(\left(\color{blue}{\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \frac{1}{\color{blue}{\sqrt{x}} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + \color{blue}{\sqrt{1 + x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 68.2

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + \sqrt{\color{blue}{1 + x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Simplified 68.2%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 96.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 97.2

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \color{blue}{\sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 97.2%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. sqrt-lowering-sqrt.f64 74.2

         \[\leadsto \left(\left(\left(1 - \color{blue}{\sqrt{x}}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Simplified 74.2%

      \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) \leq 1:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}} + \left(1 - \sqrt{x}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{y + 1}\\ \mathbf{if}\;\left(t\_2 - \sqrt{x}\right) + \left(t\_3 - \sqrt{y}\right) \leq 2 \cdot 10^{-5}:\\ \;\;\;\;t\_1 + 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(t\_2 + \left(\left(\left(\sqrt{1 + z} - \sqrt{z}\right) - \left(\sqrt{y} - t\_3\right)\right) - \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (sqrt (+ y 1.0))))
   (if (<= (+ (- t_2 (sqrt x)) (- t_3 (sqrt y))) 2e-5)
     (+ t_1 (* 0.5 (+ (sqrt (/ 1.0 x)) (+ (sqrt (/ 1.0 y)) (sqrt (/ 1.0 z))))))
     (+
      t_1
      (+
       t_2
       (- (- (- (sqrt (+ 1.0 z)) (sqrt z)) (- (sqrt y) t_3)) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + t)) - sqrt(t);
	double t_2 = sqrt((1.0 + x));
	double t_3 = sqrt((y + 1.0));
	double tmp;
	if (((t_2 - sqrt(x)) + (t_3 - sqrt(y))) <= 2e-5) {
		tmp = t_1 + (0.5 * (sqrt((1.0 / x)) + (sqrt((1.0 / y)) + sqrt((1.0 / z)))));
	} else {
		tmp = t_1 + (t_2 + (((sqrt((1.0 + z)) - sqrt(z)) - (sqrt(y) - t_3)) - sqrt(x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + t)) - sqrt(t)
    t_2 = sqrt((1.0d0 + x))
    t_3 = sqrt((y + 1.0d0))
    if (((t_2 - sqrt(x)) + (t_3 - sqrt(y))) <= 2d-5) then
        tmp = t_1 + (0.5d0 * (sqrt((1.0d0 / x)) + (sqrt((1.0d0 / y)) + sqrt((1.0d0 / z)))))
    else
        tmp = t_1 + (t_2 + (((sqrt((1.0d0 + z)) - sqrt(z)) - (sqrt(y) - t_3)) - sqrt(x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double t_2 = Math.sqrt((1.0 + x));
	double t_3 = Math.sqrt((y + 1.0));
	double tmp;
	if (((t_2 - Math.sqrt(x)) + (t_3 - Math.sqrt(y))) <= 2e-5) {
		tmp = t_1 + (0.5 * (Math.sqrt((1.0 / x)) + (Math.sqrt((1.0 / y)) + Math.sqrt((1.0 / z)))));
	} else {
		tmp = t_1 + (t_2 + (((Math.sqrt((1.0 + z)) - Math.sqrt(z)) - (Math.sqrt(y) - t_3)) - Math.sqrt(x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + t)) - math.sqrt(t)
	t_2 = math.sqrt((1.0 + x))
	t_3 = math.sqrt((y + 1.0))
	tmp = 0
	if ((t_2 - math.sqrt(x)) + (t_3 - math.sqrt(y))) <= 2e-5:
		tmp = t_1 + (0.5 * (math.sqrt((1.0 / x)) + (math.sqrt((1.0 / y)) + math.sqrt((1.0 / z)))))
	else:
		tmp = t_1 + (t_2 + (((math.sqrt((1.0 + z)) - math.sqrt(z)) - (math.sqrt(y) - t_3)) - math.sqrt(x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_2 = sqrt(Float64(1.0 + x))
	t_3 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (Float64(Float64(t_2 - sqrt(x)) + Float64(t_3 - sqrt(y))) <= 2e-5)
		tmp = Float64(t_1 + Float64(0.5 * Float64(sqrt(Float64(1.0 / x)) + Float64(sqrt(Float64(1.0 / y)) + sqrt(Float64(1.0 / z))))));
	else
		tmp = Float64(t_1 + Float64(t_2 + Float64(Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) - Float64(sqrt(y) - t_3)) - sqrt(x))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + t)) - sqrt(t);
	t_2 = sqrt((1.0 + x));
	t_3 = sqrt((y + 1.0));
	tmp = 0.0;
	if (((t_2 - sqrt(x)) + (t_3 - sqrt(y))) <= 2e-5)
		tmp = t_1 + (0.5 * (sqrt((1.0 / x)) + (sqrt((1.0 / y)) + sqrt((1.0 / z)))));
	else
		tmp = t_1 + (t_2 + (((sqrt((1.0 + z)) - sqrt(z)) - (sqrt(y) - t_3)) - sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-5], N[(t$95$1 + N[(0.5 * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(t$95$2 + N[(N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) < 2.00000000000000016e-5

   1. Initial program 70.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + t} + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \sqrt{1 + t}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{t}\right)}\right) + \sqrt{1 + t} \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \sqrt{t}\right)} + \sqrt{1 + t} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   5. Simplified 16.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      13. sqrt-lowering-sqrt.f64 29.0

          \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   8. Simplified 29.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \color{blue}{\left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       9. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       10. /-lowering-/.f64 52.7

           \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   11. Simplified 52.7%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   if 2.00000000000000016e-5 < (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)))

   1. Initial program 96.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{x + 1}} - \left(\sqrt{x} - \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{x + 1}} - \left(\sqrt{x} - \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{x + 1} - \color{blue}{\left(\sqrt{x} - \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{x + 1} - \left(\color{blue}{\sqrt{x}} - \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{x + 1} - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\left(\color{blue}{\sqrt{y + 1}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\color{blue}{\sqrt{z + 1}} - \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. +-commutative N/A

          \[\leadsto \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. sqrt-lowering-sqrt.f64 73.7

          \[\leadsto \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \color{blue}{\sqrt{z}}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 73.7%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right) \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + x} + \left(\left(\left(\sqrt{1 + z} - \sqrt{z}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \sqrt{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1} - \sqrt{y}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{1 + z}\\ t_4 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + t\_2}\right) + \left(t\_3 - \sqrt{z}\right)\right) + t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_4 + \left(\left(\left(t\_2 - \sqrt{x}\right) + t\_1\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + t\_3}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ y 1.0)) (sqrt y)))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (sqrt (+ 1.0 z)))
        (t_4 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= t_1 5e-7)
     (+
      (+ (fma 0.5 (sqrt (/ 1.0 y)) (/ 1.0 (+ (sqrt x) t_2))) (- t_3 (sqrt z)))
      t_4)
     (+
      t_4
      (+ (+ (- t_2 (sqrt x)) t_1) (/ (- (+ 1.0 z) z) (+ (sqrt z) t_3)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0)) - sqrt(y);
	double t_2 = sqrt((1.0 + x));
	double t_3 = sqrt((1.0 + z));
	double t_4 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if (t_1 <= 5e-7) {
		tmp = (fma(0.5, sqrt((1.0 / y)), (1.0 / (sqrt(x) + t_2))) + (t_3 - sqrt(z))) + t_4;
	} else {
		tmp = t_4 + (((t_2 - sqrt(x)) + t_1) + (((1.0 + z) - z) / (sqrt(z) + t_3)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(y + 1.0)) - sqrt(y))
	t_2 = sqrt(Float64(1.0 + x))
	t_3 = sqrt(Float64(1.0 + z))
	t_4 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (t_1 <= 5e-7)
		tmp = Float64(Float64(fma(0.5, sqrt(Float64(1.0 / y)), Float64(1.0 / Float64(sqrt(x) + t_2))) + Float64(t_3 - sqrt(z))) + t_4);
	else
		tmp = Float64(t_4 + Float64(Float64(Float64(t_2 - sqrt(x)) + t_1) + Float64(Float64(Float64(1.0 + z) - z) / Float64(sqrt(z) + t_3))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-7], N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision], N[(t$95$4 + N[(N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(N[(N[(1.0 + z), $MachinePrecision] - z), $MachinePrecision] / N[(N[Sqrt[z], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)) < 4.99999999999999977e-7

   1. Initial program 83.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - \color{blue}{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right) - x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - x}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\color{blue}{\sqrt{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 83.3

          \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \color{blue}{\sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 83.3%

      \[\leadsto \left(\left(\color{blue}{\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \frac{1}{\color{blue}{\sqrt{x}} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + \color{blue}{\sqrt{1 + x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 90.4

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + \sqrt{\color{blue}{1 + x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Simplified 90.4%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 4.99999999999999977e-7 < (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))

   1. Initial program 96.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z + 1}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 97.2

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \color{blue}{\sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 97.2%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{y + 1} - \sqrt{y} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{1 + z}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x} - \sqrt{x}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;t\_2 + 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(t\_1 + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 x)) (sqrt x)))
        (t_2 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= t_1 2e-5)
     (+ t_2 (* 0.5 (+ (sqrt (/ 1.0 x)) (+ (sqrt (/ 1.0 y)) (sqrt (/ 1.0 z))))))
     (+
      t_2
      (+
       (- (sqrt (+ 1.0 z)) (sqrt z))
       (+ t_1 (- (sqrt (+ y 1.0)) (sqrt y))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x)) - sqrt(x);
	double t_2 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if (t_1 <= 2e-5) {
		tmp = t_2 + (0.5 * (sqrt((1.0 / x)) + (sqrt((1.0 / y)) + sqrt((1.0 / z)))));
	} else {
		tmp = t_2 + ((sqrt((1.0 + z)) - sqrt(z)) + (t_1 + (sqrt((y + 1.0)) - sqrt(y))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x)) - sqrt(x)
    t_2 = sqrt((1.0d0 + t)) - sqrt(t)
    if (t_1 <= 2d-5) then
        tmp = t_2 + (0.5d0 * (sqrt((1.0d0 / x)) + (sqrt((1.0d0 / y)) + sqrt((1.0d0 / z)))))
    else
        tmp = t_2 + ((sqrt((1.0d0 + z)) - sqrt(z)) + (t_1 + (sqrt((y + 1.0d0)) - sqrt(y))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	double t_2 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if (t_1 <= 2e-5) {
		tmp = t_2 + (0.5 * (Math.sqrt((1.0 / x)) + (Math.sqrt((1.0 / y)) + Math.sqrt((1.0 / z)))));
	} else {
		tmp = t_2 + ((Math.sqrt((1.0 + z)) - Math.sqrt(z)) + (t_1 + (Math.sqrt((y + 1.0)) - Math.sqrt(y))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x)) - math.sqrt(x)
	t_2 = math.sqrt((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if t_1 <= 2e-5:
		tmp = t_2 + (0.5 * (math.sqrt((1.0 / x)) + (math.sqrt((1.0 / y)) + math.sqrt((1.0 / z)))))
	else:
		tmp = t_2 + ((math.sqrt((1.0 + z)) - math.sqrt(z)) + (t_1 + (math.sqrt((y + 1.0)) - math.sqrt(y))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + x)) - sqrt(x))
	t_2 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (t_1 <= 2e-5)
		tmp = Float64(t_2 + Float64(0.5 * Float64(sqrt(Float64(1.0 / x)) + Float64(sqrt(Float64(1.0 / y)) + sqrt(Float64(1.0 / z))))));
	else
		tmp = Float64(t_2 + Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(t_1 + Float64(sqrt(Float64(y + 1.0)) - sqrt(y)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x)) - sqrt(x);
	t_2 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if (t_1 <= 2e-5)
		tmp = t_2 + (0.5 * (sqrt((1.0 / x)) + (sqrt((1.0 / y)) + sqrt((1.0 / z)))));
	else
		tmp = t_2 + ((sqrt((1.0 + z)) - sqrt(z)) + (t_1 + (sqrt((y + 1.0)) - sqrt(y))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-5], N[(t$95$2 + N[(0.5 * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 2.00000000000000016e-5

   1. Initial program 82.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + t} + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \sqrt{1 + t}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{t}\right)}\right) + \sqrt{1 + t} \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \sqrt{t}\right)} + \sqrt{1 + t} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   5. Simplified 36.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      13. sqrt-lowering-sqrt.f64 19.6

          \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   8. Simplified 19.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \color{blue}{\left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       9. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

       10. /-lowering-/.f64 32.0

           \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   11. Simplified 32.0%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   if 2.00000000000000016e-5 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 96.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 91.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{1 + z}\\ t_3 := t\_2 - \sqrt{z}\\ \mathbf{if}\;t\_3 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;1 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, t\_1\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{elif}\;t\_3 \leq 0.99999997:\\ \;\;\;\;\left(\left(\left(1 + t\_1\right) + \left(t\_2 - \sqrt{x}\right)\right) - \sqrt{z}\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(3 - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0))) (t_2 (sqrt (+ 1.0 z))) (t_3 (- t_2 (sqrt z))))
   (if (<= t_3 5e-5)
     (+ 1.0 (- (fma 0.5 (sqrt (/ 1.0 z)) t_1) (+ (sqrt y) (sqrt x))))
     (if (<= t_3 0.99999997)
       (- (- (+ (+ 1.0 t_1) (- t_2 (sqrt x))) (sqrt z)) (sqrt y))
       (+
        (- (sqrt (+ 1.0 t)) (sqrt t))
        (- (- 3.0 (sqrt x)) (+ (sqrt y) (sqrt z))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((1.0 + z));
	double t_3 = t_2 - sqrt(z);
	double tmp;
	if (t_3 <= 5e-5) {
		tmp = 1.0 + (fma(0.5, sqrt((1.0 / z)), t_1) - (sqrt(y) + sqrt(x)));
	} else if (t_3 <= 0.99999997) {
		tmp = (((1.0 + t_1) + (t_2 - sqrt(x))) - sqrt(z)) - sqrt(y);
	} else {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + ((3.0 - sqrt(x)) - (sqrt(y) + sqrt(z)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = Float64(t_2 - sqrt(z))
	tmp = 0.0
	if (t_3 <= 5e-5)
		tmp = Float64(1.0 + Float64(fma(0.5, sqrt(Float64(1.0 / z)), t_1) - Float64(sqrt(y) + sqrt(x))));
	elseif (t_3 <= 0.99999997)
		tmp = Float64(Float64(Float64(Float64(1.0 + t_1) + Float64(t_2 - sqrt(x))) - sqrt(z)) - sqrt(y));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(3.0 - sqrt(x)) - Float64(sqrt(y) + sqrt(z))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 5e-5], N[(1.0 + N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.99999997], N[(N[(N[(N[(1.0 + t$95$1), $MachinePrecision] + N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 5.00000000000000024e-5

   1. Initial program 83.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 5.6

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 5.6%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) \]

      13. sqrt-lowering-sqrt.f64 4.1

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

   8. Simplified 4.1%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   10. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

       3. --lowering--.f64 N/A

          \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

       4. +-commutative N/A

          \[\leadsto 1 + \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto 1 + \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto 1 + \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto 1 + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto 1 + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto 1 + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       10. +-commutative N/A

           \[\leadsto 1 + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto 1 + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

       12. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto 1 + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) \]

       13. sqrt-lowering-sqrt.f64 32.2

           \[\leadsto 1 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{y} + \color{blue}{\sqrt{x}}\right)\right) \]

   11. Simplified 32.2%

       \[\leadsto \color{blue}{1 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   if 5.00000000000000024e-5 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 0.99999996999999996

   1. Initial program 96.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 27.6

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 27.6%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) \]

      13. sqrt-lowering-sqrt.f64 24.1

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

   8. Simplified 24.1%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   9. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \sqrt{x}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} \]

      3. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \sqrt{x}\right) - \sqrt{z}\right) - \sqrt{y}} \]

      4. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \sqrt{x}\right) - \sqrt{z}\right) - \sqrt{y}} \]

   10. Applied egg-rr 24.1%

       \[\leadsto \color{blue}{\left(\left(\left(1 + \sqrt{1 + y}\right) + \left(\sqrt{1 + z} - \sqrt{x}\right)\right) - \sqrt{z}\right) - \sqrt{y}} \]

   if 0.99999996999999996 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))

   1. Initial program 97.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate-+r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \sqrt{1 + y}\right)} - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(1 + \sqrt{1 + x}\right)} + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(1 + \color{blue}{\sqrt{1 + x}}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{\color{blue}{1 + x}}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \color{blue}{\left(\sqrt{1 + y} - \sqrt{x}\right)}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\color{blue}{\sqrt{1 + y}} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \color{blue}{\sqrt{x}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. sqrt-lowering-sqrt.f64 33.8

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 33.8%

      \[\leadsto \color{blue}{\left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + 2\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + 2\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + y}} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + y}} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 28.7

          \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 28.7%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(3 - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. associate--r+ N/A

          \[\leadsto \color{blue}{\left(\left(3 - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(3 - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \left(\color{blue}{\left(3 - \sqrt{x}\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(3 - \color{blue}{\sqrt{x}}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. +-commutative N/A

          \[\leadsto \left(\left(3 - \sqrt{x}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(3 - \sqrt{x}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(3 - \sqrt{x}\right) - \left(\color{blue}{\sqrt{z}} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       8. sqrt-lowering-sqrt.f64 23.6

          \[\leadsto \left(\left(3 - \sqrt{x}\right) - \left(\sqrt{z} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 23.6%

       \[\leadsto \color{blue}{\left(\left(3 - \sqrt{x}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + z} - \sqrt{z} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;1 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{elif}\;\sqrt{1 + z} - \sqrt{z} \leq 0.99999997:\\ \;\;\;\;\left(\left(\left(1 + \sqrt{y + 1}\right) + \left(\sqrt{1 + z} - \sqrt{x}\right)\right) - \sqrt{z}\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(3 - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 91.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y} + \sqrt{z}\\ t_2 := \sqrt{1 + z}\\ t_3 := t\_2 - \sqrt{z}\\ \mathbf{if}\;t\_3 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;1 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{elif}\;t\_3 \leq 0.99999997:\\ \;\;\;\;\left(t\_2 + \mathsf{fma}\left(y, 0.5, 2\right)\right) - \left(\sqrt{x} + t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(3 - \sqrt{x}\right) - t\_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (sqrt y) (sqrt z)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (- t_2 (sqrt z))))
   (if (<= t_3 5e-5)
     (+
      1.0
      (- (fma 0.5 (sqrt (/ 1.0 z)) (sqrt (+ y 1.0))) (+ (sqrt y) (sqrt x))))
     (if (<= t_3 0.99999997)
       (- (+ t_2 (fma y 0.5 2.0)) (+ (sqrt x) t_1))
       (+ (- (sqrt (+ 1.0 t)) (sqrt t)) (- (- 3.0 (sqrt x)) t_1))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(y) + sqrt(z);
	double t_2 = sqrt((1.0 + z));
	double t_3 = t_2 - sqrt(z);
	double tmp;
	if (t_3 <= 5e-5) {
		tmp = 1.0 + (fma(0.5, sqrt((1.0 / z)), sqrt((y + 1.0))) - (sqrt(y) + sqrt(x)));
	} else if (t_3 <= 0.99999997) {
		tmp = (t_2 + fma(y, 0.5, 2.0)) - (sqrt(x) + t_1);
	} else {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + ((3.0 - sqrt(x)) - t_1);
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(y) + sqrt(z))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = Float64(t_2 - sqrt(z))
	tmp = 0.0
	if (t_3 <= 5e-5)
		tmp = Float64(1.0 + Float64(fma(0.5, sqrt(Float64(1.0 / z)), sqrt(Float64(y + 1.0))) - Float64(sqrt(y) + sqrt(x))));
	elseif (t_3 <= 0.99999997)
		tmp = Float64(Float64(t_2 + fma(y, 0.5, 2.0)) - Float64(sqrt(x) + t_1));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(3.0 - sqrt(x)) - t_1));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 5e-5], N[(1.0 + N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.99999997], N[(N[(t$95$2 + N[(y * 0.5 + 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 5.00000000000000024e-5

   1. Initial program 83.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 5.6

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 5.6%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) \]

      13. sqrt-lowering-sqrt.f64 4.1

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

   8. Simplified 4.1%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   10. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

       3. --lowering--.f64 N/A

          \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

       4. +-commutative N/A

          \[\leadsto 1 + \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto 1 + \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto 1 + \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto 1 + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto 1 + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto 1 + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       10. +-commutative N/A

           \[\leadsto 1 + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto 1 + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

       12. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto 1 + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) \]

       13. sqrt-lowering-sqrt.f64 32.2

           \[\leadsto 1 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{y} + \color{blue}{\sqrt{x}}\right)\right) \]

   11. Simplified 32.2%

       \[\leadsto \color{blue}{1 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   if 5.00000000000000024e-5 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 0.99999996999999996

   1. Initial program 96.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 27.6

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 27.6%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) \]

      13. sqrt-lowering-sqrt.f64 24.1

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

   8. Simplified 24.1%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(\color{blue}{\left(2 + \frac{1}{2} \cdot y\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot y + 2\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{y \cdot \frac{1}{2}} + 2\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

       3. accelerator-lowering-fma.f64 24.9

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(y, 0.5, 2\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   11. Simplified 24.9%

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(y, 0.5, 2\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   if 0.99999996999999996 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))

   1. Initial program 97.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate-+r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \sqrt{1 + y}\right)} - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(1 + \sqrt{1 + x}\right)} + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(1 + \color{blue}{\sqrt{1 + x}}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{\color{blue}{1 + x}}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \color{blue}{\left(\sqrt{1 + y} - \sqrt{x}\right)}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\color{blue}{\sqrt{1 + y}} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \color{blue}{\sqrt{x}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. sqrt-lowering-sqrt.f64 33.8

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 33.8%

      \[\leadsto \color{blue}{\left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + 2\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + 2\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + y}} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + y}} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 28.7

          \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 28.7%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(3 - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. associate--r+ N/A

          \[\leadsto \color{blue}{\left(\left(3 - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(3 - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \left(\color{blue}{\left(3 - \sqrt{x}\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(3 - \color{blue}{\sqrt{x}}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. +-commutative N/A

          \[\leadsto \left(\left(3 - \sqrt{x}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(3 - \sqrt{x}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(3 - \sqrt{x}\right) - \left(\color{blue}{\sqrt{z}} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       8. sqrt-lowering-sqrt.f64 23.6

          \[\leadsto \left(\left(3 - \sqrt{x}\right) - \left(\sqrt{z} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 23.6%

       \[\leadsto \color{blue}{\left(\left(3 - \sqrt{x}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + z} - \sqrt{z} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;1 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{elif}\;\sqrt{1 + z} - \sqrt{z} \leq 0.99999997:\\ \;\;\;\;\left(\sqrt{1 + z} + \mathsf{fma}\left(y, 0.5, 2\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(3 - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 90.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y} + \sqrt{z}\\ t_2 := \sqrt{1 + z}\\ t_3 := t\_2 - \sqrt{z}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{elif}\;t\_3 \leq 0.99999997:\\ \;\;\;\;\left(t\_2 + \mathsf{fma}\left(y, 0.5, 2\right)\right) - \left(\sqrt{x} + t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(3 - \sqrt{x}\right) - t\_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (sqrt y) (sqrt z)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (- t_2 (sqrt z))))
   (if (<= t_3 0.0)
     (+ (sqrt (+ 1.0 x)) (- (sqrt (+ y 1.0)) (+ (sqrt y) (sqrt x))))
     (if (<= t_3 0.99999997)
       (- (+ t_2 (fma y 0.5 2.0)) (+ (sqrt x) t_1))
       (+ (- (sqrt (+ 1.0 t)) (sqrt t)) (- (- 3.0 (sqrt x)) t_1))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(y) + sqrt(z);
	double t_2 = sqrt((1.0 + z));
	double t_3 = t_2 - sqrt(z);
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((1.0 + x)) + (sqrt((y + 1.0)) - (sqrt(y) + sqrt(x)));
	} else if (t_3 <= 0.99999997) {
		tmp = (t_2 + fma(y, 0.5, 2.0)) - (sqrt(x) + t_1);
	} else {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + ((3.0 - sqrt(x)) - t_1);
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(y) + sqrt(z))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = Float64(t_2 - sqrt(z))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(1.0 + x)) + Float64(sqrt(Float64(y + 1.0)) - Float64(sqrt(y) + sqrt(x))));
	elseif (t_3 <= 0.99999997)
		tmp = Float64(Float64(t_2 + fma(y, 0.5, 2.0)) - Float64(sqrt(x) + t_1));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(3.0 - sqrt(x)) - t_1));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.99999997], N[(N[(t$95$2 + N[(y * 0.5 + 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 0.0

   1. Initial program 84.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 5.0

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 5.0%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\sqrt{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{\color{blue}{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) \]

      10. sqrt-lowering-sqrt.f64 31.2

          \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   8. Simplified 31.2%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 0.99999996999999996

   1. Initial program 88.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 26.3

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 26.3%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) \]

      13. sqrt-lowering-sqrt.f64 22.2

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

   8. Simplified 22.2%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(\color{blue}{\left(2 + \frac{1}{2} \cdot y\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot y + 2\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{y \cdot \frac{1}{2}} + 2\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

       3. accelerator-lowering-fma.f64 23.2

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(y, 0.5, 2\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   11. Simplified 23.2%

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(y, 0.5, 2\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   if 0.99999996999999996 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))

   1. Initial program 97.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate-+r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \sqrt{1 + y}\right)} - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(1 + \sqrt{1 + x}\right)} + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(1 + \color{blue}{\sqrt{1 + x}}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{\color{blue}{1 + x}}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \color{blue}{\left(\sqrt{1 + y} - \sqrt{x}\right)}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\color{blue}{\sqrt{1 + y}} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \color{blue}{\sqrt{x}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. sqrt-lowering-sqrt.f64 33.8

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 33.8%

      \[\leadsto \color{blue}{\left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + 2\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + 2\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + y}} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + y}} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 28.7

          \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 28.7%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(3 - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. associate--r+ N/A

          \[\leadsto \color{blue}{\left(\left(3 - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(3 - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \left(\color{blue}{\left(3 - \sqrt{x}\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(3 - \color{blue}{\sqrt{x}}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. +-commutative N/A

          \[\leadsto \left(\left(3 - \sqrt{x}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(3 - \sqrt{x}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(3 - \sqrt{x}\right) - \left(\color{blue}{\sqrt{z}} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       8. sqrt-lowering-sqrt.f64 23.6

          \[\leadsto \left(\left(3 - \sqrt{x}\right) - \left(\sqrt{z} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 23.6%

       \[\leadsto \color{blue}{\left(\left(3 - \sqrt{x}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 27.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + z} - \sqrt{z} \leq 0:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{elif}\;\sqrt{1 + z} - \sqrt{z} \leq 0.99999997:\\ \;\;\;\;\left(\sqrt{1 + z} + \mathsf{fma}\left(y, 0.5, 2\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(3 - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 95.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z} - \sqrt{z}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;t\_2 - \sqrt{x} \leq 0.99:\\ \;\;\;\;t\_3 + \left(\frac{1}{\sqrt{x} + t\_2} + t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \left(t\_1 + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(1 - \sqrt{x}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 z)) (sqrt z)))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= (- t_2 (sqrt x)) 0.99)
     (+ t_3 (+ (/ 1.0 (+ (sqrt x) t_2)) t_1))
     (+ t_3 (+ t_1 (+ (- (sqrt (+ y 1.0)) (sqrt y)) (- 1.0 (sqrt x))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z)) - sqrt(z);
	double t_2 = sqrt((1.0 + x));
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if ((t_2 - sqrt(x)) <= 0.99) {
		tmp = t_3 + ((1.0 / (sqrt(x) + t_2)) + t_1);
	} else {
		tmp = t_3 + (t_1 + ((sqrt((y + 1.0)) - sqrt(y)) + (1.0 - sqrt(x))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z)) - sqrt(z)
    t_2 = sqrt((1.0d0 + x))
    t_3 = sqrt((1.0d0 + t)) - sqrt(t)
    if ((t_2 - sqrt(x)) <= 0.99d0) then
        tmp = t_3 + ((1.0d0 / (sqrt(x) + t_2)) + t_1)
    else
        tmp = t_3 + (t_1 + ((sqrt((y + 1.0d0)) - sqrt(y)) + (1.0d0 - sqrt(x))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z)) - Math.sqrt(z);
	double t_2 = Math.sqrt((1.0 + x));
	double t_3 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if ((t_2 - Math.sqrt(x)) <= 0.99) {
		tmp = t_3 + ((1.0 / (Math.sqrt(x) + t_2)) + t_1);
	} else {
		tmp = t_3 + (t_1 + ((Math.sqrt((y + 1.0)) - Math.sqrt(y)) + (1.0 - Math.sqrt(x))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z)) - math.sqrt(z)
	t_2 = math.sqrt((1.0 + x))
	t_3 = math.sqrt((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if (t_2 - math.sqrt(x)) <= 0.99:
		tmp = t_3 + ((1.0 / (math.sqrt(x) + t_2)) + t_1)
	else:
		tmp = t_3 + (t_1 + ((math.sqrt((y + 1.0)) - math.sqrt(y)) + (1.0 - math.sqrt(x))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	t_2 = sqrt(Float64(1.0 + x))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (Float64(t_2 - sqrt(x)) <= 0.99)
		tmp = Float64(t_3 + Float64(Float64(1.0 / Float64(sqrt(x) + t_2)) + t_1));
	else
		tmp = Float64(t_3 + Float64(t_1 + Float64(Float64(sqrt(Float64(y + 1.0)) - sqrt(y)) + Float64(1.0 - sqrt(x)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z)) - sqrt(z);
	t_2 = sqrt((1.0 + x));
	t_3 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if ((t_2 - sqrt(x)) <= 0.99)
		tmp = t_3 + ((1.0 / (sqrt(x) + t_2)) + t_1);
	else
		tmp = t_3 + (t_1 + ((sqrt((y + 1.0)) - sqrt(y)) + (1.0 - sqrt(x))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.99], N[(t$95$3 + N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(t$95$1 + N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 0.98999999999999999

   1. Initial program 82.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - \color{blue}{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right) - x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - x}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\color{blue}{\sqrt{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 83.6

          \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \color{blue}{\sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 83.6%

      \[\leadsto \left(\left(\color{blue}{\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{x}} + \sqrt{1 + x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{x} + \color{blue}{\sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 47.5

         \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{\color{blue}{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Simplified 47.5%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.98999999999999999 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 97.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 96.1

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 96.1%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0.99:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(1 - \sqrt{x}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 93.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ t_3 := \sqrt{1 + x}\\ \mathbf{if}\;z \leq 7 \cdot 10^{-16}:\\ \;\;\;\;t\_2 + \left(\left(3 - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+27}:\\ \;\;\;\;t\_3 + \left(\left(t\_1 + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(t\_3 + \left(\frac{1}{\sqrt{y} + t\_1} - \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0)))
        (t_2 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_3 (sqrt (+ 1.0 x))))
   (if (<= z 7e-16)
     (+ t_2 (- (- 3.0 (sqrt x)) (+ (sqrt y) (sqrt z))))
     (if (<= z 4.6e+27)
       (+
        t_3
        (-
         (+ t_1 (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z)))))
         (+ (sqrt y) (sqrt x))))
       (+ t_2 (+ t_3 (- (/ 1.0 (+ (sqrt y) t_1)) (sqrt x))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((1.0 + t)) - sqrt(t);
	double t_3 = sqrt((1.0 + x));
	double tmp;
	if (z <= 7e-16) {
		tmp = t_2 + ((3.0 - sqrt(x)) - (sqrt(y) + sqrt(z)));
	} else if (z <= 4.6e+27) {
		tmp = t_3 + ((t_1 + (1.0 / (sqrt(z) + sqrt((1.0 + z))))) - (sqrt(y) + sqrt(x)));
	} else {
		tmp = t_2 + (t_3 + ((1.0 / (sqrt(y) + t_1)) - sqrt(x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0))
    t_2 = sqrt((1.0d0 + t)) - sqrt(t)
    t_3 = sqrt((1.0d0 + x))
    if (z <= 7d-16) then
        tmp = t_2 + ((3.0d0 - sqrt(x)) - (sqrt(y) + sqrt(z)))
    else if (z <= 4.6d+27) then
        tmp = t_3 + ((t_1 + (1.0d0 / (sqrt(z) + sqrt((1.0d0 + z))))) - (sqrt(y) + sqrt(x)))
    else
        tmp = t_2 + (t_3 + ((1.0d0 / (sqrt(y) + t_1)) - sqrt(x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0));
	double t_2 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double t_3 = Math.sqrt((1.0 + x));
	double tmp;
	if (z <= 7e-16) {
		tmp = t_2 + ((3.0 - Math.sqrt(x)) - (Math.sqrt(y) + Math.sqrt(z)));
	} else if (z <= 4.6e+27) {
		tmp = t_3 + ((t_1 + (1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z))))) - (Math.sqrt(y) + Math.sqrt(x)));
	} else {
		tmp = t_2 + (t_3 + ((1.0 / (Math.sqrt(y) + t_1)) - Math.sqrt(x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0))
	t_2 = math.sqrt((1.0 + t)) - math.sqrt(t)
	t_3 = math.sqrt((1.0 + x))
	tmp = 0
	if z <= 7e-16:
		tmp = t_2 + ((3.0 - math.sqrt(x)) - (math.sqrt(y) + math.sqrt(z)))
	elif z <= 4.6e+27:
		tmp = t_3 + ((t_1 + (1.0 / (math.sqrt(z) + math.sqrt((1.0 + z))))) - (math.sqrt(y) + math.sqrt(x)))
	else:
		tmp = t_2 + (t_3 + ((1.0 / (math.sqrt(y) + t_1)) - math.sqrt(x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_3 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (z <= 7e-16)
		tmp = Float64(t_2 + Float64(Float64(3.0 - sqrt(x)) - Float64(sqrt(y) + sqrt(z))));
	elseif (z <= 4.6e+27)
		tmp = Float64(t_3 + Float64(Float64(t_1 + Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z))))) - Float64(sqrt(y) + sqrt(x))));
	else
		tmp = Float64(t_2 + Float64(t_3 + Float64(Float64(1.0 / Float64(sqrt(y) + t_1)) - sqrt(x))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((y + 1.0));
	t_2 = sqrt((1.0 + t)) - sqrt(t);
	t_3 = sqrt((1.0 + x));
	tmp = 0.0;
	if (z <= 7e-16)
		tmp = t_2 + ((3.0 - sqrt(x)) - (sqrt(y) + sqrt(z)));
	elseif (z <= 4.6e+27)
		tmp = t_3 + ((t_1 + (1.0 / (sqrt(z) + sqrt((1.0 + z))))) - (sqrt(y) + sqrt(x)));
	else
		tmp = t_2 + (t_3 + ((1.0 / (sqrt(y) + t_1)) - sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 7e-16], N[(t$95$2 + N[(N[(3.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.6e+27], N[(t$95$3 + N[(N[(t$95$1 + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(t$95$3 + N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 7.00000000000000035e-16

   1. Initial program 97.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate-+r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \sqrt{1 + y}\right)} - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(1 + \sqrt{1 + x}\right)} + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(1 + \color{blue}{\sqrt{1 + x}}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{\color{blue}{1 + x}}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \color{blue}{\left(\sqrt{1 + y} - \sqrt{x}\right)}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\color{blue}{\sqrt{1 + y}} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \color{blue}{\sqrt{x}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. sqrt-lowering-sqrt.f64 33.8

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 33.8%

      \[\leadsto \color{blue}{\left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + 2\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + 2\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + y}} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + y}} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 28.7

          \[\leadsto \left(\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 28.7%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(3 - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. associate--r+ N/A

          \[\leadsto \color{blue}{\left(\left(3 - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(3 - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \left(\color{blue}{\left(3 - \sqrt{x}\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(3 - \color{blue}{\sqrt{x}}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. +-commutative N/A

          \[\leadsto \left(\left(3 - \sqrt{x}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(3 - \sqrt{x}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(3 - \sqrt{x}\right) - \left(\color{blue}{\sqrt{z}} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       8. sqrt-lowering-sqrt.f64 23.6

          \[\leadsto \left(\left(3 - \sqrt{x}\right) - \left(\sqrt{z} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 23.6%

       \[\leadsto \color{blue}{\left(\left(3 - \sqrt{x}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 7.00000000000000035e-16 < z < 4.6000000000000001e27

   1. Initial program 82.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z + 1}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 88.1

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \color{blue}{\sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 88.1%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\left(\color{blue}{\sqrt{1 + y}} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{\color{blue}{1 + y}} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \color{blue}{\frac{1}{\sqrt{z} + \sqrt{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\color{blue}{\sqrt{z} + \sqrt{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\color{blue}{\sqrt{z}} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \color{blue}{\sqrt{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) \]

      16. sqrt-lowering-sqrt.f64 30.1

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   7. Simplified 30.1%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   if 4.6000000000000001e27 < z

   1. Initial program 85.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 86.0

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \color{blue}{\sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 86.0%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} - \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} - \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + x}} + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + x}} + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} - \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\color{blue}{\frac{1}{\sqrt{y} + \sqrt{1 + y}}} - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\frac{1}{\color{blue}{\sqrt{y} + \sqrt{1 + y}}} - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\frac{1}{\color{blue}{\sqrt{y}} + \sqrt{1 + y}} - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\frac{1}{\sqrt{y} + \color{blue}{\sqrt{1 + y}}} - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + x} + \left(\frac{1}{\sqrt{y} + \sqrt{\color{blue}{1 + y}}} - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. sqrt-lowering-sqrt.f64 66.1

          \[\leadsto \left(\sqrt{1 + x} + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} - \color{blue}{\sqrt{x}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Simplified 66.1%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} - \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7 \cdot 10^{-16}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(3 - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+27}:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(\sqrt{y + 1} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + x} + \left(\frac{1}{\sqrt{y} + \sqrt{y + 1}} - \sqrt{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 10.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) \leq 2.5:\\ \;\;\;\;\sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<=
      (+
       (- (sqrt (+ 1.0 z)) (sqrt z))
       (+ (- (sqrt (+ 1.0 x)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))))
      2.5)
   (sqrt z)
   (- 1.0 (sqrt z))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (((sqrt((1.0 + z)) - sqrt(z)) + ((sqrt((1.0 + x)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y)))) <= 2.5) {
		tmp = sqrt(z);
	} else {
		tmp = 1.0 - sqrt(z);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((sqrt((1.0d0 + z)) - sqrt(z)) + ((sqrt((1.0d0 + x)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y)))) <= 2.5d0) then
        tmp = sqrt(z)
    else
        tmp = 1.0d0 - sqrt(z)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((Math.sqrt((1.0 + z)) - Math.sqrt(z)) + ((Math.sqrt((1.0 + x)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y)))) <= 2.5) {
		tmp = Math.sqrt(z);
	} else {
		tmp = 1.0 - Math.sqrt(z);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if ((math.sqrt((1.0 + z)) - math.sqrt(z)) + ((math.sqrt((1.0 + x)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y)))) <= 2.5:
		tmp = math.sqrt(z)
	else:
		tmp = 1.0 - math.sqrt(z)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(Float64(sqrt(Float64(1.0 + x)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y)))) <= 2.5)
		tmp = sqrt(z);
	else
		tmp = Float64(1.0 - sqrt(z));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((sqrt((1.0 + z)) - sqrt(z)) + ((sqrt((1.0 + x)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y)))) <= 2.5)
		tmp = sqrt(z);
	else
		tmp = 1.0 - sqrt(z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.5], N[Sqrt[z], $MachinePrecision], N[(1.0 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.5

   1. Initial program 88.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 11.5

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 11.5%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) \]

      13. sqrt-lowering-sqrt.f64 5.3

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

   8. Simplified 5.3%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) \]
   10. Step-by-step derivation

       1. sqrt-lowering-sqrt.f64 4.5

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) \]

   11. Simplified 4.5%

       \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) \]

   12. Taylor expanded in z around inf

       \[\leadsto \color{blue}{\sqrt{z}} \]
   13. Step-by-step derivation

       1. sqrt-lowering-sqrt.f64 7.3

          \[\leadsto \color{blue}{\sqrt{z}} \]

   14. Simplified 7.3%

       \[\leadsto \color{blue}{\sqrt{z}} \]

   if 2.5 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 99.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate-+r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \sqrt{1 + y}\right)} - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(1 + \sqrt{1 + x}\right)} + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(1 + \color{blue}{\sqrt{1 + x}}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{\color{blue}{1 + x}}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \color{blue}{\left(\sqrt{1 + y} - \sqrt{x}\right)}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\color{blue}{\sqrt{1 + y}} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \color{blue}{\sqrt{x}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. sqrt-lowering-sqrt.f64 96.3

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 96.3%

      \[\leadsto \color{blue}{\left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{t}} + \sqrt{1 + y}\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{t}}, \sqrt{1 + y}\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{t}}}, \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{t}}}, \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{t}}, \color{blue}{\sqrt{1 + y}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{t}}, \sqrt{\color{blue}{1 + y}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{t}}, \sqrt{1 + y}\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

   8. Simplified 58.2%

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, \sqrt{\frac{1}{t}}, \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto 1 + \color{blue}{-1 \cdot \sqrt{z}} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(\sqrt{z}\right)\right)} \]

       2. neg-lowering-neg.f64 N/A

          \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(\sqrt{z}\right)\right)} \]

       3. sqrt-lowering-sqrt.f64 17.5

          \[\leadsto 1 + \left(-\color{blue}{\sqrt{z}}\right) \]

   11. Simplified 17.5%

       \[\leadsto 1 + \color{blue}{\left(-\sqrt{z}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 8.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) \leq 2.5:\\ \;\;\;\;\sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 85.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 10^{+16}:\\ \;\;\;\;\left(\sqrt{1 + z} + \mathsf{fma}\left(y, 0.5, 2\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 1e+16)
   (- (+ (sqrt (+ 1.0 z)) (fma y 0.5 2.0)) (+ (sqrt x) (+ (sqrt y) (sqrt z))))
   (+ (sqrt (+ 1.0 x)) (- (sqrt (+ y 1.0)) (+ (sqrt y) (sqrt x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1e+16) {
		tmp = (sqrt((1.0 + z)) + fma(y, 0.5, 2.0)) - (sqrt(x) + (sqrt(y) + sqrt(z)));
	} else {
		tmp = sqrt((1.0 + x)) + (sqrt((y + 1.0)) - (sqrt(y) + sqrt(x)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1e+16)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) + fma(y, 0.5, 2.0)) - Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z))));
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) + Float64(sqrt(Float64(y + 1.0)) - Float64(sqrt(y) + sqrt(x))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 1e+16], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[(y * 0.5 + 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1e16

   1. Initial program 96.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 29.9

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 29.9%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) \]

      13. sqrt-lowering-sqrt.f64 19.9

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

   8. Simplified 19.9%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(\color{blue}{\left(2 + \frac{1}{2} \cdot y\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot y + 2\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{y \cdot \frac{1}{2}} + 2\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

       3. accelerator-lowering-fma.f64 19.7

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(y, 0.5, 2\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   11. Simplified 19.7%

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(y, 0.5, 2\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   if 1e16 < z

   1. Initial program 84.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 5.0

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 5.0%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\sqrt{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{\color{blue}{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) \]

      10. sqrt-lowering-sqrt.f64 31.2

          \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   8. Simplified 31.2%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{+16}:\\ \;\;\;\;\left(\sqrt{1 + z} + \mathsf{fma}\left(y, 0.5, 2\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 85.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.6 \cdot 10^{+15}:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.6e+15)
   (- (+ (sqrt (+ 1.0 z)) 2.0) (+ (sqrt x) (+ (sqrt y) (sqrt z))))
   (+ (sqrt (+ 1.0 x)) (- (sqrt (+ y 1.0)) (+ (sqrt y) (sqrt x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.6e+15) {
		tmp = (sqrt((1.0 + z)) + 2.0) - (sqrt(x) + (sqrt(y) + sqrt(z)));
	} else {
		tmp = sqrt((1.0 + x)) + (sqrt((y + 1.0)) - (sqrt(y) + sqrt(x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 1.6d+15) then
        tmp = (sqrt((1.0d0 + z)) + 2.0d0) - (sqrt(x) + (sqrt(y) + sqrt(z)))
    else
        tmp = sqrt((1.0d0 + x)) + (sqrt((y + 1.0d0)) - (sqrt(y) + sqrt(x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.6e+15) {
		tmp = (Math.sqrt((1.0 + z)) + 2.0) - (Math.sqrt(x) + (Math.sqrt(y) + Math.sqrt(z)));
	} else {
		tmp = Math.sqrt((1.0 + x)) + (Math.sqrt((y + 1.0)) - (Math.sqrt(y) + Math.sqrt(x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 1.6e+15:
		tmp = (math.sqrt((1.0 + z)) + 2.0) - (math.sqrt(x) + (math.sqrt(y) + math.sqrt(z)))
	else:
		tmp = math.sqrt((1.0 + x)) + (math.sqrt((y + 1.0)) - (math.sqrt(y) + math.sqrt(x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.6e+15)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) + 2.0) - Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z))));
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) + Float64(sqrt(Float64(y + 1.0)) - Float64(sqrt(y) + sqrt(x))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.6e+15)
		tmp = (sqrt((1.0 + z)) + 2.0) - (sqrt(x) + (sqrt(y) + sqrt(z)));
	else
		tmp = sqrt((1.0 + x)) + (sqrt((y + 1.0)) - (sqrt(y) + sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 1.6e+15], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + 2.0), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.6e15

   1. Initial program 96.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 29.9

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 29.9%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) \]

      13. sqrt-lowering-sqrt.f64 19.9

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

   8. Simplified 19.9%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(2 + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + z} + 2\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + z} + 2\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

       3. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{1 + z}} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

       4. +-lowering-+.f64 18.0

          \[\leadsto \left(\sqrt{\color{blue}{1 + z}} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   11. Simplified 18.0%

       \[\leadsto \color{blue}{\left(\sqrt{1 + z} + 2\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   if 1.6e15 < z

   1. Initial program 84.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 5.0

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 5.0%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\sqrt{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{\color{blue}{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) \]

      10. sqrt-lowering-sqrt.f64 31.2

          \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   8. Simplified 31.2%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.6 \cdot 10^{+15}:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 81.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{y} + \sqrt{x}\\ \mathbf{if}\;z \leq 0.0155:\\ \;\;\;\;\left(t\_1 + 2\right) - t\_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(t\_1 - t\_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0))) (t_2 (+ (sqrt y) (sqrt x))))
   (if (<= z 0.0155) (- (+ t_1 2.0) t_2) (+ (sqrt (+ 1.0 x)) (- t_1 t_2)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt(y) + sqrt(x);
	double tmp;
	if (z <= 0.0155) {
		tmp = (t_1 + 2.0) - t_2;
	} else {
		tmp = sqrt((1.0 + x)) + (t_1 - t_2);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0))
    t_2 = sqrt(y) + sqrt(x)
    if (z <= 0.0155d0) then
        tmp = (t_1 + 2.0d0) - t_2
    else
        tmp = sqrt((1.0d0 + x)) + (t_1 - t_2)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0));
	double t_2 = Math.sqrt(y) + Math.sqrt(x);
	double tmp;
	if (z <= 0.0155) {
		tmp = (t_1 + 2.0) - t_2;
	} else {
		tmp = Math.sqrt((1.0 + x)) + (t_1 - t_2);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0))
	t_2 = math.sqrt(y) + math.sqrt(x)
	tmp = 0
	if z <= 0.0155:
		tmp = (t_1 + 2.0) - t_2
	else:
		tmp = math.sqrt((1.0 + x)) + (t_1 - t_2)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = Float64(sqrt(y) + sqrt(x))
	tmp = 0.0
	if (z <= 0.0155)
		tmp = Float64(Float64(t_1 + 2.0) - t_2);
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) + Float64(t_1 - t_2));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((y + 1.0));
	t_2 = sqrt(y) + sqrt(x);
	tmp = 0.0;
	if (z <= 0.0155)
		tmp = (t_1 + 2.0) - t_2;
	else
		tmp = sqrt((1.0 + x)) + (t_1 - t_2);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 0.0155], N[(N[(t$95$1 + 2.0), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < 0.0155

   1. Initial program 97.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 29.5

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 29.5%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) \]

      13. sqrt-lowering-sqrt.f64 18.6

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

   8. Simplified 18.6%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) \]
   10. Step-by-step derivation

       1. sqrt-lowering-sqrt.f64 18.6

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) \]

   11. Simplified 18.6%

       \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) \]

   12. Taylor expanded in z around 0

       \[\leadsto \color{blue}{\left(2 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   13. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(2 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + y} + 2\right)} - \left(\sqrt{x} + \sqrt{y}\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + y} + 2\right)} - \left(\sqrt{x} + \sqrt{y}\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + 2\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + 2\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + 2\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + 2\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right) \]

       8. sqrt-lowering-sqrt.f64 18.6

          \[\leadsto \left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) \]

   14. Simplified 18.6%

       \[\leadsto \color{blue}{\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

   if 0.0155 < z

   1. Initial program 84.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 7.6

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 7.6%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\sqrt{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{\color{blue}{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) \]

      10. sqrt-lowering-sqrt.f64 29.9

          \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   8. Simplified 29.9%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.0155:\\ \;\;\;\;\left(\sqrt{y + 1} + 2\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 81.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{y} + \sqrt{x}\\ \mathbf{if}\;z \leq 0.005:\\ \;\;\;\;\left(t\_1 + 2\right) - t\_2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(t\_1 - t\_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0))) (t_2 (+ (sqrt y) (sqrt x))))
   (if (<= z 0.005) (- (+ t_1 2.0) t_2) (+ 1.0 (- t_1 t_2)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt(y) + sqrt(x);
	double tmp;
	if (z <= 0.005) {
		tmp = (t_1 + 2.0) - t_2;
	} else {
		tmp = 1.0 + (t_1 - t_2);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0))
    t_2 = sqrt(y) + sqrt(x)
    if (z <= 0.005d0) then
        tmp = (t_1 + 2.0d0) - t_2
    else
        tmp = 1.0d0 + (t_1 - t_2)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0));
	double t_2 = Math.sqrt(y) + Math.sqrt(x);
	double tmp;
	if (z <= 0.005) {
		tmp = (t_1 + 2.0) - t_2;
	} else {
		tmp = 1.0 + (t_1 - t_2);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0))
	t_2 = math.sqrt(y) + math.sqrt(x)
	tmp = 0
	if z <= 0.005:
		tmp = (t_1 + 2.0) - t_2
	else:
		tmp = 1.0 + (t_1 - t_2)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = Float64(sqrt(y) + sqrt(x))
	tmp = 0.0
	if (z <= 0.005)
		tmp = Float64(Float64(t_1 + 2.0) - t_2);
	else
		tmp = Float64(1.0 + Float64(t_1 - t_2));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((y + 1.0));
	t_2 = sqrt(y) + sqrt(x);
	tmp = 0.0;
	if (z <= 0.005)
		tmp = (t_1 + 2.0) - t_2;
	else
		tmp = 1.0 + (t_1 - t_2);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 0.005], N[(N[(t$95$1 + 2.0), $MachinePrecision] - t$95$2), $MachinePrecision], N[(1.0 + N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < 0.0050000000000000001

   1. Initial program 97.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 29.5

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 29.5%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) \]

      13. sqrt-lowering-sqrt.f64 18.6

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

   8. Simplified 18.6%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) \]
   10. Step-by-step derivation

       1. sqrt-lowering-sqrt.f64 18.6

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) \]

   11. Simplified 18.6%

       \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) \]

   12. Taylor expanded in z around 0

       \[\leadsto \color{blue}{\left(2 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   13. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(2 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + y} + 2\right)} - \left(\sqrt{x} + \sqrt{y}\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + y} + 2\right)} - \left(\sqrt{x} + \sqrt{y}\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + 2\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + 2\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + 2\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + 2\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right) \]

       8. sqrt-lowering-sqrt.f64 18.6

          \[\leadsto \left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) \]

   14. Simplified 18.6%

       \[\leadsto \color{blue}{\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

   if 0.0050000000000000001 < z

   1. Initial program 84.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 7.6

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 7.6%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) \]

      13. sqrt-lowering-sqrt.f64 6.0

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

   8. Simplified 6.0%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   10. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

       3. --lowering--.f64 N/A

          \[\leadsto 1 + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto 1 + \left(\color{blue}{\sqrt{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto 1 + \left(\sqrt{\color{blue}{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       6. +-commutative N/A

          \[\leadsto 1 + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto 1 + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto 1 + \left(\sqrt{1 + y} - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) \]

       9. sqrt-lowering-sqrt.f64 30.9

          \[\leadsto 1 + \left(\sqrt{1 + y} - \left(\sqrt{y} + \color{blue}{\sqrt{x}}\right)\right) \]

   11. Simplified 30.9%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.005:\\ \;\;\;\;\left(\sqrt{y + 1} + 2\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 64.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 1 + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right) \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (+ 1.0 (- (sqrt (+ y 1.0)) (+ (sqrt y) (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 1.0 + (sqrt((y + 1.0)) - (sqrt(y) + sqrt(x)));
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0 + (sqrt((y + 1.0d0)) - (sqrt(y) + sqrt(x)))
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 1.0 + (Math.sqrt((y + 1.0)) - (Math.sqrt(y) + Math.sqrt(x)));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 1.0 + (math.sqrt((y + 1.0)) - (math.sqrt(y) + math.sqrt(x)))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(1.0 + Float64(sqrt(Float64(y + 1.0)) - Float64(sqrt(y) + sqrt(x))))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 1.0 + (sqrt((y + 1.0)) - (sqrt(y) + sqrt(x)));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(1.0 + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.0%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   2. associate--l+ N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   9. --lowering--.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

   13. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

   15. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

   16. sqrt-lowering-sqrt.f64 17.1

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

5. Simplified 17.1%

   \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
7. Step-by-step derivation

   1. --lowering--.f64 N/A

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   2. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) \]

   12. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) \]

   13. sqrt-lowering-sqrt.f64 11.5

       \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

8. Simplified 11.5%

   \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

9. Taylor expanded in z around inf

   \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
10. Step-by-step derivation

    1. associate--l+ N/A

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

    3. --lowering--.f64 N/A

       \[\leadsto 1 + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

    4. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto 1 + \left(\color{blue}{\sqrt{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

    5. +-lowering-+.f64 N/A

       \[\leadsto 1 + \left(\sqrt{\color{blue}{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

    6. +-commutative N/A

       \[\leadsto 1 + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

    7. +-lowering-+.f64 N/A

       \[\leadsto 1 + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

    8. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto 1 + \left(\sqrt{1 + y} - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) \]

    9. sqrt-lowering-sqrt.f64 24.2

       \[\leadsto 1 + \left(\sqrt{1 + y} - \left(\sqrt{y} + \color{blue}{\sqrt{x}}\right)\right) \]

11. Simplified 24.2%

    \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

12. Final simplification 24.2%

    \[\leadsto 1 + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right) \]
13. Add Preprocessing

Alternative 23: 11.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1:\\ \;\;\;\;1 - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.0) (- 1.0 (sqrt z)) (* 0.5 (sqrt (/ 1.0 y)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.0) {
		tmp = 1.0 - sqrt(z);
	} else {
		tmp = 0.5 * sqrt((1.0 / y));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 1.0d0) then
        tmp = 1.0d0 - sqrt(z)
    else
        tmp = 0.5d0 * sqrt((1.0d0 / y))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.0) {
		tmp = 1.0 - Math.sqrt(z);
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 1.0:
		tmp = 1.0 - math.sqrt(z)
	else:
		tmp = 0.5 * math.sqrt((1.0 / y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.0)
		tmp = Float64(1.0 - sqrt(z));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.0)
		tmp = 1.0 - sqrt(z);
	else
		tmp = 0.5 * sqrt((1.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 1.0], N[(1.0 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 97.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate-+r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \sqrt{1 + y}\right)} - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(1 + \sqrt{1 + x}\right)} + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(1 + \color{blue}{\sqrt{1 + x}}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{\color{blue}{1 + x}}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \color{blue}{\left(\sqrt{1 + y} - \sqrt{x}\right)}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\color{blue}{\sqrt{1 + y}} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \color{blue}{\sqrt{x}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. sqrt-lowering-sqrt.f64 33.6

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 33.6%

      \[\leadsto \color{blue}{\left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{t}} + \sqrt{1 + y}\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{t}}, \sqrt{1 + y}\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{t}}}, \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{t}}}, \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{t}}, \color{blue}{\sqrt{1 + y}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{t}}, \sqrt{\color{blue}{1 + y}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{t}}, \sqrt{1 + y}\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

   8. Simplified 35.6%

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, \sqrt{\frac{1}{t}}, \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto 1 + \color{blue}{-1 \cdot \sqrt{z}} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(\sqrt{z}\right)\right)} \]

       2. neg-lowering-neg.f64 N/A

          \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(\sqrt{z}\right)\right)} \]

       3. sqrt-lowering-sqrt.f64 26.8

          \[\leadsto 1 + \left(-\color{blue}{\sqrt{z}}\right) \]

   11. Simplified 26.8%

       \[\leadsto 1 + \color{blue}{\left(-\sqrt{z}\right)} \]

   if 1 < z

   1. Initial program 84.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \sqrt{1 + x} - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x} - \sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 40.4

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 40.4%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x} - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{y}}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{y}}} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{y}}} \]

      3. /-lowering-/.f64 9.9

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{y}}} \]

   8. Simplified 9.9%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1:\\ \;\;\;\;1 - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 34.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 1 + 0.5 \cdot \sqrt{\frac{1}{t}} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (+ 1.0 (* 0.5 (sqrt (/ 1.0 t)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 1.0 + (0.5 * sqrt((1.0 / t)));
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0 + (0.5d0 * sqrt((1.0d0 / t)))
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 1.0 + (0.5 * Math.sqrt((1.0 / t)));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 1.0 + (0.5 * math.sqrt((1.0 / t)))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(1.0 + Float64(0.5 * sqrt(Float64(1.0 / t))))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 1.0 + (0.5 * sqrt((1.0 / t)));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(1.0 + N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.0%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
4. Step-by-step derivation

   1. associate--r+ N/A

      \[\leadsto \color{blue}{\left(\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. --lowering--.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. associate-+r+ N/A

      \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \sqrt{1 + y}\right)} - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. associate--l+ N/A

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\left(1 + \sqrt{1 + x}\right)} + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \left(\left(\left(1 + \color{blue}{\sqrt{1 + x}}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \left(\left(\left(1 + \sqrt{\color{blue}{1 + x}}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. --lowering--.f64 N/A

      \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \color{blue}{\left(\sqrt{1 + y} - \sqrt{x}\right)}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\color{blue}{\sqrt{1 + y}} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \color{blue}{\sqrt{x}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   14. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   15. sqrt-lowering-sqrt.f64 18.4

       \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

5. Simplified 18.4%

   \[\leadsto \color{blue}{\left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

6. Taylor expanded in t around inf

   \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
7. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   3. --lowering--.f64 N/A

      \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   4. +-lowering-+.f64 N/A

      \[\leadsto 1 + \left(\color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto 1 + \left(\left(\color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto 1 + \left(\left(\sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{t}} + \sqrt{1 + y}\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{t}}, \sqrt{1 + y}\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   9. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{t}}}, \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{t}}}, \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   11. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{t}}, \color{blue}{\sqrt{1 + y}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{t}}, \sqrt{\color{blue}{1 + y}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   13. +-lowering-+.f64 N/A

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{t}}, \sqrt{1 + y}\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

8. Simplified 26.3%

   \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, \sqrt{\frac{1}{t}}, \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

9. Taylor expanded in t around 0

   \[\leadsto 1 + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}}} \]
10. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto 1 + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}}} \]

    2. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto 1 + \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{t}}} \]

    3. /-lowering-/.f64 25.1

       \[\leadsto 1 + 0.5 \cdot \sqrt{\color{blue}{\frac{1}{t}}} \]

11. Simplified 25.1%

    \[\leadsto 1 + \color{blue}{0.5 \cdot \sqrt{\frac{1}{t}}} \]
12. Add Preprocessing

Alternative 25: 7.6% accurate, 10.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \sqrt{z} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (sqrt z))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return sqrt(z);
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = sqrt(z)
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return Math.sqrt(z);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return math.sqrt(z)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return sqrt(z)
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = sqrt(z);
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[Sqrt[z], $MachinePrecision]

Derivation

1. Initial program 90.0%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   2. associate--l+ N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   9. --lowering--.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

   13. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

   15. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

   16. sqrt-lowering-sqrt.f64 17.1

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

5. Simplified 17.1%

   \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
7. Step-by-step derivation

   1. --lowering--.f64 N/A

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   2. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) \]

   12. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) \]

   13. sqrt-lowering-sqrt.f64 11.5

       \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

8. Simplified 11.5%

   \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

9. Taylor expanded in y around inf

   \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) \]
10. Step-by-step derivation

    1. sqrt-lowering-sqrt.f64 10.8

       \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) \]

11. Simplified 10.8%

    \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) \]

12. Taylor expanded in z around inf

    \[\leadsto \color{blue}{\sqrt{z}} \]
13. Step-by-step derivation

    1. sqrt-lowering-sqrt.f64 7.2

       \[\leadsto \color{blue}{\sqrt{z}} \]

14. Simplified 7.2%

    \[\leadsto \color{blue}{\sqrt{z}} \]
15. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+
    (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))
    (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y))))
   (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z))))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))) + (1.0d0 / (sqrt((y + 1.0d0)) + sqrt(y)))) + (1.0d0 / (sqrt((z + 1.0d0)) + sqrt(z)))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x))) + (1.0 / (Math.sqrt((y + 1.0)) + Math.sqrt(y)))) + (1.0 / (Math.sqrt((z + 1.0)) + Math.sqrt(z)))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))) + (1.0 / (math.sqrt((y + 1.0)) + math.sqrt(y)))) + (1.0 / (math.sqrt((z + 1.0)) + math.sqrt(z)))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x))) + Float64(1.0 / Float64(sqrt(Float64(y + 1.0)) + sqrt(y)))) + Float64(1.0 / Float64(sqrt(Float64(z + 1.0)) + sqrt(z)))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024198 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64

  :alt
  (! :herbie-platform default (+ (+ (+ (/ 1 (+ (sqrt (+ x 1)) (sqrt x))) (/ 1 (+ (sqrt (+ y 1)) (sqrt y)))) (/ 1 (+ (sqrt (+ z 1)) (sqrt z)))) (- (sqrt (+ t 1)) (sqrt t))))

  (+ (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) (- (sqrt (+ z 1.0)) (sqrt z))) (- (sqrt (+ t 1.0)) (sqrt t))))