Main:z from

Percentage Accurate: 91.9% → 97.3%

Time: 28.5s

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: 91.9% 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: 97.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + t}\\ t_3 := t\_2 - \sqrt{t}\\ t_4 := \sqrt{1 + y}\\ t_5 := \sqrt{1 + z}\\ t_6 := t\_5 - \sqrt{z}\\ t_7 := t\_3 + \left(t\_6 + \left(\left(t\_1 - \sqrt{x}\right) + \left(t\_4 - \sqrt{y}\right)\right)\right)\\ \mathbf{if}\;t\_7 \leq 0.0001:\\ \;\;\;\;t\_3 + \left(t\_6 + \frac{1}{\sqrt{x} + t\_1}\right)\\ \mathbf{elif}\;t\_7 \leq 2:\\ \;\;\;\;t\_3 + \left(t\_1 - \left(\sqrt{x} + \frac{-1}{\sqrt{y} + t\_4}\right)\right)\\ \mathbf{elif}\;t\_7 \leq 3.002:\\ \;\;\;\;1 + \left(\left(t\_4 + \mathsf{fma}\left(0.5, \sqrt{\frac{1}{t}}, t\_5\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 + 3\right) - \left(\sqrt{t} + \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \end{array} \end{array} \]

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)))
        (t_2 (sqrt (+ 1.0 t)))
        (t_3 (- t_2 (sqrt t)))
        (t_4 (sqrt (+ 1.0 y)))
        (t_5 (sqrt (+ 1.0 z)))
        (t_6 (- t_5 (sqrt z)))
        (t_7 (+ t_3 (+ t_6 (+ (- t_1 (sqrt x)) (- t_4 (sqrt y)))))))
   (if (<= t_7 0.0001)
     (+ t_3 (+ t_6 (/ 1.0 (+ (sqrt x) t_1))))
     (if (<= t_7 2.0)
       (+ t_3 (- t_1 (+ (sqrt x) (/ -1.0 (+ (sqrt y) t_4)))))
       (if (<= t_7 3.002)
         (+
          1.0
          (-
           (+ t_4 (fma 0.5 (sqrt (/ 1.0 t)) t_5))
           (+ (sqrt x) (+ (sqrt y) (sqrt z)))))
         (- (+ t_2 3.0) (+ (sqrt t) (+ (sqrt z) (+ (sqrt x) (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));
	double t_2 = sqrt((1.0 + t));
	double t_3 = t_2 - sqrt(t);
	double t_4 = sqrt((1.0 + y));
	double t_5 = sqrt((1.0 + z));
	double t_6 = t_5 - sqrt(z);
	double t_7 = t_3 + (t_6 + ((t_1 - sqrt(x)) + (t_4 - sqrt(y))));
	double tmp;
	if (t_7 <= 0.0001) {
		tmp = t_3 + (t_6 + (1.0 / (sqrt(x) + t_1)));
	} else if (t_7 <= 2.0) {
		tmp = t_3 + (t_1 - (sqrt(x) + (-1.0 / (sqrt(y) + t_4))));
	} else if (t_7 <= 3.002) {
		tmp = 1.0 + ((t_4 + fma(0.5, sqrt((1.0 / t)), t_5)) - (sqrt(x) + (sqrt(y) + sqrt(z))));
	} else {
		tmp = (t_2 + 3.0) - (sqrt(t) + (sqrt(z) + (sqrt(x) + sqrt(y))));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = sqrt(Float64(1.0 + t))
	t_3 = Float64(t_2 - sqrt(t))
	t_4 = sqrt(Float64(1.0 + y))
	t_5 = sqrt(Float64(1.0 + z))
	t_6 = Float64(t_5 - sqrt(z))
	t_7 = Float64(t_3 + Float64(t_6 + Float64(Float64(t_1 - sqrt(x)) + Float64(t_4 - sqrt(y)))))
	tmp = 0.0
	if (t_7 <= 0.0001)
		tmp = Float64(t_3 + Float64(t_6 + Float64(1.0 / Float64(sqrt(x) + t_1))));
	elseif (t_7 <= 2.0)
		tmp = Float64(t_3 + Float64(t_1 - Float64(sqrt(x) + Float64(-1.0 / Float64(sqrt(y) + t_4)))));
	elseif (t_7 <= 3.002)
		tmp = Float64(1.0 + Float64(Float64(t_4 + fma(0.5, sqrt(Float64(1.0 / t)), t_5)) - Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z)))));
	else
		tmp = Float64(Float64(t_2 + 3.0) - Float64(sqrt(t) + Float64(sqrt(z) + Float64(sqrt(x) + 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[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$3 + N[(t$95$6 + N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$7, 0.0001], N[(t$95$3 + N[(t$95$6 + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$7, 2.0], N[(t$95$3 + N[(t$95$1 - N[(N[Sqrt[x], $MachinePrecision] + N[(-1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$7, 3.002], N[(1.0 + N[(N[(t$95$4 + N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 + 3.0), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.00000000000000005e-4

   1. Initial program 10.8%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. flip-- N/A

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

      5. lift-+.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. flip-- N/A

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

      9. frac-add N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 58.3

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

   7. Applied rewrites 58.3%

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

   8. Taylor expanded in y around inf

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

      1. lower-/.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. lower-+.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. lower-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. lower-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. lower-+.f64 56.6

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

   10. Applied rewrites 56.6%

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

   if 1.00000000000000005e-4 < (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2

   1. Initial program 97.1%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lower--.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-+.f64 97.5

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 97.5

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

   4. Applied rewrites 97.5%

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

   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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-/.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. lower-+.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. lower-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. lower-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. lower-+.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. lower-sqrt.f64 46.9

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

   7. Applied rewrites 46.9%

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

   if 2 < (+.f64 (+.f64 (+.f64 (-.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3.0019999999999998

   1. Initial program 96.4%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate--r+ N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 38.9%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-+.f64 N/A

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

   8. Applied rewrites 33.8%

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

   if 3.0019999999999998 < (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   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. lower--.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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-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. lower-+.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. lower-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. lower-+.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. lower-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. lower-sqrt.f64 93.6

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

   5. Applied rewrites 93.6%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 93.6

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

   8. Applied rewrites 93.6%

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

   9. Taylor expanded in y around 0

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

       1. lower--.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-+.f64 N/A

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

       7. lower-sqrt.f64 N/A

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

       8. associate-+r+ N/A

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

       9. lower-+.f64 N/A

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

       10. lower-+.f64 N/A

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

       11. lower-sqrt.f64 N/A

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

       12. lower-sqrt.f64 N/A

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

       13. lower-sqrt.f64 87.7

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

   11. Applied rewrites 87.7%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 0.0001:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 2:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \frac{-1}{\sqrt{y} + \sqrt{1 + y}}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 3.002:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + y} + \mathsf{fma}\left(0.5, \sqrt{\frac{1}{t}}, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} + 3\right) - \left(\sqrt{t} + \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ t_3 := \sqrt{1 + y}\\ t_4 := \sqrt{1 + z}\\ t_5 := t\_4 - \sqrt{z}\\ t_6 := t\_2 + \left(t\_5 + \left(\left(t\_1 - \sqrt{x}\right) + \left(t\_3 - \sqrt{y}\right)\right)\right)\\ \mathbf{if}\;t\_6 \leq 0.0001:\\ \;\;\;\;t\_2 + \left(t\_5 + \frac{1}{\sqrt{x} + t\_1}\right)\\ \mathbf{elif}\;t\_6 \leq 2:\\ \;\;\;\;t\_2 + \left(t\_1 - \left(\sqrt{x} + \frac{-1}{\sqrt{y} + t\_3}\right)\right)\\ \mathbf{elif}\;t\_6 \leq 2.9999999:\\ \;\;\;\;1 + \left(\left(t\_1 + \mathsf{fma}\left(0.5, y, t\_4\right)\right) - \left(\sqrt{x} + \frac{z - y}{\sqrt{z} - \sqrt{y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(\left(2 + \left(t\_3 - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \end{array} \end{array} \]

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 (+ 1.0 x)))
        (t_2 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_3 (sqrt (+ 1.0 y)))
        (t_4 (sqrt (+ 1.0 z)))
        (t_5 (- t_4 (sqrt z)))
        (t_6 (+ t_2 (+ t_5 (+ (- t_1 (sqrt x)) (- t_3 (sqrt y)))))))
   (if (<= t_6 0.0001)
     (+ t_2 (+ t_5 (/ 1.0 (+ (sqrt x) t_1))))
     (if (<= t_6 2.0)
       (+ t_2 (- t_1 (+ (sqrt x) (/ -1.0 (+ (sqrt y) t_3)))))
       (if (<= t_6 2.9999999)
         (+
          1.0
          (-
           (+ t_1 (fma 0.5 y t_4))
           (+ (sqrt x) (/ (- z y) (- (sqrt z) (sqrt y))))))
         (+ t_2 (- (+ 2.0 (- t_3 (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((1.0 + x));
	double t_2 = sqrt((1.0 + t)) - sqrt(t);
	double t_3 = sqrt((1.0 + y));
	double t_4 = sqrt((1.0 + z));
	double t_5 = t_4 - sqrt(z);
	double t_6 = t_2 + (t_5 + ((t_1 - sqrt(x)) + (t_3 - sqrt(y))));
	double tmp;
	if (t_6 <= 0.0001) {
		tmp = t_2 + (t_5 + (1.0 / (sqrt(x) + t_1)));
	} else if (t_6 <= 2.0) {
		tmp = t_2 + (t_1 - (sqrt(x) + (-1.0 / (sqrt(y) + t_3))));
	} else if (t_6 <= 2.9999999) {
		tmp = 1.0 + ((t_1 + fma(0.5, y, t_4)) - (sqrt(x) + ((z - y) / (sqrt(z) - sqrt(y)))));
	} else {
		tmp = t_2 + ((2.0 + (t_3 - 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(1.0 + x))
	t_2 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_3 = sqrt(Float64(1.0 + y))
	t_4 = sqrt(Float64(1.0 + z))
	t_5 = Float64(t_4 - sqrt(z))
	t_6 = Float64(t_2 + Float64(t_5 + Float64(Float64(t_1 - sqrt(x)) + Float64(t_3 - sqrt(y)))))
	tmp = 0.0
	if (t_6 <= 0.0001)
		tmp = Float64(t_2 + Float64(t_5 + Float64(1.0 / Float64(sqrt(x) + t_1))));
	elseif (t_6 <= 2.0)
		tmp = Float64(t_2 + Float64(t_1 - Float64(sqrt(x) + Float64(-1.0 / Float64(sqrt(y) + t_3)))));
	elseif (t_6 <= 2.9999999)
		tmp = Float64(1.0 + Float64(Float64(t_1 + fma(0.5, y, t_4)) - Float64(sqrt(x) + Float64(Float64(z - y) / Float64(sqrt(z) - sqrt(y))))));
	else
		tmp = Float64(t_2 + Float64(Float64(2.0 + Float64(t_3 - 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[(1.0 + x), $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 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$2 + N[(t$95$5 + N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 0.0001], N[(t$95$2 + N[(t$95$5 + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.0], N[(t$95$2 + N[(t$95$1 - N[(N[Sqrt[x], $MachinePrecision] + N[(-1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.9999999], N[(1.0 + N[(N[(t$95$1 + N[(0.5 * y + t$95$4), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[(z - y), $MachinePrecision] / N[(N[Sqrt[z], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(N[(2.0 + N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $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 (-.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.00000000000000005e-4

   1. Initial program 10.8%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. flip-- N/A

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

      5. lift-+.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. flip-- N/A

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

      9. frac-add N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 58.3

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

   7. Applied rewrites 58.3%

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

   8. Taylor expanded in y around inf

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

      1. lower-/.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. lower-+.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. lower-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. lower-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. lower-+.f64 56.6

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

   10. Applied rewrites 56.6%

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

   if 1.00000000000000005e-4 < (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2

   1. Initial program 97.1%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lower--.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-+.f64 97.5

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 97.5

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

   4. Applied rewrites 97.5%

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

   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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-/.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. lower-+.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. lower-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. lower-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. lower-+.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. lower-sqrt.f64 46.9

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

   7. Applied rewrites 46.9%

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

   if 2 < (+.f64 (+.f64 (+.f64 (-.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2.99999990000000016

   1. Initial program 91.7%

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

   3. 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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-sqrt.f64 43.9

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

   5. Applied rewrites 43.9%

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

   6. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-sqrt.f64 32.5

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

   8. Applied rewrites 32.5%

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, y, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. +-commutative N/A

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

      4. flip-+ N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 31.5

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

   10. Applied rewrites 31.5%

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

   if 2.99999990000000016 < (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   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. 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. lower--.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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-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. lower-+.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. lower-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. lower-+.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. lower-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. lower-sqrt.f64 51.8

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 48.0

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

   8. Applied rewrites 48.0%

      \[\leadsto \left(\color{blue}{\left(2 + \left(\sqrt{1 + y} - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 0.0001:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 2:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \frac{-1}{\sqrt{y} + \sqrt{1 + y}}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 2.9999999:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, y, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \frac{z - y}{\sqrt{z} - \sqrt{y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(2 + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := t\_1 - \sqrt{x}\\ t_3 := \sqrt{1 + t}\\ t_4 := \sqrt{1 + y}\\ t_5 := \sqrt{x} + \sqrt{y}\\ t_6 := \sqrt{1 + z}\\ t_7 := \left(t\_3 - \sqrt{t}\right) + \left(\left(t\_6 - \sqrt{z}\right) + \left(t\_2 + \left(t\_4 - \sqrt{y}\right)\right)\right)\\ \mathbf{if}\;t\_7 \leq 1:\\ \;\;\;\;t\_2 + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \mathbf{elif}\;t\_7 \leq 2.0002:\\ \;\;\;\;\left(t\_1 + \mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, t\_4\right)\right) - t\_5\\ \mathbf{elif}\;t\_7 \leq 3:\\ \;\;\;\;\left(2 + \mathsf{fma}\left(y, 0.5, t\_6\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_3 + 3\right) - \left(\sqrt{t} + \left(\sqrt{z} + t\_5\right)\right)\\ \end{array} \end{array} \]

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)))
        (t_2 (- t_1 (sqrt x)))
        (t_3 (sqrt (+ 1.0 t)))
        (t_4 (sqrt (+ 1.0 y)))
        (t_5 (+ (sqrt x) (sqrt y)))
        (t_6 (sqrt (+ 1.0 z)))
        (t_7
         (+ (- t_3 (sqrt t)) (+ (- t_6 (sqrt z)) (+ t_2 (- t_4 (sqrt y)))))))
   (if (<= t_7 1.0)
     (+ t_2 (* 0.5 (sqrt (/ 1.0 t))))
     (if (<= t_7 2.0002)
       (- (+ t_1 (fma 0.5 (sqrt (/ 1.0 z)) t_4)) t_5)
       (if (<= t_7 3.0)
         (- (+ 2.0 (fma y 0.5 t_6)) (+ (sqrt x) (+ (sqrt y) (sqrt z))))
         (- (+ t_3 3.0) (+ (sqrt t) (+ (sqrt z) t_5))))))))

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));
	double t_2 = t_1 - sqrt(x);
	double t_3 = sqrt((1.0 + t));
	double t_4 = sqrt((1.0 + y));
	double t_5 = sqrt(x) + sqrt(y);
	double t_6 = sqrt((1.0 + z));
	double t_7 = (t_3 - sqrt(t)) + ((t_6 - sqrt(z)) + (t_2 + (t_4 - sqrt(y))));
	double tmp;
	if (t_7 <= 1.0) {
		tmp = t_2 + (0.5 * sqrt((1.0 / t)));
	} else if (t_7 <= 2.0002) {
		tmp = (t_1 + fma(0.5, sqrt((1.0 / z)), t_4)) - t_5;
	} else if (t_7 <= 3.0) {
		tmp = (2.0 + fma(y, 0.5, t_6)) - (sqrt(x) + (sqrt(y) + sqrt(z)));
	} else {
		tmp = (t_3 + 3.0) - (sqrt(t) + (sqrt(z) + t_5));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = Float64(t_1 - sqrt(x))
	t_3 = sqrt(Float64(1.0 + t))
	t_4 = sqrt(Float64(1.0 + y))
	t_5 = Float64(sqrt(x) + sqrt(y))
	t_6 = sqrt(Float64(1.0 + z))
	t_7 = Float64(Float64(t_3 - sqrt(t)) + Float64(Float64(t_6 - sqrt(z)) + Float64(t_2 + Float64(t_4 - sqrt(y)))))
	tmp = 0.0
	if (t_7 <= 1.0)
		tmp = Float64(t_2 + Float64(0.5 * sqrt(Float64(1.0 / t))));
	elseif (t_7 <= 2.0002)
		tmp = Float64(Float64(t_1 + fma(0.5, sqrt(Float64(1.0 / z)), t_4)) - t_5);
	elseif (t_7 <= 3.0)
		tmp = Float64(Float64(2.0 + fma(y, 0.5, t_6)) - Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z))));
	else
		tmp = Float64(Float64(t_3 + 3.0) - Float64(sqrt(t) + Float64(sqrt(z) + t_5)));
	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[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$7 = N[(N[(t$95$3 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$6 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$7, 1.0], N[(t$95$2 + N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$7, 2.0002], N[(N[(t$95$1 + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision] - t$95$5), $MachinePrecision], If[LessEqual[t$95$7, 3.0], N[(N[(2.0 + N[(y * 0.5 + t$95$6), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$3 + 3.0), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1

   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 t around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate--r+ N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 21.1%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 22.1

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

   8. Applied rewrites 22.1%

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

   if 1 < (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2.00019999999999998

   1. Initial program 97.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-sqrt.f64 15.4

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

   5. Applied rewrites 15.4%

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

   6. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-sqrt.f64 25.2

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

   8. Applied rewrites 25.2%

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

   if 2.00019999999999998 < (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3

   1. Initial program 97.7%

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

   3. 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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-sqrt.f64 38.0

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

   5. Applied rewrites 38.0%

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

   6. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-sqrt.f64 38.2

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

   8. Applied rewrites 38.2%

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

   9. Taylor expanded in x around 0

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-sqrt.f64 N/A

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

       7. lower-+.f64 N/A

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

       8. lower-+.f64 N/A

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

       9. lower-sqrt.f64 N/A

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

       10. lower-+.f64 N/A

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

       11. lower-sqrt.f64 N/A

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

       12. lower-sqrt.f64 29.1

           \[\leadsto \left(2 + \mathsf{fma}\left(y, 0.5, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

   11. Applied rewrites 29.1%

       \[\leadsto \color{blue}{\left(2 + \mathsf{fma}\left(y, 0.5, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   if 3 < (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   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. lower--.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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-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. lower-+.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. lower-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. lower-+.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. lower-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. lower-sqrt.f64 92.1

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

   5. Applied rewrites 92.1%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 88.2

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

   8. Applied rewrites 88.2%

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

   9. Taylor expanded in y around 0

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

       1. lower--.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-+.f64 N/A

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

       7. lower-sqrt.f64 N/A

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

       8. associate-+r+ N/A

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

       9. lower-+.f64 N/A

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

       10. lower-+.f64 N/A

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

       11. lower-sqrt.f64 N/A

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

       12. lower-sqrt.f64 N/A

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

       13. lower-sqrt.f64 82.6

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

   11. Applied rewrites 82.6%

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

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 1:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 2.0002:\\ \;\;\;\;\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 3:\\ \;\;\;\;\left(2 + \mathsf{fma}\left(y, 0.5, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} + 3\right) - \left(\sqrt{t} + \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x} + \sqrt{y}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{1 + t}\\ t_4 := \sqrt{1 + y}\\ t_5 := \sqrt{1 + z}\\ t_6 := \left(t\_3 - \sqrt{t}\right) + \left(\left(t\_5 - \sqrt{z}\right) + \left(\left(t\_2 - \sqrt{x}\right) + \left(t\_4 - \sqrt{y}\right)\right)\right)\\ \mathbf{if}\;t\_6 \leq 2:\\ \;\;\;\;t\_2 + \left(t\_4 - t\_1\right)\\ \mathbf{elif}\;t\_6 \leq 3:\\ \;\;\;\;\left(2 + \mathsf{fma}\left(y, 0.5, t\_5\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_3 + 3\right) - \left(\sqrt{t} + \left(\sqrt{z} + t\_1\right)\right)\\ \end{array} \end{array} \]

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 x) (sqrt y)))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (sqrt (+ 1.0 t)))
        (t_4 (sqrt (+ 1.0 y)))
        (t_5 (sqrt (+ 1.0 z)))
        (t_6
         (+
          (- t_3 (sqrt t))
          (+ (- t_5 (sqrt z)) (+ (- t_2 (sqrt x)) (- t_4 (sqrt y)))))))
   (if (<= t_6 2.0)
     (+ t_2 (- t_4 t_1))
     (if (<= t_6 3.0)
       (- (+ 2.0 (fma y 0.5 t_5)) (+ (sqrt x) (+ (sqrt y) (sqrt z))))
       (- (+ t_3 3.0) (+ (sqrt t) (+ (sqrt z) t_1)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(x) + sqrt(y);
	double t_2 = sqrt((1.0 + x));
	double t_3 = sqrt((1.0 + t));
	double t_4 = sqrt((1.0 + y));
	double t_5 = sqrt((1.0 + z));
	double t_6 = (t_3 - sqrt(t)) + ((t_5 - sqrt(z)) + ((t_2 - sqrt(x)) + (t_4 - sqrt(y))));
	double tmp;
	if (t_6 <= 2.0) {
		tmp = t_2 + (t_4 - t_1);
	} else if (t_6 <= 3.0) {
		tmp = (2.0 + fma(y, 0.5, t_5)) - (sqrt(x) + (sqrt(y) + sqrt(z)));
	} else {
		tmp = (t_3 + 3.0) - (sqrt(t) + (sqrt(z) + t_1));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(x) + sqrt(y))
	t_2 = sqrt(Float64(1.0 + x))
	t_3 = sqrt(Float64(1.0 + t))
	t_4 = sqrt(Float64(1.0 + y))
	t_5 = sqrt(Float64(1.0 + z))
	t_6 = Float64(Float64(t_3 - sqrt(t)) + Float64(Float64(t_5 - sqrt(z)) + Float64(Float64(t_2 - sqrt(x)) + Float64(t_4 - sqrt(y)))))
	tmp = 0.0
	if (t_6 <= 2.0)
		tmp = Float64(t_2 + Float64(t_4 - t_1));
	elseif (t_6 <= 3.0)
		tmp = Float64(Float64(2.0 + fma(y, 0.5, t_5)) - Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z))));
	else
		tmp = Float64(Float64(t_3 + 3.0) - Float64(sqrt(t) + Float64(sqrt(z) + 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[x], $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 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$3 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$5 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 2.0], N[(t$95$2 + N[(t$95$4 - t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 3.0], N[(N[(2.0 + N[(y * 0.5 + t$95$5), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$3 + 3.0), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2

   1. Initial program 90.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-sqrt.f64 10.3

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

   5. Applied rewrites 10.3%

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

   6. Taylor expanded in 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. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 25.1

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

   8. Applied rewrites 25.1%

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

   if 2 < (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3

   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. 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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-sqrt.f64 37.5

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

   5. Applied rewrites 37.5%

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

   6. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-sqrt.f64 37.6

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

   8. Applied rewrites 37.6%

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

   9. Taylor expanded in x around 0

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-sqrt.f64 N/A

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

       7. lower-+.f64 N/A

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

       8. lower-+.f64 N/A

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

       9. lower-sqrt.f64 N/A

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

       10. lower-+.f64 N/A

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

       11. lower-sqrt.f64 N/A

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

       12. lower-sqrt.f64 28.2

           \[\leadsto \left(2 + \mathsf{fma}\left(y, 0.5, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

   11. Applied rewrites 28.2%

       \[\leadsto \color{blue}{\left(2 + \mathsf{fma}\left(y, 0.5, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   if 3 < (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   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. lower--.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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-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. lower-+.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. lower-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. lower-+.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. lower-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. lower-sqrt.f64 92.1

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

   5. Applied rewrites 92.1%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 88.2

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

   8. Applied rewrites 88.2%

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

   9. Taylor expanded in y around 0

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

       1. lower--.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-+.f64 N/A

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

       7. lower-sqrt.f64 N/A

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

       8. associate-+r+ N/A

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

       9. lower-+.f64 N/A

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

       10. lower-+.f64 N/A

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

       11. lower-sqrt.f64 N/A

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

       12. lower-sqrt.f64 N/A

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

       13. lower-sqrt.f64 82.6

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

   11. Applied rewrites 82.6%

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

4. Final simplification 29.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 2:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 3:\\ \;\;\;\;\left(2 + \mathsf{fma}\left(y, 0.5, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} + 3\right) - \left(\sqrt{t} + \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ t_4 := \sqrt{1 + x}\\ t_5 := \left(t\_2 - \sqrt{z}\right) + \left(\left(t\_4 - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right)\\ \mathbf{if}\;t\_5 \leq 2:\\ \;\;\;\;t\_3 + \left(t\_4 - \left(\sqrt{x} + \frac{-1}{\sqrt{y} + t\_1}\right)\right)\\ \mathbf{elif}\;t\_5 \leq 2.9999999:\\ \;\;\;\;1 + \left(\left(t\_4 + \mathsf{fma}\left(0.5, y, t\_2\right)\right) - \left(\sqrt{x} + \frac{z - y}{\sqrt{z} - \sqrt{y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \left(\left(2 + \left(t\_1 - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \end{array} \end{array} \]

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 (+ 1.0 y)))
        (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 2.0)
     (+ t_3 (- t_4 (+ (sqrt x) (/ -1.0 (+ (sqrt y) t_1)))))
     (if (<= t_5 2.9999999)
       (+
        1.0
        (-
         (+ t_4 (fma 0.5 y t_2))
         (+ (sqrt x) (/ (- z y) (- (sqrt z) (sqrt y))))))
       (+ t_3 (- (+ 2.0 (- t_1 (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((1.0 + y));
	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 <= 2.0) {
		tmp = t_3 + (t_4 - (sqrt(x) + (-1.0 / (sqrt(y) + t_1))));
	} else if (t_5 <= 2.9999999) {
		tmp = 1.0 + ((t_4 + fma(0.5, y, t_2)) - (sqrt(x) + ((z - y) / (sqrt(z) - sqrt(y)))));
	} else {
		tmp = t_3 + ((2.0 + (t_1 - 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(1.0 + y))
	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 <= 2.0)
		tmp = Float64(t_3 + Float64(t_4 - Float64(sqrt(x) + Float64(-1.0 / Float64(sqrt(y) + t_1)))));
	elseif (t_5 <= 2.9999999)
		tmp = Float64(1.0 + Float64(Float64(t_4 + fma(0.5, y, t_2)) - Float64(sqrt(x) + Float64(Float64(z - y) / Float64(sqrt(z) - sqrt(y))))));
	else
		tmp = Float64(t_3 + Float64(Float64(2.0 + Float64(t_1 - 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[(1.0 + y), $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, 2.0], N[(t$95$3 + N[(t$95$4 - N[(N[Sqrt[x], $MachinePrecision] + N[(-1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.9999999], N[(1.0 + N[(N[(t$95$4 + N[(0.5 * y + t$95$2), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[(z - y), $MachinePrecision] / N[(N[Sqrt[z], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(N[(2.0 + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $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))) < 2

   1. Initial program 92.0%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lower--.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-+.f64 92.3

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 92.3

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

   4. Applied rewrites 92.3%

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

   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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-/.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. lower-+.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. lower-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. lower-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. lower-+.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. lower-sqrt.f64 45.0

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

   7. Applied rewrites 45.0%

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

   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))) < 2.99999990000000016

   1. Initial program 91.8%

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

   3. Taylor expanded in 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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-sqrt.f64 63.5

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

   5. Applied rewrites 63.5%

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

   6. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-sqrt.f64 40.2

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

   8. Applied rewrites 40.2%

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, y, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. +-commutative N/A

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

      4. flip-+ N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 40.2

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

   10. Applied rewrites 40.2%

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

   if 2.99999990000000016 < (+.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 98.7%

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

   3. 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. lower--.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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-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. lower-+.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. lower-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. lower-+.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. lower-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. lower-sqrt.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 98.7

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

   8. Applied rewrites 98.7%

      \[\leadsto \left(\color{blue}{\left(2 + \left(\sqrt{1 + y} - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 2:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \frac{-1}{\sqrt{y} + \sqrt{1 + y}}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 2.9999999:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, y, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \frac{z - y}{\sqrt{z} - \sqrt{y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(2 + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + x}\\ t_3 := t\_2 - \sqrt{x}\\ t_4 := \left(\sqrt{1 + z} - \sqrt{z}\right) + \left(t\_3 + \left(t\_1 - \sqrt{y}\right)\right)\\ \mathbf{if}\;t\_4 \leq 1:\\ \;\;\;\;t\_3 + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \mathbf{elif}\;t\_4 \leq 2.5:\\ \;\;\;\;\left(t\_2 + \mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, t\_1\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(2 + \left(t\_1 - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \end{array} \end{array} \]

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 (+ 1.0 y)))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (- t_2 (sqrt x)))
        (t_4 (+ (- (sqrt (+ 1.0 z)) (sqrt z)) (+ t_3 (- t_1 (sqrt y))))))
   (if (<= t_4 1.0)
     (+ t_3 (* 0.5 (sqrt (/ 1.0 t))))
     (if (<= t_4 2.5)
       (- (+ t_2 (fma 0.5 (sqrt (/ 1.0 z)) t_1)) (+ (sqrt x) (sqrt y)))
       (+
        (- (sqrt (+ 1.0 t)) (sqrt t))
        (- (+ 2.0 (- t_1 (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((1.0 + y));
	double t_2 = sqrt((1.0 + x));
	double t_3 = t_2 - sqrt(x);
	double t_4 = (sqrt((1.0 + z)) - sqrt(z)) + (t_3 + (t_1 - sqrt(y)));
	double tmp;
	if (t_4 <= 1.0) {
		tmp = t_3 + (0.5 * sqrt((1.0 / t)));
	} else if (t_4 <= 2.5) {
		tmp = (t_2 + fma(0.5, sqrt((1.0 / z)), t_1)) - (sqrt(x) + sqrt(y));
	} else {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + ((2.0 + (t_1 - 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(1.0 + y))
	t_2 = sqrt(Float64(1.0 + x))
	t_3 = Float64(t_2 - sqrt(x))
	t_4 = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(t_3 + Float64(t_1 - sqrt(y))))
	tmp = 0.0
	if (t_4 <= 1.0)
		tmp = Float64(t_3 + Float64(0.5 * sqrt(Float64(1.0 / t))));
	elseif (t_4 <= 2.5)
		tmp = Float64(Float64(t_2 + fma(0.5, sqrt(Float64(1.0 / z)), t_1)) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(2.0 + Float64(t_1 - 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[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 1.0], N[(t$95$3 + N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.5], N[(N[(t$95$2 + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $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

   1. Initial program 87.3%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate--r+ N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 16.5%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 16.8

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

   8. Applied rewrites 16.8%

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

   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))) < 2.5

   1. Initial program 97.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-sqrt.f64 20.4

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

   5. Applied rewrites 20.4%

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

   6. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-sqrt.f64 27.6

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

   8. Applied rewrites 27.6%

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

   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 98.7%

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

   3. 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. lower--.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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-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. lower-+.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. lower-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. lower-+.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. lower-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. lower-sqrt.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 98.0

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

   8. Applied rewrites 98.0%

      \[\leadsto \left(\color{blue}{\left(2 + \left(\sqrt{1 + y} - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 1:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 2.5:\\ \;\;\;\;\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(2 + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + z} - \sqrt{z}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ t_4 := \sqrt{1 + x}\\ \mathbf{if}\;t\_2 + \left(\left(t\_4 - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right) \leq 0.9995:\\ \;\;\;\;t\_3 + \left(t\_2 + \frac{\sqrt{y} + \left(\sqrt{x} + 2\right)}{\sqrt{y} \cdot \left(\sqrt{x} + t\_4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \left(t\_2 + \left(\left(\mathsf{fma}\left(x, 0.5, 1\right) + \frac{1}{\sqrt{y} + t\_1}\right) - \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

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 y)))
        (t_2 (- (sqrt (+ 1.0 z)) (sqrt z)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_4 (sqrt (+ 1.0 x))))
   (if (<= (+ t_2 (+ (- t_4 (sqrt x)) (- t_1 (sqrt y)))) 0.9995)
     (+
      t_3
      (+ t_2 (/ (+ (sqrt y) (+ (sqrt x) 2.0)) (* (sqrt y) (+ (sqrt x) t_4)))))
     (+
      t_3
      (+ t_2 (- (+ (fma x 0.5 1.0) (/ 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((1.0 + y));
	double t_2 = sqrt((1.0 + z)) - sqrt(z);
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double t_4 = sqrt((1.0 + x));
	double tmp;
	if ((t_2 + ((t_4 - sqrt(x)) + (t_1 - sqrt(y)))) <= 0.9995) {
		tmp = t_3 + (t_2 + ((sqrt(y) + (sqrt(x) + 2.0)) / (sqrt(y) * (sqrt(x) + t_4))));
	} else {
		tmp = t_3 + (t_2 + ((fma(x, 0.5, 1.0) + (1.0 / (sqrt(y) + t_1))) - sqrt(x)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_4 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(t_2 + Float64(Float64(t_4 - sqrt(x)) + Float64(t_1 - sqrt(y)))) <= 0.9995)
		tmp = Float64(t_3 + Float64(t_2 + Float64(Float64(sqrt(y) + Float64(sqrt(x) + 2.0)) / Float64(sqrt(y) * Float64(sqrt(x) + t_4)))));
	else
		tmp = Float64(t_3 + Float64(t_2 + Float64(Float64(fma(x, 0.5, 1.0) + Float64(1.0 / Float64(sqrt(y) + t_1))) - 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[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[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]}, If[LessEqual[N[(t$95$2 + N[(N[(t$95$4 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.9995], N[(t$95$3 + N[(t$95$2 + N[(N[(N[Sqrt[y], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[y], $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(t$95$2 + N[(N[(N[(x * 0.5 + 1.0), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $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))) < 0.99950000000000006

   1. Initial program 64.5%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. flip-- N/A

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

      5. lift-+.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. flip-- N/A

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

      9. frac-add N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 67.6%

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

   5. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 77.0

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

   7. Applied rewrites 77.0%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-+.f64 82.0

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

   10. Applied rewrites 82.0%

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

   11. Taylor expanded in y around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-+.f64 83.4

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

   13. Applied rewrites 83.4%

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

   if 0.99950000000000006 < (+.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 97.3%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lower--.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-+.f64 97.5

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 97.5

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 60.4

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

   7. Applied rewrites 60.4%

      \[\leadsto \left(\color{blue}{\left(\left(\mathsf{fma}\left(x, 0.5, 1\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 0.9995:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\sqrt{y} + \left(\sqrt{x} + 2\right)}{\sqrt{y} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\mathsf{fma}\left(x, 0.5, 1\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + y}\\ t_3 := \sqrt{1 + z}\\ \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(t\_3 - \sqrt{z}\right) + \left(\left(t\_1 - \sqrt{x}\right) + \left(t\_2 - \sqrt{y}\right)\right)\right) \leq 2:\\ \;\;\;\;t\_1 + \left(t\_2 - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 + \mathsf{fma}\left(y, 0.5, t\_3\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \end{array} \end{array} \]

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 (+ 1.0 x))) (t_2 (sqrt (+ 1.0 y))) (t_3 (sqrt (+ 1.0 z))))
   (if (<=
        (+
         (- (sqrt (+ 1.0 t)) (sqrt t))
         (+ (- t_3 (sqrt z)) (+ (- t_1 (sqrt x)) (- t_2 (sqrt y)))))
        2.0)
     (+ t_1 (- t_2 (+ (sqrt x) (sqrt y))))
     (- (+ 2.0 (fma y 0.5 t_3)) (+ (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((1.0 + x));
	double t_2 = sqrt((1.0 + y));
	double t_3 = sqrt((1.0 + z));
	double tmp;
	if (((sqrt((1.0 + t)) - sqrt(t)) + ((t_3 - sqrt(z)) + ((t_1 - sqrt(x)) + (t_2 - sqrt(y))))) <= 2.0) {
		tmp = t_1 + (t_2 - (sqrt(x) + sqrt(y)));
	} else {
		tmp = (2.0 + fma(y, 0.5, t_3)) - (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(1.0 + x))
	t_2 = sqrt(Float64(1.0 + y))
	t_3 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(t_3 - sqrt(z)) + Float64(Float64(t_1 - sqrt(x)) + Float64(t_2 - sqrt(y))))) <= 2.0)
		tmp = Float64(t_1 + Float64(t_2 - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(Float64(2.0 + fma(y, 0.5, t_3)) - Float64(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[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[(t$95$1 + N[(t$95$2 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(y * 0.5 + t$95$3), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2

   1. Initial program 90.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-sqrt.f64 10.3

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

   5. Applied rewrites 10.3%

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

   6. Taylor expanded in 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. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 25.1

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

   8. Applied rewrites 25.1%

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

   if 2 < (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 97.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-sqrt.f64 33.7

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

   5. Applied rewrites 33.7%

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

   6. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-sqrt.f64 33.6

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

   8. Applied rewrites 33.6%

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

   9. Taylor expanded in x around 0

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-sqrt.f64 N/A

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

       7. lower-+.f64 N/A

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

       8. lower-+.f64 N/A

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

       9. lower-sqrt.f64 N/A

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

       10. lower-+.f64 N/A

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

       11. lower-sqrt.f64 N/A

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

       12. lower-sqrt.f64 26.1

           \[\leadsto \left(2 + \mathsf{fma}\left(y, 0.5, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

   11. Applied rewrites 26.1%

       \[\leadsto \color{blue}{\left(2 + \mathsf{fma}\left(y, 0.5, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.4%

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

Alternative 9: 97.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y} + \sqrt{1 + y}\\ \left(\frac{\left(1 + \sqrt{x}\right) + t\_1}{t\_1 \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \end{array} \end{array} \]

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 (+ 1.0 y)))))
   (+
    (+
     (/ (+ (+ 1.0 (sqrt x)) t_1) (* t_1 (+ (sqrt x) (sqrt (+ 1.0 x)))))
     (- (sqrt (+ 1.0 z)) (sqrt z)))
    (- (sqrt (+ 1.0 t)) (sqrt t)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(y) + sqrt((1.0 + y));
	return ((((1.0 + sqrt(x)) + t_1) / (t_1 * (sqrt(x) + sqrt((1.0 + x))))) + (sqrt((1.0 + z)) - sqrt(z))) + (sqrt((1.0 + t)) - sqrt(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
    real(8) :: t_1
    t_1 = sqrt(y) + sqrt((1.0d0 + y))
    code = ((((1.0d0 + sqrt(x)) + t_1) / (t_1 * (sqrt(x) + sqrt((1.0d0 + x))))) + (sqrt((1.0d0 + z)) - sqrt(z))) + (sqrt((1.0d0 + t)) - sqrt(t))
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) + Math.sqrt((1.0 + y));
	return ((((1.0 + Math.sqrt(x)) + t_1) / (t_1 * (Math.sqrt(x) + Math.sqrt((1.0 + x))))) + (Math.sqrt((1.0 + z)) - Math.sqrt(z))) + (Math.sqrt((1.0 + t)) - Math.sqrt(t));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt(y) + math.sqrt((1.0 + y))
	return ((((1.0 + math.sqrt(x)) + t_1) / (t_1 * (math.sqrt(x) + math.sqrt((1.0 + x))))) + (math.sqrt((1.0 + z)) - math.sqrt(z))) + (math.sqrt((1.0 + t)) - math.sqrt(t))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(y) + sqrt(Float64(1.0 + y)))
	return Float64(Float64(Float64(Float64(Float64(1.0 + sqrt(x)) + t_1) / Float64(t_1 * Float64(sqrt(x) + sqrt(Float64(1.0 + x))))) + Float64(sqrt(Float64(1.0 + z)) - sqrt(z))) + Float64(sqrt(Float64(1.0 + t)) - sqrt(t)))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	t_1 = sqrt(y) + sqrt((1.0 + y));
	tmp = ((((1.0 + sqrt(x)) + t_1) / (t_1 * (sqrt(x) + sqrt((1.0 + x))))) + (sqrt((1.0 + z)) - sqrt(z))) + (sqrt((1.0 + t)) - sqrt(t));
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[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(1.0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] / N[(t$95$1 * N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 92.7%

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

   1. lift-+.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. flip-- N/A

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

   5. lift-+.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. flip-- N/A

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

   9. frac-add N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 93.4%

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

5. Taylor expanded in x around 0

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

   1. associate-+r+ N/A

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

   2. lower-+.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-+.f64 73.9

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

7. Applied rewrites 73.9%

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

8. Final simplification 73.9%

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

Alternative 10: 97.3% accurate, 0.7× speedup

Math

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

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 z)) (sqrt z)))
        (t_2 (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y)))))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= (- (sqrt (+ 1.0 x)) (sqrt x)) 0.5)
     (+ t_3 (+ t_1 (fma 0.5 (sqrt (/ 1.0 x)) t_2)))
     (+ t_3 (+ t_1 (- (+ (fma x 0.5 1.0) t_2) (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 = 1.0 / (sqrt(y) + sqrt((1.0 + y)));
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if ((sqrt((1.0 + x)) - sqrt(x)) <= 0.5) {
		tmp = t_3 + (t_1 + fma(0.5, sqrt((1.0 / x)), t_2));
	} else {
		tmp = t_3 + (t_1 + ((fma(x, 0.5, 1.0) + t_2) - 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 = Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y))))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (Float64(sqrt(Float64(1.0 + x)) - sqrt(x)) <= 0.5)
		tmp = Float64(t_3 + Float64(t_1 + fma(0.5, sqrt(Float64(1.0 / x)), t_2)));
	else
		tmp = Float64(t_3 + Float64(t_1 + Float64(Float64(fma(x, 0.5, 1.0) + t_2) - 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 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.5], N[(t$95$3 + N[(t$95$1 + N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(t$95$1 + N[(N[(N[(x * 0.5 + 1.0), $MachinePrecision] + t$95$2), $MachinePrecision] - N[Sqrt[x], $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)) < 0.5

   1. Initial program 87.3%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lower--.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-+.f64 87.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 87.9

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

   4. Applied rewrites 87.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 94.8

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

   7. Applied rewrites 94.8%

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

   if 0.5 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lower--.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-+.f64 97.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 98.0

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

   7. Applied rewrites 98.0%

      \[\leadsto \left(\color{blue}{\left(\left(\mathsf{fma}\left(x, 0.5, 1\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.4%

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

Alternative 11: 97.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z} - \sqrt{z}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;t\_2 - \sqrt{x} \leq 0.9995:\\ \;\;\;\;t\_3 + \left(t\_1 + \frac{1}{\sqrt{x} + t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \left(t\_1 + \left(\left(\mathsf{fma}\left(x, 0.5, 1\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

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 z)) (sqrt z)))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= (- t_2 (sqrt x)) 0.9995)
     (+ t_3 (+ t_1 (/ 1.0 (+ (sqrt x) t_2))))
     (+
      t_3
      (+
       t_1
       (-
        (+ (fma x 0.5 1.0) (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 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 + 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)) <= 0.9995) {
		tmp = t_3 + (t_1 + (1.0 / (sqrt(x) + t_2)));
	} else {
		tmp = t_3 + (t_1 + ((fma(x, 0.5, 1.0) + (1.0 / (sqrt(y) + sqrt((1.0 + 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 + 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)) <= 0.9995)
		tmp = Float64(t_3 + Float64(t_1 + Float64(1.0 / Float64(sqrt(x) + t_2))));
	else
		tmp = Float64(t_3 + Float64(t_1 + Float64(Float64(fma(x, 0.5, 1.0) + Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + 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 + 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], 0.9995], N[(t$95$3 + N[(t$95$1 + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(t$95$1 + N[(N[(N[(x * 0.5 + 1.0), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $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)) < 0.99950000000000006

   1. Initial program 87.3%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. flip-- N/A

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

      5. lift-+.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. flip-- N/A

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

      9. frac-add N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 88.7%

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

   5. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 49.3

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

   7. Applied rewrites 49.3%

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

   8. Taylor expanded in y around inf

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

      1. lower-/.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. lower-+.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. lower-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. lower-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. lower-+.f64 46.2

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

   10. Applied rewrites 46.2%

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

   if 0.99950000000000006 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lower--.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-+.f64 97.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 98.0

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

   7. Applied rewrites 98.0%

      \[\leadsto \left(\color{blue}{\left(\left(\mathsf{fma}\left(x, 0.5, 1\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0.9995:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\mathsf{fma}\left(x, 0.5, 1\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{1 + x}\\ \mathbf{if}\;\left(t\_2 - \sqrt{z}\right) + \left(\left(t\_3 - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right) \leq 2:\\ \;\;\;\;t\_3 + \left(t\_1 - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 + t\_2\right) + \left(t\_3 - \sqrt{z}\right)\\ \end{array} \end{array} \]

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 y))) (t_2 (sqrt (+ 1.0 z))) (t_3 (sqrt (+ 1.0 x))))
   (if (<= (+ (- t_2 (sqrt z)) (+ (- t_3 (sqrt x)) (- t_1 (sqrt y)))) 2.0)
     (+ t_3 (- t_1 (+ (sqrt x) (sqrt y))))
     (+ (+ t_1 t_2) (- t_3 (sqrt z))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((1.0 + z));
	double t_3 = sqrt((1.0 + x));
	double tmp;
	if (((t_2 - sqrt(z)) + ((t_3 - sqrt(x)) + (t_1 - sqrt(y)))) <= 2.0) {
		tmp = t_3 + (t_1 - (sqrt(x) + sqrt(y)));
	} else {
		tmp = (t_1 + t_2) + (t_3 - 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) :: tmp
    t_1 = sqrt((1.0d0 + y))
    t_2 = sqrt((1.0d0 + z))
    t_3 = sqrt((1.0d0 + x))
    if (((t_2 - sqrt(z)) + ((t_3 - sqrt(x)) + (t_1 - sqrt(y)))) <= 2.0d0) then
        tmp = t_3 + (t_1 - (sqrt(x) + sqrt(y)))
    else
        tmp = (t_1 + t_2) + (t_3 - 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((1.0 + y));
	double t_2 = Math.sqrt((1.0 + z));
	double t_3 = Math.sqrt((1.0 + x));
	double tmp;
	if (((t_2 - Math.sqrt(z)) + ((t_3 - Math.sqrt(x)) + (t_1 - Math.sqrt(y)))) <= 2.0) {
		tmp = t_3 + (t_1 - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = (t_1 + t_2) + (t_3 - 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((1.0 + y))
	t_2 = math.sqrt((1.0 + z))
	t_3 = math.sqrt((1.0 + x))
	tmp = 0
	if ((t_2 - math.sqrt(z)) + ((t_3 - math.sqrt(x)) + (t_1 - math.sqrt(y)))) <= 2.0:
		tmp = t_3 + (t_1 - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = (t_1 + t_2) + (t_3 - 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(1.0 + y))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(Float64(t_2 - sqrt(z)) + Float64(Float64(t_3 - sqrt(x)) + Float64(t_1 - sqrt(y)))) <= 2.0)
		tmp = Float64(t_3 + Float64(t_1 - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(Float64(t_1 + t_2) + Float64(t_3 - 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((1.0 + y));
	t_2 = sqrt((1.0 + z));
	t_3 = sqrt((1.0 + x));
	tmp = 0.0;
	if (((t_2 - sqrt(z)) + ((t_3 - sqrt(x)) + (t_1 - sqrt(y)))) <= 2.0)
		tmp = t_3 + (t_1 - (sqrt(x) + sqrt(y)));
	else
		tmp = (t_1 + t_2) + (t_3 - 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[(1.0 + y), $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]}, If[LessEqual[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], 2.0], N[(t$95$3 + N[(t$95$1 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 + t$95$2), $MachinePrecision] + N[(t$95$3 - N[Sqrt[z], $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))) < 2

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-sqrt.f64 10.5

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

   5. Applied rewrites 10.5%

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

   6. Taylor expanded in 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. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 23.3

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

   8. Applied rewrites 23.3%

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

   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 97.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-sqrt.f64 59.7

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

   5. Applied rewrites 59.7%

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

   6. Taylor expanded in z around inf

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

      1. lower-sqrt.f64 50.0

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

   8. Applied rewrites 50.0%

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\sqrt{z}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.9%

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

Alternative 13: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;x \leq 3.8 \cdot 10^{-7}:\\ \;\;\;\;t\_1 + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\mathsf{fma}\left(x, 0.5, 1\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(\frac{\sqrt{y} + \left(\sqrt{x} + 2\right)}{\left(1 + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + 0.5 \cdot \sqrt{\frac{1}{z}}\right)\\ \end{array} \end{array} \]

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))))
   (if (<= x 3.8e-7)
     (+
      t_1
      (+
       (- (sqrt (+ 1.0 z)) (sqrt z))
       (- (+ (fma x 0.5 1.0) (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y))))) (sqrt x))))
     (+
      t_1
      (+
       (/
        (+ (sqrt y) (+ (sqrt x) 2.0))
        (* (+ 1.0 (sqrt y)) (+ (sqrt x) (sqrt (+ 1.0 x)))))
       (* 0.5 (sqrt (/ 1.0 z))))))))

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 tmp;
	if (x <= 3.8e-7) {
		tmp = t_1 + ((sqrt((1.0 + z)) - sqrt(z)) + ((fma(x, 0.5, 1.0) + (1.0 / (sqrt(y) + sqrt((1.0 + y))))) - sqrt(x)));
	} else {
		tmp = t_1 + (((sqrt(y) + (sqrt(x) + 2.0)) / ((1.0 + sqrt(y)) * (sqrt(x) + sqrt((1.0 + x))))) + (0.5 * sqrt((1.0 / z))));
	}
	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))
	tmp = 0.0
	if (x <= 3.8e-7)
		tmp = Float64(t_1 + Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(Float64(fma(x, 0.5, 1.0) + Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y))))) - sqrt(x))));
	else
		tmp = Float64(t_1 + Float64(Float64(Float64(sqrt(y) + Float64(sqrt(x) + 2.0)) / Float64(Float64(1.0 + sqrt(y)) * Float64(sqrt(x) + sqrt(Float64(1.0 + x))))) + Float64(0.5 * sqrt(Float64(1.0 / 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[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3.8e-7], N[(t$95$1 + N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x * 0.5 + 1.0), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(N[(N[Sqrt[y], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.80000000000000015e-7

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lower--.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-+.f64 97.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 98.0

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

   7. Applied rewrites 98.0%

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

   if 3.80000000000000015e-7 < x

   1. Initial program 87.3%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. flip-- N/A

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

      5. lift-+.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. flip-- N/A

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

      9. frac-add N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 88.7%

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

   5. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 49.3

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

   7. Applied rewrites 49.3%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

         \[\leadsto \left(\frac{\left(2 + \sqrt{x}\right) + \sqrt{y}}{\color{blue}{\left(1 + \sqrt{y}\right)} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(2 + \sqrt{x}\right) + \sqrt{y}}{\left(1 + \color{blue}{\sqrt{y}}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{\left(2 + \sqrt{x}\right) + \sqrt{y}}{\left(1 + \sqrt{y}\right) \cdot \color{blue}{\left(\sqrt{x} + \sqrt{1 + x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\left(2 + \sqrt{x}\right) + \sqrt{y}}{\left(1 + \sqrt{y}\right) \cdot \left(\color{blue}{\sqrt{x}} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\left(2 + \sqrt{x}\right) + \sqrt{y}}{\left(1 + \sqrt{y}\right) \cdot \left(\sqrt{x} + \color{blue}{\sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-+.f64 51.1

          \[\leadsto \left(\frac{\left(2 + \sqrt{x}\right) + \sqrt{y}}{\left(1 + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{\color{blue}{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Applied rewrites 51.1%

       \[\leadsto \left(\color{blue}{\frac{\left(2 + \sqrt{x}\right) + \sqrt{y}}{\left(1 + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Taylor expanded in z around inf

       \[\leadsto \left(\frac{\left(2 + \sqrt{x}\right) + \sqrt{y}}{\left(1 + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(\frac{\left(2 + \sqrt{x}\right) + \sqrt{y}}{\left(1 + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\left(2 + \sqrt{x}\right) + \sqrt{y}}{\left(1 + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. lower-/.f64 34.4

          \[\leadsto \left(\frac{\left(2 + \sqrt{x}\right) + \sqrt{y}}{\left(1 + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + 0.5 \cdot \sqrt{\color{blue}{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   13. Applied rewrites 34.4%

       \[\leadsto \left(\frac{\left(2 + \sqrt{x}\right) + \sqrt{y}}{\left(1 + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \color{blue}{0.5 \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{-7}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\mathsf{fma}\left(x, 0.5, 1\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{\sqrt{y} + \left(\sqrt{x} + 2\right)}{\left(1 + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + 0.5 \cdot \sqrt{\frac{1}{z}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 96.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z} - \sqrt{z}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;y \leq 2.3 \cdot 10^{+17}:\\ \;\;\;\;t\_3 + \left(t\_1 + \left(\left(t\_2 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \left(t\_1 + \frac{1}{\sqrt{x} + t\_2}\right)\\ \end{array} \end{array} \]

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 (+ 1.0 z)) (sqrt z)))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= y 2.3e+17)
     (+ t_3 (+ t_1 (+ (- t_2 (sqrt x)) (- (sqrt (+ 1.0 y)) (sqrt y)))))
     (+ t_3 (+ t_1 (/ 1.0 (+ (sqrt x) t_2)))))))

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 (y <= 2.3e+17) {
		tmp = t_3 + (t_1 + ((t_2 - sqrt(x)) + (sqrt((1.0 + y)) - sqrt(y))));
	} else {
		tmp = t_3 + (t_1 + (1.0 / (sqrt(x) + 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) :: 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 (y <= 2.3d+17) then
        tmp = t_3 + (t_1 + ((t_2 - sqrt(x)) + (sqrt((1.0d0 + y)) - sqrt(y))))
    else
        tmp = t_3 + (t_1 + (1.0d0 / (sqrt(x) + 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((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 (y <= 2.3e+17) {
		tmp = t_3 + (t_1 + ((t_2 - Math.sqrt(x)) + (Math.sqrt((1.0 + y)) - Math.sqrt(y))));
	} else {
		tmp = t_3 + (t_1 + (1.0 / (Math.sqrt(x) + t_2)));
	}
	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 y <= 2.3e+17:
		tmp = t_3 + (t_1 + ((t_2 - math.sqrt(x)) + (math.sqrt((1.0 + y)) - math.sqrt(y))))
	else:
		tmp = t_3 + (t_1 + (1.0 / (math.sqrt(x) + t_2)))
	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 (y <= 2.3e+17)
		tmp = Float64(t_3 + Float64(t_1 + Float64(Float64(t_2 - sqrt(x)) + Float64(sqrt(Float64(1.0 + y)) - sqrt(y)))));
	else
		tmp = Float64(t_3 + Float64(t_1 + Float64(1.0 / Float64(sqrt(x) + 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((1.0 + z)) - sqrt(z);
	t_2 = sqrt((1.0 + x));
	t_3 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if (y <= 2.3e+17)
		tmp = t_3 + (t_1 + ((t_2 - sqrt(x)) + (sqrt((1.0 + y)) - sqrt(y))));
	else
		tmp = t_3 + (t_1 + (1.0 / (sqrt(x) + 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[(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[y, 2.3e+17], N[(t$95$3 + N[(t$95$1 + N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(t$95$1 + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.3e17

   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

   if 2.3e17 < y

   1. Initial program 87.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{x + 1}} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{x + 1}} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{\color{blue}{y + 1}} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\color{blue}{\sqrt{y + 1}} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. flip-- N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. frac-add N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right) + \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right) + \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 87.9%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(\left(x + 1\right) - x, \sqrt{1 + y} + \sqrt{y}, \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\left(1 + y\right) - y\right)\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\frac{\color{blue}{1 + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{1 + y}\right)\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right)} + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \color{blue}{\left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\color{blue}{\sqrt{y}} + \sqrt{1 + y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \color{blue}{\sqrt{1 + y}}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 93.7

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{\color{blue}{1 + y}}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 93.7%

      \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

      1. lower-/.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. lower-+.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. lower-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. lower-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. lower-+.f64 93.4

         \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{\color{blue}{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Applied rewrites 93.4%

       \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{+17}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 97.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z} - \sqrt{z}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ t_4 := \sqrt{1 + y}\\ \mathbf{if}\;y \leq 2.85 \cdot 10^{-8}:\\ \;\;\;\;t\_3 + \left(t\_1 + \left(\left(1 - \sqrt{x}\right) - \left(\sqrt{y} - t\_4\right)\right)\right)\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+29}:\\ \;\;\;\;t\_3 + \left(t\_2 - \left(\sqrt{x} + \frac{-1}{\sqrt{y} + t\_4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \left(t\_1 + \frac{1}{\sqrt{x} + t\_2}\right)\\ \end{array} \end{array} \]

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 (+ 1.0 z)) (sqrt z)))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_4 (sqrt (+ 1.0 y))))
   (if (<= y 2.85e-8)
     (+ t_3 (+ t_1 (- (- 1.0 (sqrt x)) (- (sqrt y) t_4))))
     (if (<= y 5.3e+29)
       (+ t_3 (- t_2 (+ (sqrt x) (/ -1.0 (+ (sqrt y) t_4)))))
       (+ t_3 (+ t_1 (/ 1.0 (+ (sqrt x) t_2))))))))

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 t_4 = sqrt((1.0 + y));
	double tmp;
	if (y <= 2.85e-8) {
		tmp = t_3 + (t_1 + ((1.0 - sqrt(x)) - (sqrt(y) - t_4)));
	} else if (y <= 5.3e+29) {
		tmp = t_3 + (t_2 - (sqrt(x) + (-1.0 / (sqrt(y) + t_4))));
	} else {
		tmp = t_3 + (t_1 + (1.0 / (sqrt(x) + 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) :: t_3
    real(8) :: t_4
    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)
    t_4 = sqrt((1.0d0 + y))
    if (y <= 2.85d-8) then
        tmp = t_3 + (t_1 + ((1.0d0 - sqrt(x)) - (sqrt(y) - t_4)))
    else if (y <= 5.3d+29) then
        tmp = t_3 + (t_2 - (sqrt(x) + ((-1.0d0) / (sqrt(y) + t_4))))
    else
        tmp = t_3 + (t_1 + (1.0d0 / (sqrt(x) + 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((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 t_4 = Math.sqrt((1.0 + y));
	double tmp;
	if (y <= 2.85e-8) {
		tmp = t_3 + (t_1 + ((1.0 - Math.sqrt(x)) - (Math.sqrt(y) - t_4)));
	} else if (y <= 5.3e+29) {
		tmp = t_3 + (t_2 - (Math.sqrt(x) + (-1.0 / (Math.sqrt(y) + t_4))));
	} else {
		tmp = t_3 + (t_1 + (1.0 / (Math.sqrt(x) + t_2)));
	}
	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)
	t_4 = math.sqrt((1.0 + y))
	tmp = 0
	if y <= 2.85e-8:
		tmp = t_3 + (t_1 + ((1.0 - math.sqrt(x)) - (math.sqrt(y) - t_4)))
	elif y <= 5.3e+29:
		tmp = t_3 + (t_2 - (math.sqrt(x) + (-1.0 / (math.sqrt(y) + t_4))))
	else:
		tmp = t_3 + (t_1 + (1.0 / (math.sqrt(x) + t_2)))
	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))
	t_4 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (y <= 2.85e-8)
		tmp = Float64(t_3 + Float64(t_1 + Float64(Float64(1.0 - sqrt(x)) - Float64(sqrt(y) - t_4))));
	elseif (y <= 5.3e+29)
		tmp = Float64(t_3 + Float64(t_2 - Float64(sqrt(x) + Float64(-1.0 / Float64(sqrt(y) + t_4)))));
	else
		tmp = Float64(t_3 + Float64(t_1 + Float64(1.0 / Float64(sqrt(x) + 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((1.0 + z)) - sqrt(z);
	t_2 = sqrt((1.0 + x));
	t_3 = sqrt((1.0 + t)) - sqrt(t);
	t_4 = sqrt((1.0 + y));
	tmp = 0.0;
	if (y <= 2.85e-8)
		tmp = t_3 + (t_1 + ((1.0 - sqrt(x)) - (sqrt(y) - t_4)));
	elseif (y <= 5.3e+29)
		tmp = t_3 + (t_2 - (sqrt(x) + (-1.0 / (sqrt(y) + t_4))));
	else
		tmp = t_3 + (t_1 + (1.0 / (sqrt(x) + 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[(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]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 2.85e-8], N[(t$95$3 + N[(t$95$1 + N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] - t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.3e+29], N[(t$95$3 + N[(t$95$2 - N[(N[Sqrt[x], $MachinePrecision] + N[(-1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(t$95$1 + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.85000000000000004e-8

   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 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. lower--.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. lower--.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. lower-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. lower-+.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. lower-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. lower--.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. lower-sqrt.f64 48.9

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 48.9%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 2.85000000000000004e-8 < y < 5.3e29

   1. Initial program 88.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{y + 1}} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\color{blue}{\sqrt{y + 1}} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{\sqrt{y}} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \sqrt{y} \cdot \color{blue}{\sqrt{y}}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-+.f64 96.4

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{y + 1}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. lower-+.f64 96.4

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 96.4%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-/.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. lower-+.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. lower-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. lower-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. lower-+.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. lower-sqrt.f64 34.8

          \[\leadsto \left(\sqrt{1 + x} + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} - \color{blue}{\sqrt{x}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 34.8%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} - \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 5.3e29 < y

   1. Initial program 87.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{x + 1}} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{x + 1}} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{\color{blue}{y + 1}} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\color{blue}{\sqrt{y + 1}} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. flip-- N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. frac-add N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right) + \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right) + \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 88.0%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(\left(x + 1\right) - x, \sqrt{1 + y} + \sqrt{y}, \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\left(1 + y\right) - y\right)\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\frac{\color{blue}{1 + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{1 + y}\right)\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right)} + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \color{blue}{\left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\color{blue}{\sqrt{y}} + \sqrt{1 + y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \color{blue}{\sqrt{1 + y}}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 93.8

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{\color{blue}{1 + y}}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 93.8%

      \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

      1. lower-/.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. lower-+.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. lower-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. lower-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. lower-+.f64 93.6

         \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{\color{blue}{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Applied rewrites 93.6%

       \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.85 \cdot 10^{-8}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(1 - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{1 + y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+29}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \frac{-1}{\sqrt{y} + \sqrt{1 + y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 96.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;y \leq 9 \cdot 10^{-37}:\\ \;\;\;\;t\_3 + \left(\left(1 + t\_1\right) + \left(t\_2 - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+29}:\\ \;\;\;\;t\_3 + \left(t\_1 - \left(\sqrt{x} + \frac{-1}{\sqrt{y} + \sqrt{1 + y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \left(\left(t\_2 - \sqrt{z}\right) + \frac{1}{\sqrt{x} + t\_1}\right)\\ \end{array} \end{array} \]

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 (+ 1.0 x)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= y 9e-37)
     (+ t_3 (+ (+ 1.0 t_1) (- t_2 (+ (sqrt x) (+ (sqrt y) (sqrt z))))))
     (if (<= y 5.3e+29)
       (+ t_3 (- t_1 (+ (sqrt x) (/ -1.0 (+ (sqrt y) (sqrt (+ 1.0 y)))))))
       (+ t_3 (+ (- t_2 (sqrt z)) (/ 1.0 (+ (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((1.0 + x));
	double t_2 = sqrt((1.0 + z));
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if (y <= 9e-37) {
		tmp = t_3 + ((1.0 + t_1) + (t_2 - (sqrt(x) + (sqrt(y) + sqrt(z)))));
	} else if (y <= 5.3e+29) {
		tmp = t_3 + (t_1 - (sqrt(x) + (-1.0 / (sqrt(y) + sqrt((1.0 + y))))));
	} else {
		tmp = t_3 + ((t_2 - sqrt(z)) + (1.0 / (sqrt(x) + t_1)));
	}
	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 + x))
    t_2 = sqrt((1.0d0 + z))
    t_3 = sqrt((1.0d0 + t)) - sqrt(t)
    if (y <= 9d-37) then
        tmp = t_3 + ((1.0d0 + t_1) + (t_2 - (sqrt(x) + (sqrt(y) + sqrt(z)))))
    else if (y <= 5.3d+29) then
        tmp = t_3 + (t_1 - (sqrt(x) + ((-1.0d0) / (sqrt(y) + sqrt((1.0d0 + y))))))
    else
        tmp = t_3 + ((t_2 - sqrt(z)) + (1.0d0 / (sqrt(x) + t_1)))
    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));
	double t_2 = Math.sqrt((1.0 + z));
	double t_3 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if (y <= 9e-37) {
		tmp = t_3 + ((1.0 + t_1) + (t_2 - (Math.sqrt(x) + (Math.sqrt(y) + Math.sqrt(z)))));
	} else if (y <= 5.3e+29) {
		tmp = t_3 + (t_1 - (Math.sqrt(x) + (-1.0 / (Math.sqrt(y) + Math.sqrt((1.0 + y))))));
	} else {
		tmp = t_3 + ((t_2 - Math.sqrt(z)) + (1.0 / (Math.sqrt(x) + t_1)));
	}
	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))
	t_2 = math.sqrt((1.0 + z))
	t_3 = math.sqrt((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if y <= 9e-37:
		tmp = t_3 + ((1.0 + t_1) + (t_2 - (math.sqrt(x) + (math.sqrt(y) + math.sqrt(z)))))
	elif y <= 5.3e+29:
		tmp = t_3 + (t_1 - (math.sqrt(x) + (-1.0 / (math.sqrt(y) + math.sqrt((1.0 + y))))))
	else:
		tmp = t_3 + ((t_2 - math.sqrt(z)) + (1.0 / (math.sqrt(x) + t_1)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (y <= 9e-37)
		tmp = Float64(t_3 + Float64(Float64(1.0 + t_1) + Float64(t_2 - Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z))))));
	elseif (y <= 5.3e+29)
		tmp = Float64(t_3 + Float64(t_1 - Float64(sqrt(x) + Float64(-1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y)))))));
	else
		tmp = Float64(t_3 + Float64(Float64(t_2 - sqrt(z)) + Float64(1.0 / Float64(sqrt(x) + t_1))));
	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));
	t_2 = sqrt((1.0 + z));
	t_3 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if (y <= 9e-37)
		tmp = t_3 + ((1.0 + t_1) + (t_2 - (sqrt(x) + (sqrt(y) + sqrt(z)))));
	elseif (y <= 5.3e+29)
		tmp = t_3 + (t_1 - (sqrt(x) + (-1.0 / (sqrt(y) + sqrt((1.0 + y))))));
	else
		tmp = t_3 + ((t_2 - sqrt(z)) + (1.0 / (sqrt(x) + t_1)));
	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[(1.0 + x), $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]}, If[LessEqual[y, 9e-37], N[(t$95$3 + N[(N[(1.0 + t$95$1), $MachinePrecision] + N[(t$95$2 - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.3e+29], N[(t$95$3 + N[(t$95$1 - N[(N[Sqrt[x], $MachinePrecision] + N[(-1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 9.00000000000000081e-37

   1. Initial program 97.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + x}\right)} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower--.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \color{blue}{\left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\color{blue}{\sqrt{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{\color{blue}{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-sqrt.f64 52.9

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 52.9%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 9.00000000000000081e-37 < y < 5.3e29

   1. Initial program 92.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{y + 1}} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\color{blue}{\sqrt{y + 1}} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{\sqrt{y}} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \sqrt{y} \cdot \color{blue}{\sqrt{y}}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-+.f64 97.3

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{y + 1}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. lower-+.f64 97.3

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 97.3%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-/.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. lower-+.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. lower-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. lower-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. lower-+.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. lower-sqrt.f64 41.3

          \[\leadsto \left(\sqrt{1 + x} + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} - \color{blue}{\sqrt{x}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 41.3%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} - \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 5.3e29 < y

   1. Initial program 87.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{x + 1}} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{x + 1}} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{\color{blue}{y + 1}} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\color{blue}{\sqrt{y + 1}} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. flip-- N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. frac-add N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right) + \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right) + \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 88.0%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(\left(x + 1\right) - x, \sqrt{1 + y} + \sqrt{y}, \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\left(1 + y\right) - y\right)\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\frac{\color{blue}{1 + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{1 + y}\right)\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right)} + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \color{blue}{\left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\color{blue}{\sqrt{y}} + \sqrt{1 + y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \color{blue}{\sqrt{1 + y}}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 93.8

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{\color{blue}{1 + y}}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 93.8%

      \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

      1. lower-/.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. lower-+.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. lower-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. lower-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. lower-+.f64 93.6

         \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{\color{blue}{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Applied rewrites 93.6%

       \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-37}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+29}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \frac{-1}{\sqrt{y} + \sqrt{1 + y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 56.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;\sqrt{1 + y} - \sqrt{y} \leq 0.1:\\ \;\;\;\;1 + \left(\sqrt{z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\mathsf{fma}\left(y, 0.5, \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

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 (<= (- (sqrt (+ 1.0 y)) (sqrt y)) 0.1)
   (+ 1.0 (- (sqrt z) (+ (sqrt x) (+ (sqrt y) (sqrt z)))))
   (+ 1.0 (- (fma y 0.5 (sqrt (+ 1.0 x))) (+ (sqrt x) (sqrt y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((sqrt((1.0 + y)) - sqrt(y)) <= 0.1) {
		tmp = 1.0 + (sqrt(z) - (sqrt(x) + (sqrt(y) + sqrt(z))));
	} else {
		tmp = 1.0 + (fma(y, 0.5, sqrt((1.0 + x))) - (sqrt(x) + sqrt(y)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) <= 0.1)
		tmp = Float64(1.0 + Float64(sqrt(z) - Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z)))));
	else
		tmp = Float64(1.0 + Float64(fma(y, 0.5, sqrt(Float64(1.0 + x))) - Float64(sqrt(x) + 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_] := If[LessEqual[N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], 0.1], N[(1.0 + N[(N[Sqrt[z], $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * 0.5 + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $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)) < 0.10000000000000001

   1. Initial program 87.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-sqrt.f64 4.5

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Applied rewrites 4.5%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{\left(\frac{1}{2} \cdot y + \sqrt{1 + z}\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{\mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \color{blue}{\sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{\color{blue}{1 + z}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      15. lower-sqrt.f64 11.8

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, y, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   8. Applied rewrites 11.8%

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, y, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto 1 + \left(\color{blue}{\sqrt{z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]
   10. Step-by-step derivation

       1. lower-sqrt.f64 10.2

          \[\leadsto 1 + \left(\color{blue}{\sqrt{z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   11. Applied rewrites 10.2%

       \[\leadsto 1 + \left(\color{blue}{\sqrt{z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   if 0.10000000000000001 < (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))

   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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-sqrt.f64 28.0

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Applied rewrites 28.0%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{\left(\frac{1}{2} \cdot y + \sqrt{1 + z}\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{\mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \color{blue}{\sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{\color{blue}{1 + z}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      15. lower-sqrt.f64 41.3

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, y, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   8. Applied rewrites 41.3%

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, y, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   10. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

       3. lower--.f64 N/A

          \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

       4. +-commutative N/A

          \[\leadsto 1 + \left(\color{blue}{\left(\frac{1}{2} \cdot y + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto 1 + \left(\left(\color{blue}{y \cdot \frac{1}{2}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto 1 + \left(\color{blue}{\mathsf{fma}\left(y, \frac{1}{2}, \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       7. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\mathsf{fma}\left(y, \frac{1}{2}, \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       8. lower-+.f64 N/A

          \[\leadsto 1 + \left(\mathsf{fma}\left(y, \frac{1}{2}, \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       9. lower-+.f64 N/A

          \[\leadsto 1 + \left(\mathsf{fma}\left(y, \frac{1}{2}, \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]

       10. lower-sqrt.f64 N/A

           \[\leadsto 1 + \left(\mathsf{fma}\left(y, \frac{1}{2}, \sqrt{1 + x}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) \]

       11. lower-sqrt.f64 45.2

           \[\leadsto 1 + \left(\mathsf{fma}\left(y, 0.5, \sqrt{1 + x}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   11. Applied rewrites 45.2%

       \[\leadsto \color{blue}{1 + \left(\mathsf{fma}\left(y, 0.5, \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + y} - \sqrt{y} \leq 0.1:\\ \;\;\;\;1 + \left(\sqrt{z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\mathsf{fma}\left(y, 0.5, \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 48.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;\sqrt{1 + y} - \sqrt{y} \leq 0.1:\\ \;\;\;\;1 + \left(\mathsf{fma}\left(y, 0.5, \sqrt{1 + z}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\mathsf{fma}\left(y, 0.5, \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

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 (<= (- (sqrt (+ 1.0 y)) (sqrt y)) 0.1)
   (+ 1.0 (- (fma y 0.5 (sqrt (+ 1.0 z))) (+ (sqrt y) (sqrt z))))
   (+ 1.0 (- (fma y 0.5 (sqrt (+ 1.0 x))) (+ (sqrt x) (sqrt y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((sqrt((1.0 + y)) - sqrt(y)) <= 0.1) {
		tmp = 1.0 + (fma(y, 0.5, sqrt((1.0 + z))) - (sqrt(y) + sqrt(z)));
	} else {
		tmp = 1.0 + (fma(y, 0.5, sqrt((1.0 + x))) - (sqrt(x) + sqrt(y)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) <= 0.1)
		tmp = Float64(1.0 + Float64(fma(y, 0.5, sqrt(Float64(1.0 + z))) - Float64(sqrt(y) + sqrt(z))));
	else
		tmp = Float64(1.0 + Float64(fma(y, 0.5, sqrt(Float64(1.0 + x))) - Float64(sqrt(x) + 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_] := If[LessEqual[N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], 0.1], N[(1.0 + N[(N[(y * 0.5 + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * 0.5 + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $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)) < 0.10000000000000001

   1. Initial program 87.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-sqrt.f64 4.5

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Applied rewrites 4.5%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{\left(\frac{1}{2} \cdot y + \sqrt{1 + z}\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{\mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \color{blue}{\sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{\color{blue}{1 + z}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      15. lower-sqrt.f64 11.8

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, y, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   8. Applied rewrites 11.8%

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, y, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]
   10. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

       3. lower--.f64 N/A

          \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

       4. +-commutative N/A

          \[\leadsto 1 + \left(\color{blue}{\left(\frac{1}{2} \cdot y + \sqrt{1 + z}\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto 1 + \left(\left(\color{blue}{y \cdot \frac{1}{2}} + \sqrt{1 + z}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto 1 + \left(\color{blue}{\mathsf{fma}\left(y, \frac{1}{2}, \sqrt{1 + z}\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) \]

       7. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\mathsf{fma}\left(y, \frac{1}{2}, \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) \]

       8. lower-+.f64 N/A

          \[\leadsto 1 + \left(\mathsf{fma}\left(y, \frac{1}{2}, \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) \]

       9. lower-+.f64 N/A

          \[\leadsto 1 + \left(\mathsf{fma}\left(y, \frac{1}{2}, \sqrt{1 + z}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) \]

       10. lower-sqrt.f64 N/A

           \[\leadsto 1 + \left(\mathsf{fma}\left(y, \frac{1}{2}, \sqrt{1 + z}\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) \]

       11. lower-sqrt.f64 10.3

           \[\leadsto 1 + \left(\mathsf{fma}\left(y, 0.5, \sqrt{1 + z}\right) - \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

   11. Applied rewrites 10.3%

       \[\leadsto \color{blue}{1 + \left(\mathsf{fma}\left(y, 0.5, \sqrt{1 + z}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   if 0.10000000000000001 < (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))

   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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-sqrt.f64 28.0

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Applied rewrites 28.0%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{\left(\frac{1}{2} \cdot y + \sqrt{1 + z}\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{\mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \color{blue}{\sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{\color{blue}{1 + z}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      15. lower-sqrt.f64 41.3

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, y, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   8. Applied rewrites 41.3%

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, y, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   10. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

       3. lower--.f64 N/A

          \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

       4. +-commutative N/A

          \[\leadsto 1 + \left(\color{blue}{\left(\frac{1}{2} \cdot y + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto 1 + \left(\left(\color{blue}{y \cdot \frac{1}{2}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto 1 + \left(\color{blue}{\mathsf{fma}\left(y, \frac{1}{2}, \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       7. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\mathsf{fma}\left(y, \frac{1}{2}, \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       8. lower-+.f64 N/A

          \[\leadsto 1 + \left(\mathsf{fma}\left(y, \frac{1}{2}, \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       9. lower-+.f64 N/A

          \[\leadsto 1 + \left(\mathsf{fma}\left(y, \frac{1}{2}, \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]

       10. lower-sqrt.f64 N/A

           \[\leadsto 1 + \left(\mathsf{fma}\left(y, \frac{1}{2}, \sqrt{1 + x}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) \]

       11. lower-sqrt.f64 45.2

           \[\leadsto 1 + \left(\mathsf{fma}\left(y, 0.5, \sqrt{1 + x}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   11. Applied rewrites 45.2%

       \[\leadsto \color{blue}{1 + \left(\mathsf{fma}\left(y, 0.5, \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + y} - \sqrt{y} \leq 0.1:\\ \;\;\;\;1 + \left(\mathsf{fma}\left(y, 0.5, \sqrt{1 + z}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\mathsf{fma}\left(y, 0.5, \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 82.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x} + \sqrt{y}\\ t_2 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 0.8:\\ \;\;\;\;\left(t\_2 + 2\right) - \left(\sqrt{z} + t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(t\_2 - t\_1\right)\\ \end{array} \end{array} \]

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 x) (sqrt y))) (t_2 (sqrt (+ 1.0 y))))
   (if (<= z 0.8)
     (- (+ t_2 2.0) (+ (sqrt z) t_1))
     (+ (sqrt (+ 1.0 x)) (- t_2 t_1)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(x) + sqrt(y);
	double t_2 = sqrt((1.0 + y));
	double tmp;
	if (z <= 0.8) {
		tmp = (t_2 + 2.0) - (sqrt(z) + t_1);
	} else {
		tmp = sqrt((1.0 + x)) + (t_2 - t_1);
	}
	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(x) + sqrt(y)
    t_2 = sqrt((1.0d0 + y))
    if (z <= 0.8d0) then
        tmp = (t_2 + 2.0d0) - (sqrt(z) + t_1)
    else
        tmp = sqrt((1.0d0 + x)) + (t_2 - t_1)
    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(x) + Math.sqrt(y);
	double t_2 = Math.sqrt((1.0 + y));
	double tmp;
	if (z <= 0.8) {
		tmp = (t_2 + 2.0) - (Math.sqrt(z) + t_1);
	} else {
		tmp = Math.sqrt((1.0 + x)) + (t_2 - t_1);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt(x) + math.sqrt(y)
	t_2 = math.sqrt((1.0 + y))
	tmp = 0
	if z <= 0.8:
		tmp = (t_2 + 2.0) - (math.sqrt(z) + t_1)
	else:
		tmp = math.sqrt((1.0 + x)) + (t_2 - t_1)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(x) + sqrt(y))
	t_2 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (z <= 0.8)
		tmp = Float64(Float64(t_2 + 2.0) - Float64(sqrt(z) + t_1));
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) + Float64(t_2 - t_1));
	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(x) + sqrt(y);
	t_2 = sqrt((1.0 + y));
	tmp = 0.0;
	if (z <= 0.8)
		tmp = (t_2 + 2.0) - (sqrt(z) + t_1);
	else
		tmp = sqrt((1.0 + x)) + (t_2 - t_1);
	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[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 0.8], N[(N[(t$95$2 + 2.0), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(t$95$2 - t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < 0.80000000000000004

   1. Initial program 97.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. 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. lower--.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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-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. lower-+.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. lower-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. lower-+.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. lower-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. lower-sqrt.f64 35.3

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 35.3%

      \[\leadsto \color{blue}{\left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(2 + \sqrt{1 + y}\right) - \sqrt{x}\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(2 + \left(\sqrt{1 + y} - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 + \left(\sqrt{1 + y} - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \left(\left(2 + \color{blue}{\left(\sqrt{1 + y} - \sqrt{x}\right)}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(2 + \left(\color{blue}{\sqrt{1 + y}} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(2 + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 30.5

         \[\leadsto \left(\left(2 + \left(\sqrt{1 + y} - \color{blue}{\sqrt{x}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 30.5%

      \[\leadsto \left(\color{blue}{\left(2 + \left(\sqrt{1 + y} - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(2 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(2 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{\left(2 + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \left(2 + \color{blue}{\sqrt{1 + y}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

       4. lower-+.f64 N/A

          \[\leadsto \left(2 + \sqrt{\color{blue}{1 + y}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

       5. associate-+r+ N/A

          \[\leadsto \left(2 + \sqrt{1 + y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)} \]

       6. lower-+.f64 N/A

          \[\leadsto \left(2 + \sqrt{1 + y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)} \]

       7. lower-+.f64 N/A

          \[\leadsto \left(2 + \sqrt{1 + y}\right) - \left(\color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} + \sqrt{z}\right) \]

       8. lower-sqrt.f64 N/A

          \[\leadsto \left(2 + \sqrt{1 + y}\right) - \left(\left(\color{blue}{\sqrt{x}} + \sqrt{y}\right) + \sqrt{z}\right) \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \left(2 + \sqrt{1 + y}\right) - \left(\left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) + \sqrt{z}\right) \]

       10. lower-sqrt.f64 18.5

           \[\leadsto \left(2 + \sqrt{1 + y}\right) - \left(\left(\sqrt{x} + \sqrt{y}\right) + \color{blue}{\sqrt{z}}\right) \]

   11. Applied rewrites 18.5%

       \[\leadsto \color{blue}{\left(2 + \sqrt{1 + y}\right) - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)} \]

   if 0.80000000000000004 < z

   1. Initial program 88.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-sqrt.f64 4.4

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Applied rewrites 4.4%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   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. lower-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\sqrt{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{\color{blue}{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) \]

      10. lower-sqrt.f64 31.0

          \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   8. Applied rewrites 31.0%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.8:\\ \;\;\;\;\left(\sqrt{1 + y} + 2\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 64.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (+ (sqrt (+ 1.0 x)) (- (sqrt (+ 1.0 y)) (+ (sqrt x) (sqrt y)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return sqrt((1.0 + x)) + (sqrt((1.0 + y)) - (sqrt(x) + sqrt(y)));
}

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((1.0d0 + x)) + (sqrt((1.0d0 + y)) - (sqrt(x) + sqrt(y)))
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((1.0 + x)) + (Math.sqrt((1.0 + y)) - (Math.sqrt(x) + Math.sqrt(y)));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return math.sqrt((1.0 + x)) + (math.sqrt((1.0 + y)) - (math.sqrt(x) + math.sqrt(y)))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(sqrt(Float64(1.0 + x)) + Float64(sqrt(Float64(1.0 + y)) - Float64(sqrt(x) + sqrt(y))))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = sqrt((1.0 + x)) + (sqrt((1.0 + y)) - (sqrt(x) + sqrt(y)));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.7%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. 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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-sqrt.f64 17.3

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

5. Applied rewrites 17.3%

   \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

6. Taylor expanded in 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. lower-+.f64 N/A

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   4. lower-+.f64 N/A

      \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   5. lower--.f64 N/A

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\sqrt{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   7. lower-+.f64 N/A

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{\color{blue}{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   8. lower-+.f64 N/A

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]

   9. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) \]

   10. lower-sqrt.f64 22.8

       \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

8. Applied rewrites 22.8%

   \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
9. Add Preprocessing

Alternative 21: 42.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 1 + \left(\mathsf{fma}\left(y, 0.5, \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \end{array} \]

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 (- (fma y 0.5 (sqrt (+ 1.0 x))) (+ (sqrt x) (sqrt y)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 1.0 + (fma(y, 0.5, sqrt((1.0 + x))) - (sqrt(x) + sqrt(y)));
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(1.0 + Float64(fma(y, 0.5, sqrt(Float64(1.0 + x))) - Float64(sqrt(x) + sqrt(y))))
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[(y * 0.5 + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.7%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. 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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-sqrt.f64 17.3

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

5. Applied rewrites 17.3%

   \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
7. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   2. lower-+.f64 N/A

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   3. lower--.f64 N/A

      \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   4. lower-+.f64 N/A

      \[\leadsto 1 + \left(\color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   5. lower-sqrt.f64 N/A

      \[\leadsto 1 + \left(\left(\color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   6. lower-+.f64 N/A

      \[\leadsto 1 + \left(\left(\sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{\left(\frac{1}{2} \cdot y + \sqrt{1 + z}\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{\mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   9. lower-sqrt.f64 N/A

      \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \color{blue}{\sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   10. lower-+.f64 N/A

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{\color{blue}{1 + z}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   11. lower-+.f64 N/A

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

   12. lower-sqrt.f64 N/A

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   13. lower-+.f64 N/A

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

   14. lower-sqrt.f64 N/A

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

   15. lower-sqrt.f64 27.8

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, y, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

8. Applied rewrites 27.8%

   \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, y, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

9. Taylor expanded in z around inf

   \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
10. Step-by-step derivation

    1. associate--l+ N/A

       \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

    2. lower-+.f64 N/A

       \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

    3. lower--.f64 N/A

       \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

    4. +-commutative N/A

       \[\leadsto 1 + \left(\color{blue}{\left(\frac{1}{2} \cdot y + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

    5. *-commutative N/A

       \[\leadsto 1 + \left(\left(\color{blue}{y \cdot \frac{1}{2}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

    6. lower-fma.f64 N/A

       \[\leadsto 1 + \left(\color{blue}{\mathsf{fma}\left(y, \frac{1}{2}, \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

    7. lower-sqrt.f64 N/A

       \[\leadsto 1 + \left(\mathsf{fma}\left(y, \frac{1}{2}, \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

    8. lower-+.f64 N/A

       \[\leadsto 1 + \left(\mathsf{fma}\left(y, \frac{1}{2}, \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

    9. lower-+.f64 N/A

       \[\leadsto 1 + \left(\mathsf{fma}\left(y, \frac{1}{2}, \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]

    10. lower-sqrt.f64 N/A

        \[\leadsto 1 + \left(\mathsf{fma}\left(y, \frac{1}{2}, \sqrt{1 + x}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) \]

    11. lower-sqrt.f64 28.6

        \[\leadsto 1 + \left(\mathsf{fma}\left(y, 0.5, \sqrt{1 + x}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

11. Applied rewrites 28.6%

    \[\leadsto \color{blue}{1 + \left(\mathsf{fma}\left(y, 0.5, \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
12. Add Preprocessing

Alternative 22: 14.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 1 + y \cdot \left(0.5 - \sqrt{\frac{1}{y}}\right) \end{array} \]

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 (* y (- 0.5 (sqrt (/ 1.0 y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 1.0 + (y * (0.5 - sqrt((1.0 / y))));
}

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 + (y * (0.5d0 - sqrt((1.0d0 / y))))
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 + (y * (0.5 - Math.sqrt((1.0 / y))));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 1.0 + (y * (0.5 - math.sqrt((1.0 / y))))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(1.0 + Float64(y * Float64(0.5 - sqrt(Float64(1.0 / y)))))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 1.0 + (y * (0.5 - sqrt((1.0 / y))));
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[(y * N[(0.5 - N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.7%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. 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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-sqrt.f64 17.3

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

5. Applied rewrites 17.3%

   \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
7. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   2. lower-+.f64 N/A

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   3. lower--.f64 N/A

      \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   4. lower-+.f64 N/A

      \[\leadsto 1 + \left(\color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   5. lower-sqrt.f64 N/A

      \[\leadsto 1 + \left(\left(\color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   6. lower-+.f64 N/A

      \[\leadsto 1 + \left(\left(\sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{\left(\frac{1}{2} \cdot y + \sqrt{1 + z}\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{\mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   9. lower-sqrt.f64 N/A

      \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \color{blue}{\sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   10. lower-+.f64 N/A

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{\color{blue}{1 + z}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   11. lower-+.f64 N/A

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

   12. lower-sqrt.f64 N/A

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   13. lower-+.f64 N/A

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

   14. lower-sqrt.f64 N/A

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(\frac{1}{2}, y, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

   15. lower-sqrt.f64 27.8

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, y, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

8. Applied rewrites 27.8%

   \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, y, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

9. Taylor expanded in y around inf

   \[\leadsto 1 + \color{blue}{y \cdot \left(\frac{1}{2} - \sqrt{\frac{1}{y}}\right)} \]
10. Step-by-step derivation

    1. sub-neg N/A

       \[\leadsto 1 + y \cdot \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\sqrt{\frac{1}{y}}\right)\right)\right)} \]

    2. neg-mul-1 N/A

       \[\leadsto 1 + y \cdot \left(\frac{1}{2} + \color{blue}{-1 \cdot \sqrt{\frac{1}{y}}}\right) \]

    3. rem-square-sqrt N/A

       \[\leadsto 1 + y \cdot \left(\frac{1}{2} + \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{y}}\right) \]

    4. unpow2 N/A

       \[\leadsto 1 + y \cdot \left(\frac{1}{2} + \color{blue}{{\left(\sqrt{-1}\right)}^{2}} \cdot \sqrt{\frac{1}{y}}\right) \]

    5. *-commutative N/A

       \[\leadsto 1 + y \cdot \left(\frac{1}{2} + \color{blue}{\sqrt{\frac{1}{y}} \cdot {\left(\sqrt{-1}\right)}^{2}}\right) \]

    6. lower-*.f64 N/A

       \[\leadsto 1 + \color{blue}{y \cdot \left(\frac{1}{2} + \sqrt{\frac{1}{y}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

    7. *-commutative N/A

       \[\leadsto 1 + y \cdot \left(\frac{1}{2} + \color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{y}}}\right) \]

    8. unpow2 N/A

       \[\leadsto 1 + y \cdot \left(\frac{1}{2} + \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{y}}\right) \]

    9. rem-square-sqrt N/A

       \[\leadsto 1 + y \cdot \left(\frac{1}{2} + \color{blue}{-1} \cdot \sqrt{\frac{1}{y}}\right) \]

    10. neg-mul-1 N/A

        \[\leadsto 1 + y \cdot \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\sqrt{\frac{1}{y}}\right)\right)}\right) \]

    11. sub-neg N/A

        \[\leadsto 1 + y \cdot \color{blue}{\left(\frac{1}{2} - \sqrt{\frac{1}{y}}\right)} \]

    12. lower--.f64 N/A

        \[\leadsto 1 + y \cdot \color{blue}{\left(\frac{1}{2} - \sqrt{\frac{1}{y}}\right)} \]

    13. lower-sqrt.f64 N/A

        \[\leadsto 1 + y \cdot \left(\frac{1}{2} - \color{blue}{\sqrt{\frac{1}{y}}}\right) \]

    14. lower-/.f64 19.5

        \[\leadsto 1 + y \cdot \left(0.5 - \sqrt{\color{blue}{\frac{1}{y}}}\right) \]

11. Applied rewrites 19.5%

    \[\leadsto 1 + \color{blue}{y \cdot \left(0.5 - \sqrt{\frac{1}{y}}\right)} \]
12. Add Preprocessing

Alternative 23: 7.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 0.5 \cdot \sqrt{\frac{1}{z}} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (* 0.5 (sqrt (/ 1.0 z))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 0.5 * sqrt((1.0 / 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 = 0.5d0 * sqrt((1.0d0 / z))
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 0.5 * Math.sqrt((1.0 / z));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 0.5 * math.sqrt((1.0 / z))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(0.5 * sqrt(Float64(1.0 / z)))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 0.5 * sqrt((1.0 / 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[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.7%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. 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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower-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. lower-sqrt.f64 17.3

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

5. Applied rewrites 17.3%

   \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

6. Taylor expanded in z around inf

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\frac{-1}{8} \cdot \sqrt{\frac{1}{{z}^{3}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
7. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\frac{-1}{8} \cdot \sqrt{\frac{1}{{z}^{3}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

8. Applied rewrites 12.4%

   \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \mathsf{fma}\left(-0.125, \sqrt{\frac{1}{z \cdot \left(z \cdot z\right)}}, 0.5 \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

9. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{z} + \frac{1}{2} \cdot \sqrt{{z}^{3}}}{{z}^{2}}} \]
10. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{z} + \frac{1}{2} \cdot \sqrt{{z}^{3}}}{{z}^{2}}} \]

    2. +-commutative N/A

       \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \sqrt{{z}^{3}} + \frac{-1}{8} \cdot \sqrt{z}}}{{z}^{2}} \]

    3. lower-fma.f64 N/A

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{{z}^{3}}, \frac{-1}{8} \cdot \sqrt{z}\right)}}{{z}^{2}} \]

    4. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{{z}^{3}}}, \frac{-1}{8} \cdot \sqrt{z}\right)}{{z}^{2}} \]

    5. cube-mult N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{z \cdot \left(z \cdot z\right)}}, \frac{-1}{8} \cdot \sqrt{z}\right)}{{z}^{2}} \]

    6. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \sqrt{z \cdot \color{blue}{{z}^{2}}}, \frac{-1}{8} \cdot \sqrt{z}\right)}{{z}^{2}} \]

    7. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{z \cdot {z}^{2}}}, \frac{-1}{8} \cdot \sqrt{z}\right)}{{z}^{2}} \]

    8. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \sqrt{z \cdot \color{blue}{\left(z \cdot z\right)}}, \frac{-1}{8} \cdot \sqrt{z}\right)}{{z}^{2}} \]

    9. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \sqrt{z \cdot \color{blue}{\left(z \cdot z\right)}}, \frac{-1}{8} \cdot \sqrt{z}\right)}{{z}^{2}} \]

    10. *-commutative N/A

        \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \sqrt{z \cdot \left(z \cdot z\right)}, \color{blue}{\sqrt{z} \cdot \frac{-1}{8}}\right)}{{z}^{2}} \]

    11. lower-*.f64 N/A

        \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \sqrt{z \cdot \left(z \cdot z\right)}, \color{blue}{\sqrt{z} \cdot \frac{-1}{8}}\right)}{{z}^{2}} \]

    12. lower-sqrt.f64 N/A

        \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \sqrt{z \cdot \left(z \cdot z\right)}, \color{blue}{\sqrt{z}} \cdot \frac{-1}{8}\right)}{{z}^{2}} \]

    13. unpow2 N/A

        \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \sqrt{z \cdot \left(z \cdot z\right)}, \sqrt{z} \cdot \frac{-1}{8}\right)}{\color{blue}{z \cdot z}} \]

    14. lower-*.f64 3.1

        \[\leadsto \frac{\mathsf{fma}\left(0.5, \sqrt{z \cdot \left(z \cdot z\right)}, \sqrt{z} \cdot -0.125\right)}{\color{blue}{z \cdot z}} \]

11. Applied rewrites 3.1%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, \sqrt{z \cdot \left(z \cdot z\right)}, \sqrt{z} \cdot -0.125\right)}{z \cdot z}} \]

12. Taylor expanded in z around inf

    \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}} \]
13. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}} \]

    2. lower-sqrt.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{z}}} \]

    3. lower-/.f64 7.7

       \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{z}}} \]

14. Applied rewrites 7.7%

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{z}}} \]
15. Add Preprocessing

Alternative 24: 6.2% accurate, 5.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \frac{0.5}{\sqrt{t}} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (/ 0.5 (sqrt t)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 0.5 / sqrt(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 = 0.5d0 / sqrt(t)
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 0.5 / Math.sqrt(t);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 0.5 / math.sqrt(t)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(0.5 / sqrt(t))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 0.5 / sqrt(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[(0.5 / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.7%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
4. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \sqrt{1 + x}} \]

   3. +-commutative N/A

      \[\leadsto \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)}\right) + \sqrt{1 + x} \]

   4. associate--r+ N/A

      \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \sqrt{x}\right)} + \sqrt{1 + x} \]

   5. associate-+l- N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

5. Applied rewrites 26.3%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{t}}, \sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

6. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}}} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{t}}} \]

   3. lower-/.f64 7.1

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{t}}} \]

8. Applied rewrites 7.1%

   \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{t}}} \]
9. Step-by-step derivation

   1. sqrt-div N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{t}}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \frac{\color{blue}{1}}{\sqrt{t}} \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{1}{\color{blue}{\sqrt{t}}} \]

   4. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\sqrt{t}}} \]

   5. lower-/.f64 7.1

      \[\leadsto \color{blue}{\frac{0.5}{\sqrt{t}}} \]

10. Applied rewrites 7.1%

    \[\leadsto \color{blue}{\frac{0.5}{\sqrt{t}}} \]
11. Add Preprocessing

Alternative 25: 1.9% accurate, 8.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ -\sqrt{x} \end{array} \]

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 x)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return -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 = -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 -Math.sqrt(x);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return -math.sqrt(x)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(-sqrt(x))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = -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[Sqrt[x], $MachinePrecision])

Derivation

1. Initial program 92.7%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. 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. lower--.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. lower-+.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. lower-+.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. lower-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. lower-+.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. lower--.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. lower-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. lower-+.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. lower-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. lower-+.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. lower-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. lower-sqrt.f64 18.8

       \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

5. Applied rewrites 18.8%

   \[\leadsto \color{blue}{\left(\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

6. Taylor expanded in x around 0

   \[\leadsto \left(\color{blue}{\left(\left(2 + \sqrt{1 + y}\right) - \sqrt{x}\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
7. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \left(\color{blue}{\left(2 + \left(\sqrt{1 + y} - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. lower-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(2 + \left(\sqrt{1 + y} - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. lower--.f64 N/A

      \[\leadsto \left(\left(2 + \color{blue}{\left(\sqrt{1 + y} - \sqrt{x}\right)}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(2 + \left(\color{blue}{\sqrt{1 + y}} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\left(2 + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. lower-sqrt.f64 16.7

      \[\leadsto \left(\left(2 + \left(\sqrt{1 + y} - \color{blue}{\sqrt{x}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

8. Applied rewrites 16.7%

   \[\leadsto \left(\color{blue}{\left(2 + \left(\sqrt{1 + y} - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

9. Taylor expanded in x around inf

   \[\leadsto \color{blue}{-1 \cdot \sqrt{x}} \]
10. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{x}\right)} \]

    2. lower-neg.f64 N/A

       \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{x}\right)} \]

    3. lower-sqrt.f64 1.6

       \[\leadsto -\color{blue}{\sqrt{x}} \]

11. Applied rewrites 1.6%

    \[\leadsto \color{blue}{-\sqrt{x}} \]
12. 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 2024214 
(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))))