Main:z from

Percentage Accurate: 91.3% → 99.4%

Time: 24.7s

Alternatives: 24

Speedup: 0.8×

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: 91.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{x + 1}\\ t_3 := t\_2 - \sqrt{x}\\ t_4 := \sqrt{t + 1}\\ t_5 := t\_4 - \sqrt{t}\\ t_6 := \sqrt{1 + y}\\ t_7 := \left(t\_3 + \left(t\_6 - \sqrt{y}\right)\right) + t\_1\\ \mathbf{if}\;t\_7 \leq 0.04:\\ \;\;\;\;t\_5 + \left(t\_1 + \mathsf{fma}\left(\sqrt{\frac{1}{y}}, 0.5, \frac{1}{\sqrt{x} + t\_2}\right)\right)\\ \mathbf{elif}\;t\_7 \leq 2.0001:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \frac{1}{t\_6 + \sqrt{y}}\right) + t\_2\right) - \sqrt{x}\right) + t\_5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{y}\right) + t\_3\right) + t\_1\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + t\_4}\\ \end{array} \end{array} \]

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 (+ z 1.0)) (sqrt z)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (- t_2 (sqrt x)))
        (t_4 (sqrt (+ t 1.0)))
        (t_5 (- t_4 (sqrt t)))
        (t_6 (sqrt (+ 1.0 y)))
        (t_7 (+ (+ t_3 (- t_6 (sqrt y))) t_1)))
   (if (<= t_7 0.04)
     (+ t_5 (+ t_1 (fma (sqrt (/ 1.0 y)) 0.5 (/ 1.0 (+ (sqrt x) t_2)))))
     (if (<= t_7 2.0001)
       (+
        (-
         (+ (fma (sqrt (/ 1.0 z)) 0.5 (/ 1.0 (+ t_6 (sqrt y)))) t_2)
         (sqrt x))
        t_5)
       (+
        (+ (+ (- 1.0 (sqrt y)) t_3) t_1)
        (/ (- (+ t 1.0) t) (+ (sqrt t) t_4)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = sqrt((x + 1.0));
	double t_3 = t_2 - sqrt(x);
	double t_4 = sqrt((t + 1.0));
	double t_5 = t_4 - sqrt(t);
	double t_6 = sqrt((1.0 + y));
	double t_7 = (t_3 + (t_6 - sqrt(y))) + t_1;
	double tmp;
	if (t_7 <= 0.04) {
		tmp = t_5 + (t_1 + fma(sqrt((1.0 / y)), 0.5, (1.0 / (sqrt(x) + t_2))));
	} else if (t_7 <= 2.0001) {
		tmp = ((fma(sqrt((1.0 / z)), 0.5, (1.0 / (t_6 + sqrt(y)))) + t_2) - sqrt(x)) + t_5;
	} else {
		tmp = (((1.0 - sqrt(y)) + t_3) + t_1) + (((t + 1.0) - t) / (sqrt(t) + t_4));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = Float64(t_2 - sqrt(x))
	t_4 = sqrt(Float64(t + 1.0))
	t_5 = Float64(t_4 - sqrt(t))
	t_6 = sqrt(Float64(1.0 + y))
	t_7 = Float64(Float64(t_3 + Float64(t_6 - sqrt(y))) + t_1)
	tmp = 0.0
	if (t_7 <= 0.04)
		tmp = Float64(t_5 + Float64(t_1 + fma(sqrt(Float64(1.0 / y)), 0.5, Float64(1.0 / Float64(sqrt(x) + t_2)))));
	elseif (t_7 <= 2.0001)
		tmp = Float64(Float64(Float64(fma(sqrt(Float64(1.0 / z)), 0.5, Float64(1.0 / Float64(t_6 + sqrt(y)))) + t_2) - sqrt(x)) + t_5);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - sqrt(y)) + t_3) + t_1) + Float64(Float64(Float64(t + 1.0) - t) / Float64(sqrt(t) + t_4)));
	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[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$7 = N[(N[(t$95$3 + N[(t$95$6 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$7, 0.04], N[(t$95$5 + N[(t$95$1 + N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] * 0.5 + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$7, 2.0001], N[(N[(N[(N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] * 0.5 + N[(1.0 / N[(t$95$6 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision], N[(N[(N[(N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(N[(N[(t + 1.0), $MachinePrecision] - t), $MachinePrecision] / N[(N[Sqrt[t], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.0400000000000000008

   1. Initial program 75.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 77.2

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

   4. Applied rewrites 77.2%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 84.2%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. lift-/.f64 N/A

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

      5. frac-add N/A

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

      6. lower-/.f64 N/A

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

   9. Applied rewrites 84.3%

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

   10. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-+.f64 N/A

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

       8. lower-sqrt.f64 N/A

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

       9. lower-+.f64 N/A

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

       10. lower-sqrt.f64 87.5

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

   12. Applied rewrites 87.5%

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

   if 0.0400000000000000008 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.00010000000000021

   1. Initial program 96.2%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 96.5

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   7. Applied rewrites 41.7%

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

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

   1. Initial program 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 y around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 91.3

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

   5. Applied rewrites 91.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 92.0

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

   7. Applied rewrites 92.0%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right) \leq 0.04:\\ \;\;\;\;\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \mathsf{fma}\left(\sqrt{\frac{1}{y}}, 0.5, \frac{1}{\sqrt{x} + \sqrt{x + 1}}\right)\right)\\ \mathbf{elif}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right) \leq 2.0001:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{z + 1}\\ t_3 := \sqrt{x + 1}\\ t_4 := \left(\left(\left(t\_3 - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right) + \left(t\_2 - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{if}\;t\_4 \leq 1.0001:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, -\sqrt{x}\right) + t\_3\\ \mathbf{elif}\;t\_4 \leq 2.0001:\\ \;\;\;\;\left(\left(t\_1 - \left(\sqrt{x} + \sqrt{y}\right)\right) + \frac{0.5}{\sqrt{z}}\right) + t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_1 + 1\right) + t\_2\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\\ \end{array} \end{array} \]

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 (+ z 1.0)))
        (t_3 (sqrt (+ x 1.0)))
        (t_4
         (+
          (+ (+ (- t_3 (sqrt x)) (- t_1 (sqrt y))) (- t_2 (sqrt z)))
          (- (sqrt (+ t 1.0)) (sqrt t)))))
   (if (<= t_4 1.0001)
     (+ (fma 0.5 (+ (sqrt (/ 1.0 z)) (sqrt (/ 1.0 y))) (- (sqrt x))) t_3)
     (if (<= t_4 2.0001)
       (+ (+ (- t_1 (+ (sqrt x) (sqrt y))) (/ 0.5 (sqrt z))) t_3)
       (- (+ (+ t_1 1.0) t_2) (+ (+ (sqrt z) (sqrt y)) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((z + 1.0));
	double t_3 = sqrt((x + 1.0));
	double t_4 = (((t_3 - sqrt(x)) + (t_1 - sqrt(y))) + (t_2 - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	double tmp;
	if (t_4 <= 1.0001) {
		tmp = fma(0.5, (sqrt((1.0 / z)) + sqrt((1.0 / y))), -sqrt(x)) + t_3;
	} else if (t_4 <= 2.0001) {
		tmp = ((t_1 - (sqrt(x) + sqrt(y))) + (0.5 / sqrt(z))) + t_3;
	} else {
		tmp = ((t_1 + 1.0) + t_2) - ((sqrt(z) + sqrt(y)) + sqrt(x));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = sqrt(Float64(z + 1.0))
	t_3 = sqrt(Float64(x + 1.0))
	t_4 = Float64(Float64(Float64(Float64(t_3 - sqrt(x)) + Float64(t_1 - sqrt(y))) + Float64(t_2 - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
	tmp = 0.0
	if (t_4 <= 1.0001)
		tmp = Float64(fma(0.5, Float64(sqrt(Float64(1.0 / z)) + sqrt(Float64(1.0 / y))), Float64(-sqrt(x))) + t_3);
	elseif (t_4 <= 2.0001)
		tmp = Float64(Float64(Float64(t_1 - Float64(sqrt(x) + sqrt(y))) + Float64(0.5 / sqrt(z))) + t_3);
	else
		tmp = Float64(Float64(Float64(t_1 + 1.0) + t_2) - Float64(Float64(sqrt(z) + sqrt(y)) + sqrt(x)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 1.0001], N[(N[(0.5 * N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + (-N[Sqrt[x], $MachinePrecision])), $MachinePrecision] + t$95$3), $MachinePrecision], If[LessEqual[t$95$4, 2.0001], N[(N[(N[(t$95$1 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], N[(N[(N[(t$95$1 + 1.0), $MachinePrecision] + t$95$2), $MachinePrecision] - N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $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))) < 1.00009999999999999

   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. 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. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-sqrt.f64 3.2

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

   5. Applied rewrites 3.2%

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

   7. Applied rewrites 27.0%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 27.7%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 28.5%

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

   if 1.00009999999999999 < (+.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))) < 2.00010000000000021

   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

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-sqrt.f64 8.8

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

   5. Applied rewrites 8.8%

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

   7. Applied rewrites 16.3%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 26.7%

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

   12. Applied rewrites 26.0%

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

   if 2.00010000000000021 < (+.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 97.8%

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

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-sqrt.f64 28.9

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

   5. Applied rewrites 28.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 2.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 27.1%

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

4. Final simplification 27.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \leq 1.0001:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, -\sqrt{x}\right) + \sqrt{x + 1}\\ \mathbf{elif}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \leq 2.0001:\\ \;\;\;\;\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \frac{0.5}{\sqrt{z}}\right) + \sqrt{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{1 + y} + 1\right) + \sqrt{z + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{\frac{1}{z}}\\ t_3 := \sqrt{z + 1}\\ t_4 := \sqrt{x + 1}\\ t_5 := \left(\left(\left(t\_4 - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right) + \left(t\_3 - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{if}\;t\_5 \leq 1.0001:\\ \;\;\;\;\mathsf{fma}\left(0.5, t\_2 + \sqrt{\frac{1}{y}}, -\sqrt{x}\right) + t\_4\\ \mathbf{elif}\;t\_5 \leq 2.0001:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, t\_2, t\_1\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_1 + 1\right) + t\_3\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\\ \end{array} \end{array} \]

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 (+ z 1.0)))
        (t_4 (sqrt (+ x 1.0)))
        (t_5
         (+
          (+ (+ (- t_4 (sqrt x)) (- t_1 (sqrt y))) (- t_3 (sqrt z)))
          (- (sqrt (+ t 1.0)) (sqrt t)))))
   (if (<= t_5 1.0001)
     (+ (fma 0.5 (+ t_2 (sqrt (/ 1.0 y))) (- (sqrt x))) t_4)
     (if (<= t_5 2.0001)
       (+ (- (fma 0.5 t_2 t_1) (+ (sqrt x) (sqrt y))) 1.0)
       (- (+ (+ t_1 1.0) t_3) (+ (+ (sqrt z) (sqrt y)) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((1.0 / z));
	double t_3 = sqrt((z + 1.0));
	double t_4 = sqrt((x + 1.0));
	double t_5 = (((t_4 - sqrt(x)) + (t_1 - sqrt(y))) + (t_3 - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	double tmp;
	if (t_5 <= 1.0001) {
		tmp = fma(0.5, (t_2 + sqrt((1.0 / y))), -sqrt(x)) + t_4;
	} else if (t_5 <= 2.0001) {
		tmp = (fma(0.5, t_2, t_1) - (sqrt(x) + sqrt(y))) + 1.0;
	} else {
		tmp = ((t_1 + 1.0) + t_3) - ((sqrt(z) + sqrt(y)) + sqrt(x));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = sqrt(Float64(1.0 / z))
	t_3 = sqrt(Float64(z + 1.0))
	t_4 = sqrt(Float64(x + 1.0))
	t_5 = Float64(Float64(Float64(Float64(t_4 - sqrt(x)) + Float64(t_1 - sqrt(y))) + Float64(t_3 - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
	tmp = 0.0
	if (t_5 <= 1.0001)
		tmp = Float64(fma(0.5, Float64(t_2 + sqrt(Float64(1.0 / y))), Float64(-sqrt(x))) + t_4);
	elseif (t_5 <= 2.0001)
		tmp = Float64(Float64(fma(0.5, t_2, t_1) - Float64(sqrt(x) + sqrt(y))) + 1.0);
	else
		tmp = Float64(Float64(Float64(t_1 + 1.0) + t_3) - Float64(Float64(sqrt(z) + sqrt(y)) + sqrt(x)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[(t$95$4 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 1.0001], N[(N[(0.5 * N[(t$95$2 + N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + (-N[Sqrt[x], $MachinePrecision])), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$5, 2.0001], N[(N[(N[(0.5 * t$95$2 + t$95$1), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(t$95$1 + 1.0), $MachinePrecision] + t$95$3), $MachinePrecision] - N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $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))) < 1.00009999999999999

   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. 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. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-sqrt.f64 3.2

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

   5. Applied rewrites 3.2%

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

   7. Applied rewrites 27.0%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 27.7%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 28.5%

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

   if 1.00009999999999999 < (+.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))) < 2.00010000000000021

   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

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-sqrt.f64 8.8

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

   5. Applied rewrites 8.8%

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

   7. Applied rewrites 16.3%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 26.7%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 26.2%

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

   if 2.00010000000000021 < (+.f64 (+.f64 (+.f64 (-.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 97.8%

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

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-sqrt.f64 28.9

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

   5. Applied rewrites 28.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 2.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 27.1%

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

4. Final simplification 27.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \leq 1.0001:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, -\sqrt{x}\right) + \sqrt{x + 1}\\ \mathbf{elif}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \leq 2.0001:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{1 + y} + 1\right) + \sqrt{z + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.7% accurate, 0.4× speedup

Math

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

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 (+ z 1.0)))
        (t_3 (sqrt (+ x 1.0)))
        (t_4
         (+
          (+ (+ (- t_3 (sqrt x)) (- t_1 (sqrt y))) (- t_2 (sqrt z)))
          (- (sqrt (+ t 1.0)) (sqrt t)))))
   (if (<= t_4 1.0)
     (+ (- (sqrt x)) t_3)
     (if (<= t_4 2.0001)
       (+ (- (fma 0.5 (sqrt (/ 1.0 z)) t_1) (+ (sqrt x) (sqrt y))) 1.0)
       (- (+ (+ t_1 1.0) t_2) (+ (+ (sqrt z) (sqrt y)) (sqrt x)))))))

C

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

Julia

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

Wolfram

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

   1. Initial program 89.5%

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

   3. Taylor expanded in 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. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-sqrt.f64 3.2

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

   5. Applied rewrites 3.2%

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

   7. Applied rewrites 27.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \sqrt{x + 1} + -1 \cdot \color{blue}{\sqrt{x}} \]
   9. Step-by-step derivation

   10. Applied rewrites 27.0%

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

   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))) < 2.00010000000000021

   1. Initial program 94.5%

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

   3. Taylor expanded in 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. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-sqrt.f64 8.4

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

   5. Applied rewrites 8.4%

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

   7. Applied rewrites 17.1%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 27.5%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 26.4%

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

   if 2.00010000000000021 < (+.f64 (+.f64 (+.f64 (-.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 97.8%

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

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-sqrt.f64 28.9

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

   5. Applied rewrites 28.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 2.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 27.1%

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

4. Final simplification 26.8%

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

Alternative 5: 86.7% accurate, 0.4× speedup

Math

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

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 (+ z 1.0)))
        (t_3 (sqrt (+ x 1.0)))
        (t_4
         (+
          (+ (+ (- t_3 (sqrt x)) (- t_1 (sqrt y))) (- t_2 (sqrt z)))
          (- (sqrt (+ t 1.0)) (sqrt t)))))
   (if (<= t_4 1.0)
     (+ (- (sqrt x)) t_3)
     (if (<= t_4 2.0001)
       (+ (- (fma 0.5 (sqrt (/ 1.0 z)) t_1) (+ (sqrt x) (sqrt y))) 1.0)
       (+ (- (+ t_2 t_3) (+ (+ (sqrt z) (sqrt y)) (sqrt x))) 1.0)))))

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((z + 1.0));
	double t_3 = sqrt((x + 1.0));
	double t_4 = (((t_3 - sqrt(x)) + (t_1 - sqrt(y))) + (t_2 - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	double tmp;
	if (t_4 <= 1.0) {
		tmp = -sqrt(x) + t_3;
	} else if (t_4 <= 2.0001) {
		tmp = (fma(0.5, sqrt((1.0 / z)), t_1) - (sqrt(x) + sqrt(y))) + 1.0;
	} else {
		tmp = ((t_2 + t_3) - ((sqrt(z) + sqrt(y)) + sqrt(x))) + 1.0;
	}
	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(z + 1.0))
	t_3 = sqrt(Float64(x + 1.0))
	t_4 = Float64(Float64(Float64(Float64(t_3 - sqrt(x)) + Float64(t_1 - sqrt(y))) + Float64(t_2 - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
	tmp = 0.0
	if (t_4 <= 1.0)
		tmp = Float64(Float64(-sqrt(x)) + t_3);
	elseif (t_4 <= 2.0001)
		tmp = Float64(Float64(fma(0.5, sqrt(Float64(1.0 / z)), t_1) - Float64(sqrt(x) + sqrt(y))) + 1.0);
	else
		tmp = Float64(Float64(Float64(t_2 + t_3) - Float64(Float64(sqrt(z) + sqrt(y)) + sqrt(x))) + 1.0);
	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[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 1.0], N[((-N[Sqrt[x], $MachinePrecision]) + t$95$3), $MachinePrecision], If[LessEqual[t$95$4, 2.0001], N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(t$95$2 + t$95$3), $MachinePrecision] - N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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))) < 1

   1. Initial program 89.5%

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

   3. Taylor expanded in 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. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-sqrt.f64 3.2

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

   5. Applied rewrites 3.2%

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

   7. Applied rewrites 27.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \sqrt{x + 1} + -1 \cdot \color{blue}{\sqrt{x}} \]
   9. Step-by-step derivation

   10. Applied rewrites 27.0%

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

   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))) < 2.00010000000000021

   1. Initial program 94.5%

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

   3. Taylor expanded in 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. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-sqrt.f64 8.4

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

   5. Applied rewrites 8.4%

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

   7. Applied rewrites 17.1%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 27.5%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 26.4%

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

   if 2.00010000000000021 < (+.f64 (+.f64 (+.f64 (-.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 97.8%

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

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-sqrt.f64 28.9

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

   5. Applied rewrites 28.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 31.8%

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

4. Final simplification 28.3%

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

Alternative 6: 67.3% accurate, 0.4× speedup

Math

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

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 x) (sqrt y)))
        (t_3 (sqrt (+ x 1.0)))
        (t_4
         (+
          (+
           (+ (- t_3 (sqrt x)) (- t_1 (sqrt y)))
           (- (sqrt (+ z 1.0)) (sqrt z)))
          (- (sqrt (+ t 1.0)) (sqrt t)))))
   (if (<= t_4 1.0)
     (+ (- (sqrt x)) t_3)
     (if (<= t_4 2.5)
       (+ (- (fma 0.5 (sqrt (/ 1.0 z)) t_1) t_2) 1.0)
       (+ (- t_1 t_2) t_3)))))

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(x) + sqrt(y);
	double t_3 = sqrt((x + 1.0));
	double t_4 = (((t_3 - sqrt(x)) + (t_1 - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	double tmp;
	if (t_4 <= 1.0) {
		tmp = -sqrt(x) + t_3;
	} else if (t_4 <= 2.5) {
		tmp = (fma(0.5, sqrt((1.0 / z)), t_1) - t_2) + 1.0;
	} else {
		tmp = (t_1 - t_2) + t_3;
	}
	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(x) + sqrt(y))
	t_3 = sqrt(Float64(x + 1.0))
	t_4 = Float64(Float64(Float64(Float64(t_3 - sqrt(x)) + Float64(t_1 - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
	tmp = 0.0
	if (t_4 <= 1.0)
		tmp = Float64(Float64(-sqrt(x)) + t_3);
	elseif (t_4 <= 2.5)
		tmp = Float64(Float64(fma(0.5, sqrt(Float64(1.0 / z)), t_1) - t_2) + 1.0);
	else
		tmp = Float64(Float64(t_1 - t_2) + t_3);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - 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]}, If[LessEqual[t$95$4, 1.0], N[((-N[Sqrt[x], $MachinePrecision]) + t$95$3), $MachinePrecision], If[LessEqual[t$95$4, 2.5], N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(t$95$1 - t$95$2), $MachinePrecision] + t$95$3), $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))) < 1

   1. Initial program 89.5%

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

   3. Taylor expanded in 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. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-sqrt.f64 3.2

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

   5. Applied rewrites 3.2%

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

   7. Applied rewrites 27.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \sqrt{x + 1} + -1 \cdot \color{blue}{\sqrt{x}} \]
   9. Step-by-step derivation

   10. Applied rewrites 27.0%

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

   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))) < 2.5

   1. Initial program 94.2%

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

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-sqrt.f64 9.6

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

   5. Applied rewrites 9.6%

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

   7. Applied rewrites 18.0%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 26.5%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 25.3%

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

   if 2.5 < (+.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 98.5%

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

   3. Taylor expanded in 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. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-sqrt.f64 29.1

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

   5. Applied rewrites 29.1%

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

   7. Applied rewrites 35.1%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 17.3%

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

4. Final simplification 23.4%

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

Alternative 7: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1}\\ t_2 := t\_1 - \sqrt{t}\\ t_3 := \sqrt{z + 1} - \sqrt{z}\\ t_4 := \sqrt{x + 1}\\ t_5 := t\_4 - \sqrt{x}\\ t_6 := \sqrt{1 + y}\\ t_7 := \left(t\_5 + \left(t\_6 - \sqrt{y}\right)\right) + t\_3\\ t_8 := \frac{1}{t\_6 + \sqrt{y}}\\ \mathbf{if}\;t\_7 \leq 0:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{x}}, 0.5, t\_8\right) + t\_3\right) + t\_2\\ \mathbf{elif}\;t\_7 \leq 2.0001:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, t\_8\right) + t\_4\right) - \sqrt{x}\right) + t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{y}\right) + t\_5\right) + t\_3\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + t\_1}\\ \end{array} \end{array} \]

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 (+ t 1.0)))
        (t_2 (- t_1 (sqrt t)))
        (t_3 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_4 (sqrt (+ x 1.0)))
        (t_5 (- t_4 (sqrt x)))
        (t_6 (sqrt (+ 1.0 y)))
        (t_7 (+ (+ t_5 (- t_6 (sqrt y))) t_3))
        (t_8 (/ 1.0 (+ t_6 (sqrt y)))))
   (if (<= t_7 0.0)
     (+ (+ (fma (sqrt (/ 1.0 x)) 0.5 t_8) t_3) t_2)
     (if (<= t_7 2.0001)
       (+ (- (+ (fma (sqrt (/ 1.0 z)) 0.5 t_8) t_4) (sqrt x)) t_2)
       (+
        (+ (+ (- 1.0 (sqrt y)) t_5) t_3)
        (/ (- (+ t 1.0) t) (+ (sqrt t) t_1)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((t + 1.0));
	double t_2 = t_1 - sqrt(t);
	double t_3 = sqrt((z + 1.0)) - sqrt(z);
	double t_4 = sqrt((x + 1.0));
	double t_5 = t_4 - sqrt(x);
	double t_6 = sqrt((1.0 + y));
	double t_7 = (t_5 + (t_6 - sqrt(y))) + t_3;
	double t_8 = 1.0 / (t_6 + sqrt(y));
	double tmp;
	if (t_7 <= 0.0) {
		tmp = (fma(sqrt((1.0 / x)), 0.5, t_8) + t_3) + t_2;
	} else if (t_7 <= 2.0001) {
		tmp = ((fma(sqrt((1.0 / z)), 0.5, t_8) + t_4) - sqrt(x)) + t_2;
	} else {
		tmp = (((1.0 - sqrt(y)) + t_5) + t_3) + (((t + 1.0) - t) / (sqrt(t) + t_1));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(t + 1.0))
	t_2 = Float64(t_1 - sqrt(t))
	t_3 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_4 = sqrt(Float64(x + 1.0))
	t_5 = Float64(t_4 - sqrt(x))
	t_6 = sqrt(Float64(1.0 + y))
	t_7 = Float64(Float64(t_5 + Float64(t_6 - sqrt(y))) + t_3)
	t_8 = Float64(1.0 / Float64(t_6 + sqrt(y)))
	tmp = 0.0
	if (t_7 <= 0.0)
		tmp = Float64(Float64(fma(sqrt(Float64(1.0 / x)), 0.5, t_8) + t_3) + t_2);
	elseif (t_7 <= 2.0001)
		tmp = Float64(Float64(Float64(fma(sqrt(Float64(1.0 / z)), 0.5, t_8) + t_4) - sqrt(x)) + t_2);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - sqrt(y)) + t_5) + t_3) + Float64(Float64(Float64(t + 1.0) - t) / Float64(sqrt(t) + 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[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$7 = N[(N[(t$95$5 + N[(t$95$6 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]}, Block[{t$95$8 = N[(1.0 / N[(t$95$6 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$7, 0.0], N[(N[(N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.5 + t$95$8), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[t$95$7, 2.0001], N[(N[(N[(N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] * 0.5 + t$95$8), $MachinePrecision] + t$95$4), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], N[(N[(N[(N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision] + t$95$3), $MachinePrecision] + N[(N[(N[(t + 1.0), $MachinePrecision] - t), $MachinePrecision] / N[(N[Sqrt[t], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.0

   1. Initial program 75.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 75.0

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

   4. Applied rewrites 75.0%

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

   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. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 87.6

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

   7. Applied rewrites 87.6%

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

   if 0.0 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.00010000000000021

   1. Initial program 96.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 96.5

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   7. Applied rewrites 41.6%

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

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

   1. Initial program 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 y around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 91.3

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

   5. Applied rewrites 91.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 92.0

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

   7. Applied rewrites 92.0%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right) \leq 0:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{x}}, 0.5, \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right) \leq 2.0001:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{x + 1}\\ t_3 := t\_2 - \sqrt{x}\\ t_4 := \sqrt{t + 1}\\ t_5 := t\_4 - \sqrt{t}\\ t_6 := \sqrt{1 + y}\\ t_7 := \left(t\_3 + \left(t\_6 - \sqrt{y}\right)\right) + t\_1\\ \mathbf{if}\;t\_7 \leq 0.04:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + t\_2} + t\_1\right) + t\_5\\ \mathbf{elif}\;t\_7 \leq 2.0001:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \frac{1}{t\_6 + \sqrt{y}}\right) + t\_2\right) - \sqrt{x}\right) + t\_5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{y}\right) + t\_3\right) + t\_1\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + t\_4}\\ \end{array} \end{array} \]

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 (+ z 1.0)) (sqrt z)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (- t_2 (sqrt x)))
        (t_4 (sqrt (+ t 1.0)))
        (t_5 (- t_4 (sqrt t)))
        (t_6 (sqrt (+ 1.0 y)))
        (t_7 (+ (+ t_3 (- t_6 (sqrt y))) t_1)))
   (if (<= t_7 0.04)
     (+ (+ (/ 1.0 (+ (sqrt x) t_2)) t_1) t_5)
     (if (<= t_7 2.0001)
       (+
        (-
         (+ (fma (sqrt (/ 1.0 z)) 0.5 (/ 1.0 (+ t_6 (sqrt y)))) t_2)
         (sqrt x))
        t_5)
       (+
        (+ (+ (- 1.0 (sqrt y)) t_3) t_1)
        (/ (- (+ t 1.0) t) (+ (sqrt t) t_4)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = sqrt((x + 1.0));
	double t_3 = t_2 - sqrt(x);
	double t_4 = sqrt((t + 1.0));
	double t_5 = t_4 - sqrt(t);
	double t_6 = sqrt((1.0 + y));
	double t_7 = (t_3 + (t_6 - sqrt(y))) + t_1;
	double tmp;
	if (t_7 <= 0.04) {
		tmp = ((1.0 / (sqrt(x) + t_2)) + t_1) + t_5;
	} else if (t_7 <= 2.0001) {
		tmp = ((fma(sqrt((1.0 / z)), 0.5, (1.0 / (t_6 + sqrt(y)))) + t_2) - sqrt(x)) + t_5;
	} else {
		tmp = (((1.0 - sqrt(y)) + t_3) + t_1) + (((t + 1.0) - t) / (sqrt(t) + t_4));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = Float64(t_2 - sqrt(x))
	t_4 = sqrt(Float64(t + 1.0))
	t_5 = Float64(t_4 - sqrt(t))
	t_6 = sqrt(Float64(1.0 + y))
	t_7 = Float64(Float64(t_3 + Float64(t_6 - sqrt(y))) + t_1)
	tmp = 0.0
	if (t_7 <= 0.04)
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(x) + t_2)) + t_1) + t_5);
	elseif (t_7 <= 2.0001)
		tmp = Float64(Float64(Float64(fma(sqrt(Float64(1.0 / z)), 0.5, Float64(1.0 / Float64(t_6 + sqrt(y)))) + t_2) - sqrt(x)) + t_5);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - sqrt(y)) + t_3) + t_1) + Float64(Float64(Float64(t + 1.0) - t) / Float64(sqrt(t) + t_4)));
	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[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$7 = N[(N[(t$95$3 + N[(t$95$6 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$7, 0.04], N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$5), $MachinePrecision], If[LessEqual[t$95$7, 2.0001], N[(N[(N[(N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] * 0.5 + N[(1.0 / N[(t$95$6 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision], N[(N[(N[(N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(N[(N[(t + 1.0), $MachinePrecision] - t), $MachinePrecision] / N[(N[Sqrt[t], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.0400000000000000008

   1. Initial program 75.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 77.2

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

   4. Applied rewrites 77.2%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 84.2%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. lift-/.f64 N/A

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

      5. frac-add N/A

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

      6. lower-/.f64 N/A

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

   9. Applied rewrites 84.3%

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

   10. Taylor expanded in y around inf

       \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   11. 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. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-sqrt.f64 77.6

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

   12. Applied rewrites 77.6%

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

   if 0.0400000000000000008 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.00010000000000021

   1. Initial program 96.2%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 96.5

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   7. Applied rewrites 41.7%

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

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

   1. Initial program 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 y around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 91.3

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

   5. Applied rewrites 91.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 92.0

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

   7. Applied rewrites 92.0%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right) \leq 0.04:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{x + 1}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right) \leq 2.0001:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := t\_1 - \sqrt{z}\\ t_3 := \sqrt{x + 1}\\ t_4 := t\_3 - \sqrt{x}\\ t_5 := \sqrt{t + 1} - \sqrt{t}\\ t_6 := \sqrt{1 + y}\\ t_7 := \left(t\_4 + \left(t\_6 - \sqrt{y}\right)\right) + t\_2\\ \mathbf{if}\;t\_7 \leq 0.04:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + t\_3} + t\_2\right) + t\_5\\ \mathbf{elif}\;t\_7 \leq 1.9999999999999996:\\ \;\;\;\;\left(\left(\frac{1}{t\_6 + \sqrt{y}} + t\_3\right) - \sqrt{x}\right) + t\_5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{y}\right) + t\_4\right) + \frac{\left(z + 1\right) - z}{\sqrt{z} + t\_1}\right) + t\_5\\ \end{array} \end{array} \]

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 (+ z 1.0)))
        (t_2 (- t_1 (sqrt z)))
        (t_3 (sqrt (+ x 1.0)))
        (t_4 (- t_3 (sqrt x)))
        (t_5 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_6 (sqrt (+ 1.0 y)))
        (t_7 (+ (+ t_4 (- t_6 (sqrt y))) t_2)))
   (if (<= t_7 0.04)
     (+ (+ (/ 1.0 (+ (sqrt x) t_3)) t_2) t_5)
     (if (<= t_7 1.9999999999999996)
       (+ (- (+ (/ 1.0 (+ t_6 (sqrt y))) t_3) (sqrt x)) t_5)
       (+
        (+ (+ (- 1.0 (sqrt y)) t_4) (/ (- (+ z 1.0) z) (+ (sqrt z) t_1)))
        t_5)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0));
	double t_2 = t_1 - sqrt(z);
	double t_3 = sqrt((x + 1.0));
	double t_4 = t_3 - sqrt(x);
	double t_5 = sqrt((t + 1.0)) - sqrt(t);
	double t_6 = sqrt((1.0 + y));
	double t_7 = (t_4 + (t_6 - sqrt(y))) + t_2;
	double tmp;
	if (t_7 <= 0.04) {
		tmp = ((1.0 / (sqrt(x) + t_3)) + t_2) + t_5;
	} else if (t_7 <= 1.9999999999999996) {
		tmp = (((1.0 / (t_6 + sqrt(y))) + t_3) - sqrt(x)) + t_5;
	} else {
		tmp = (((1.0 - sqrt(y)) + t_4) + (((z + 1.0) - z) / (sqrt(z) + t_1))) + t_5;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0))
    t_2 = t_1 - sqrt(z)
    t_3 = sqrt((x + 1.0d0))
    t_4 = t_3 - sqrt(x)
    t_5 = sqrt((t + 1.0d0)) - sqrt(t)
    t_6 = sqrt((1.0d0 + y))
    t_7 = (t_4 + (t_6 - sqrt(y))) + t_2
    if (t_7 <= 0.04d0) then
        tmp = ((1.0d0 / (sqrt(x) + t_3)) + t_2) + t_5
    else if (t_7 <= 1.9999999999999996d0) then
        tmp = (((1.0d0 / (t_6 + sqrt(y))) + t_3) - sqrt(x)) + t_5
    else
        tmp = (((1.0d0 - sqrt(y)) + t_4) + (((z + 1.0d0) - z) / (sqrt(z) + t_1))) + t_5
    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((z + 1.0));
	double t_2 = t_1 - Math.sqrt(z);
	double t_3 = Math.sqrt((x + 1.0));
	double t_4 = t_3 - Math.sqrt(x);
	double t_5 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_6 = Math.sqrt((1.0 + y));
	double t_7 = (t_4 + (t_6 - Math.sqrt(y))) + t_2;
	double tmp;
	if (t_7 <= 0.04) {
		tmp = ((1.0 / (Math.sqrt(x) + t_3)) + t_2) + t_5;
	} else if (t_7 <= 1.9999999999999996) {
		tmp = (((1.0 / (t_6 + Math.sqrt(y))) + t_3) - Math.sqrt(x)) + t_5;
	} else {
		tmp = (((1.0 - Math.sqrt(y)) + t_4) + (((z + 1.0) - z) / (Math.sqrt(z) + t_1))) + t_5;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0))
	t_2 = t_1 - math.sqrt(z)
	t_3 = math.sqrt((x + 1.0))
	t_4 = t_3 - math.sqrt(x)
	t_5 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_6 = math.sqrt((1.0 + y))
	t_7 = (t_4 + (t_6 - math.sqrt(y))) + t_2
	tmp = 0
	if t_7 <= 0.04:
		tmp = ((1.0 / (math.sqrt(x) + t_3)) + t_2) + t_5
	elif t_7 <= 1.9999999999999996:
		tmp = (((1.0 / (t_6 + math.sqrt(y))) + t_3) - math.sqrt(x)) + t_5
	else:
		tmp = (((1.0 - math.sqrt(y)) + t_4) + (((z + 1.0) - z) / (math.sqrt(z) + t_1))) + t_5
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	t_2 = Float64(t_1 - sqrt(z))
	t_3 = sqrt(Float64(x + 1.0))
	t_4 = Float64(t_3 - sqrt(x))
	t_5 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_6 = sqrt(Float64(1.0 + y))
	t_7 = Float64(Float64(t_4 + Float64(t_6 - sqrt(y))) + t_2)
	tmp = 0.0
	if (t_7 <= 0.04)
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(x) + t_3)) + t_2) + t_5);
	elseif (t_7 <= 1.9999999999999996)
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(t_6 + sqrt(y))) + t_3) - sqrt(x)) + t_5);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - sqrt(y)) + t_4) + Float64(Float64(Float64(z + 1.0) - z) / Float64(sqrt(z) + t_1))) + t_5);
	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((z + 1.0));
	t_2 = t_1 - sqrt(z);
	t_3 = sqrt((x + 1.0));
	t_4 = t_3 - sqrt(x);
	t_5 = sqrt((t + 1.0)) - sqrt(t);
	t_6 = sqrt((1.0 + y));
	t_7 = (t_4 + (t_6 - sqrt(y))) + t_2;
	tmp = 0.0;
	if (t_7 <= 0.04)
		tmp = ((1.0 / (sqrt(x) + t_3)) + t_2) + t_5;
	elseif (t_7 <= 1.9999999999999996)
		tmp = (((1.0 / (t_6 + sqrt(y))) + t_3) - sqrt(x)) + t_5;
	else
		tmp = (((1.0 - sqrt(y)) + t_4) + (((z + 1.0) - z) / (sqrt(z) + t_1))) + t_5;
	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[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$7 = N[(N[(t$95$4 + N[(t$95$6 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]}, If[LessEqual[t$95$7, 0.04], N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + t$95$5), $MachinePrecision], If[LessEqual[t$95$7, 1.9999999999999996], N[(N[(N[(N[(1.0 / N[(t$95$6 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision], N[(N[(N[(N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] + N[(N[(N[(z + 1.0), $MachinePrecision] - z), $MachinePrecision] / N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.0400000000000000008

   1. Initial program 75.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 77.2

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

   4. Applied rewrites 77.2%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 84.2%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. lift-/.f64 N/A

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

      5. frac-add N/A

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

      6. lower-/.f64 N/A

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

   9. Applied rewrites 84.3%

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

   10. Taylor expanded in y around inf

       \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   11. 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. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-sqrt.f64 77.6

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

   12. Applied rewrites 77.6%

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

   if 0.0400000000000000008 < (+.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))) < 1.9999999999999996

   1. Initial program 94.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 95.0

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

   4. Applied rewrites 95.0%

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

   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. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 43.6

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

   7. Applied rewrites 43.6%

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

   if 1.9999999999999996 < (+.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 98.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 74.1

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

   5. Applied rewrites 74.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 74.4

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

   7. Applied rewrites 74.4%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right) \leq 0.04:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{x + 1}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right) \leq 1.9999999999999996:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \sqrt{x + 1}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{x + 1}\\ t_3 := t\_2 - \sqrt{x}\\ t_4 := \sqrt{t + 1}\\ t_5 := t\_4 - \sqrt{t}\\ t_6 := \sqrt{1 + y}\\ t_7 := \left(t\_3 + \left(t\_6 - \sqrt{y}\right)\right) + t\_1\\ \mathbf{if}\;t\_7 \leq 0.04:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + t\_2} + t\_1\right) + t\_5\\ \mathbf{elif}\;t\_7 \leq 1.9999999999999996:\\ \;\;\;\;\left(\left(\frac{1}{t\_6 + \sqrt{y}} + t\_2\right) - \sqrt{x}\right) + t\_5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{y}\right) + t\_3\right) + t\_1\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + t\_4}\\ \end{array} \end{array} \]

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 (+ z 1.0)) (sqrt z)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (- t_2 (sqrt x)))
        (t_4 (sqrt (+ t 1.0)))
        (t_5 (- t_4 (sqrt t)))
        (t_6 (sqrt (+ 1.0 y)))
        (t_7 (+ (+ t_3 (- t_6 (sqrt y))) t_1)))
   (if (<= t_7 0.04)
     (+ (+ (/ 1.0 (+ (sqrt x) t_2)) t_1) t_5)
     (if (<= t_7 1.9999999999999996)
       (+ (- (+ (/ 1.0 (+ t_6 (sqrt y))) t_2) (sqrt x)) t_5)
       (+
        (+ (+ (- 1.0 (sqrt y)) t_3) t_1)
        (/ (- (+ t 1.0) t) (+ (sqrt t) t_4)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = sqrt((x + 1.0));
	double t_3 = t_2 - sqrt(x);
	double t_4 = sqrt((t + 1.0));
	double t_5 = t_4 - sqrt(t);
	double t_6 = sqrt((1.0 + y));
	double t_7 = (t_3 + (t_6 - sqrt(y))) + t_1;
	double tmp;
	if (t_7 <= 0.04) {
		tmp = ((1.0 / (sqrt(x) + t_2)) + t_1) + t_5;
	} else if (t_7 <= 1.9999999999999996) {
		tmp = (((1.0 / (t_6 + sqrt(y))) + t_2) - sqrt(x)) + t_5;
	} else {
		tmp = (((1.0 - sqrt(y)) + t_3) + t_1) + (((t + 1.0) - t) / (sqrt(t) + t_4));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0)) - sqrt(z)
    t_2 = sqrt((x + 1.0d0))
    t_3 = t_2 - sqrt(x)
    t_4 = sqrt((t + 1.0d0))
    t_5 = t_4 - sqrt(t)
    t_6 = sqrt((1.0d0 + y))
    t_7 = (t_3 + (t_6 - sqrt(y))) + t_1
    if (t_7 <= 0.04d0) then
        tmp = ((1.0d0 / (sqrt(x) + t_2)) + t_1) + t_5
    else if (t_7 <= 1.9999999999999996d0) then
        tmp = (((1.0d0 / (t_6 + sqrt(y))) + t_2) - sqrt(x)) + t_5
    else
        tmp = (((1.0d0 - sqrt(y)) + t_3) + t_1) + (((t + 1.0d0) - t) / (sqrt(t) + t_4))
    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((z + 1.0)) - Math.sqrt(z);
	double t_2 = Math.sqrt((x + 1.0));
	double t_3 = t_2 - Math.sqrt(x);
	double t_4 = Math.sqrt((t + 1.0));
	double t_5 = t_4 - Math.sqrt(t);
	double t_6 = Math.sqrt((1.0 + y));
	double t_7 = (t_3 + (t_6 - Math.sqrt(y))) + t_1;
	double tmp;
	if (t_7 <= 0.04) {
		tmp = ((1.0 / (Math.sqrt(x) + t_2)) + t_1) + t_5;
	} else if (t_7 <= 1.9999999999999996) {
		tmp = (((1.0 / (t_6 + Math.sqrt(y))) + t_2) - Math.sqrt(x)) + t_5;
	} else {
		tmp = (((1.0 - Math.sqrt(y)) + t_3) + t_1) + (((t + 1.0) - t) / (Math.sqrt(t) + t_4));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0)) - math.sqrt(z)
	t_2 = math.sqrt((x + 1.0))
	t_3 = t_2 - math.sqrt(x)
	t_4 = math.sqrt((t + 1.0))
	t_5 = t_4 - math.sqrt(t)
	t_6 = math.sqrt((1.0 + y))
	t_7 = (t_3 + (t_6 - math.sqrt(y))) + t_1
	tmp = 0
	if t_7 <= 0.04:
		tmp = ((1.0 / (math.sqrt(x) + t_2)) + t_1) + t_5
	elif t_7 <= 1.9999999999999996:
		tmp = (((1.0 / (t_6 + math.sqrt(y))) + t_2) - math.sqrt(x)) + t_5
	else:
		tmp = (((1.0 - math.sqrt(y)) + t_3) + t_1) + (((t + 1.0) - t) / (math.sqrt(t) + t_4))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = Float64(t_2 - sqrt(x))
	t_4 = sqrt(Float64(t + 1.0))
	t_5 = Float64(t_4 - sqrt(t))
	t_6 = sqrt(Float64(1.0 + y))
	t_7 = Float64(Float64(t_3 + Float64(t_6 - sqrt(y))) + t_1)
	tmp = 0.0
	if (t_7 <= 0.04)
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(x) + t_2)) + t_1) + t_5);
	elseif (t_7 <= 1.9999999999999996)
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(t_6 + sqrt(y))) + t_2) - sqrt(x)) + t_5);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - sqrt(y)) + t_3) + t_1) + Float64(Float64(Float64(t + 1.0) - t) / Float64(sqrt(t) + t_4)));
	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((z + 1.0)) - sqrt(z);
	t_2 = sqrt((x + 1.0));
	t_3 = t_2 - sqrt(x);
	t_4 = sqrt((t + 1.0));
	t_5 = t_4 - sqrt(t);
	t_6 = sqrt((1.0 + y));
	t_7 = (t_3 + (t_6 - sqrt(y))) + t_1;
	tmp = 0.0;
	if (t_7 <= 0.04)
		tmp = ((1.0 / (sqrt(x) + t_2)) + t_1) + t_5;
	elseif (t_7 <= 1.9999999999999996)
		tmp = (((1.0 / (t_6 + sqrt(y))) + t_2) - sqrt(x)) + t_5;
	else
		tmp = (((1.0 - sqrt(y)) + t_3) + t_1) + (((t + 1.0) - t) / (sqrt(t) + t_4));
	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[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$7 = N[(N[(t$95$3 + N[(t$95$6 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$7, 0.04], N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$5), $MachinePrecision], If[LessEqual[t$95$7, 1.9999999999999996], N[(N[(N[(N[(1.0 / N[(t$95$6 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision], N[(N[(N[(N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(N[(N[(t + 1.0), $MachinePrecision] - t), $MachinePrecision] / N[(N[Sqrt[t], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.0400000000000000008

   1. Initial program 75.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 77.2

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

   4. Applied rewrites 77.2%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 84.2%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. lift-/.f64 N/A

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

      5. frac-add N/A

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

      6. lower-/.f64 N/A

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

   9. Applied rewrites 84.3%

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

   10. Taylor expanded in y around inf

       \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   11. 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. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-sqrt.f64 77.6

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

   12. Applied rewrites 77.6%

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

   if 0.0400000000000000008 < (+.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))) < 1.9999999999999996

   1. Initial program 94.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 95.0

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

   4. Applied rewrites 95.0%

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

   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. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 43.6

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

   7. Applied rewrites 43.6%

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

   if 1.9999999999999996 < (+.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 98.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 74.1

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

   5. Applied rewrites 74.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 74.3

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

   7. Applied rewrites 74.3%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right) \leq 0.04:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{x + 1}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right) \leq 1.9999999999999996:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \sqrt{x + 1}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{t + 1} - \sqrt{t}\\ t_4 := \sqrt{1 + y}\\ t_5 := \left(\left(t\_2 - \sqrt{x}\right) + \left(t\_4 - \sqrt{y}\right)\right) + t\_1\\ \mathbf{if}\;t\_5 \leq 0.04:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + t\_2} + t\_1\right) + t\_3\\ \mathbf{elif}\;t\_5 \leq 1.99985:\\ \;\;\;\;\left(\left(\frac{1}{t\_4 + \sqrt{y}} + t\_2\right) - \sqrt{x}\right) + t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\mathsf{fma}\left(0.5, x, 1\right) + t\_4\right) - \sqrt{y}\right) - \sqrt{x}\right) + t\_1\right) + t\_3\\ \end{array} \end{array} \]

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 (+ z 1.0)) (sqrt z)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_4 (sqrt (+ 1.0 y)))
        (t_5 (+ (+ (- t_2 (sqrt x)) (- t_4 (sqrt y))) t_1)))
   (if (<= t_5 0.04)
     (+ (+ (/ 1.0 (+ (sqrt x) t_2)) t_1) t_3)
     (if (<= t_5 1.99985)
       (+ (- (+ (/ 1.0 (+ t_4 (sqrt y))) t_2) (sqrt x)) t_3)
       (+ (+ (- (- (+ (fma 0.5 x 1.0) t_4) (sqrt y)) (sqrt x)) t_1) t_3)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = sqrt((x + 1.0));
	double t_3 = sqrt((t + 1.0)) - sqrt(t);
	double t_4 = sqrt((1.0 + y));
	double t_5 = ((t_2 - sqrt(x)) + (t_4 - sqrt(y))) + t_1;
	double tmp;
	if (t_5 <= 0.04) {
		tmp = ((1.0 / (sqrt(x) + t_2)) + t_1) + t_3;
	} else if (t_5 <= 1.99985) {
		tmp = (((1.0 / (t_4 + sqrt(y))) + t_2) - sqrt(x)) + t_3;
	} else {
		tmp = ((((fma(0.5, x, 1.0) + t_4) - sqrt(y)) - sqrt(x)) + t_1) + t_3;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_4 = sqrt(Float64(1.0 + y))
	t_5 = Float64(Float64(Float64(t_2 - sqrt(x)) + Float64(t_4 - sqrt(y))) + t_1)
	tmp = 0.0
	if (t_5 <= 0.04)
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(x) + t_2)) + t_1) + t_3);
	elseif (t_5 <= 1.99985)
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(t_4 + sqrt(y))) + t_2) - sqrt(x)) + t_3);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(fma(0.5, x, 1.0) + t_4) - sqrt(y)) - sqrt(x)) + t_1) + t_3);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$5, 0.04], N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision], If[LessEqual[t$95$5, 1.99985], N[(N[(N[(N[(1.0 / N[(t$95$4 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.5 * x + 1.0), $MachinePrecision] + t$95$4), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.0400000000000000008

   1. Initial program 75.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 77.2

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

   4. Applied rewrites 77.2%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 84.2%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. lift-/.f64 N/A

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

      5. frac-add N/A

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

      6. lower-/.f64 N/A

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

   9. Applied rewrites 84.3%

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

   10. Taylor expanded in y around inf

       \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   11. 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. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-sqrt.f64 77.6

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

   12. Applied rewrites 77.6%

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

   if 0.0400000000000000008 < (+.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))) < 1.9998499999999999

   1. Initial program 94.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 95.0

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

   4. Applied rewrites 95.0%

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

   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. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 43.1

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

   7. Applied rewrites 43.1%

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

   if 1.9998499999999999 < (+.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 97.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-sqrt.f64 58.3

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

   5. Applied rewrites 58.3%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right) \leq 0.04:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{x + 1}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right) \leq 1.99985:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \sqrt{x + 1}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\mathsf{fma}\left(0.5, x, 1\right) + \sqrt{1 + y}\right) - \sqrt{y}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{t + 1} - \sqrt{t}\\ t_4 := \sqrt{1 + y}\\ t_5 := \left(\left(t\_2 - \sqrt{x}\right) + \left(t\_4 - \sqrt{y}\right)\right) + t\_1\\ \mathbf{if}\;t\_5 \leq 0.04:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + t\_2} + t\_1\right) + t\_3\\ \mathbf{elif}\;t\_5 \leq 1.9999999999999996:\\ \;\;\;\;\left(\left(\frac{1}{t\_4 + \sqrt{y}} + t\_2\right) - \sqrt{x}\right) + t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{x}\right) + \left(1 - \sqrt{y}\right)\right) + t\_1\right) + t\_3\\ \end{array} \end{array} \]

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 (+ z 1.0)) (sqrt z)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_4 (sqrt (+ 1.0 y)))
        (t_5 (+ (+ (- t_2 (sqrt x)) (- t_4 (sqrt y))) t_1)))
   (if (<= t_5 0.04)
     (+ (+ (/ 1.0 (+ (sqrt x) t_2)) t_1) t_3)
     (if (<= t_5 1.9999999999999996)
       (+ (- (+ (/ 1.0 (+ t_4 (sqrt y))) t_2) (sqrt x)) t_3)
       (+ (+ (+ (- 1.0 (sqrt x)) (- 1.0 (sqrt y))) t_1) t_3)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = sqrt((x + 1.0));
	double t_3 = sqrt((t + 1.0)) - sqrt(t);
	double t_4 = sqrt((1.0 + y));
	double t_5 = ((t_2 - sqrt(x)) + (t_4 - sqrt(y))) + t_1;
	double tmp;
	if (t_5 <= 0.04) {
		tmp = ((1.0 / (sqrt(x) + t_2)) + t_1) + t_3;
	} else if (t_5 <= 1.9999999999999996) {
		tmp = (((1.0 / (t_4 + sqrt(y))) + t_2) - sqrt(x)) + t_3;
	} else {
		tmp = (((1.0 - sqrt(x)) + (1.0 - sqrt(y))) + t_1) + t_3;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0)) - sqrt(z)
    t_2 = sqrt((x + 1.0d0))
    t_3 = sqrt((t + 1.0d0)) - sqrt(t)
    t_4 = sqrt((1.0d0 + y))
    t_5 = ((t_2 - sqrt(x)) + (t_4 - sqrt(y))) + t_1
    if (t_5 <= 0.04d0) then
        tmp = ((1.0d0 / (sqrt(x) + t_2)) + t_1) + t_3
    else if (t_5 <= 1.9999999999999996d0) then
        tmp = (((1.0d0 / (t_4 + sqrt(y))) + t_2) - sqrt(x)) + t_3
    else
        tmp = (((1.0d0 - sqrt(x)) + (1.0d0 - sqrt(y))) + t_1) + t_3
    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((z + 1.0)) - Math.sqrt(z);
	double t_2 = Math.sqrt((x + 1.0));
	double t_3 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_4 = Math.sqrt((1.0 + y));
	double t_5 = ((t_2 - Math.sqrt(x)) + (t_4 - Math.sqrt(y))) + t_1;
	double tmp;
	if (t_5 <= 0.04) {
		tmp = ((1.0 / (Math.sqrt(x) + t_2)) + t_1) + t_3;
	} else if (t_5 <= 1.9999999999999996) {
		tmp = (((1.0 / (t_4 + Math.sqrt(y))) + t_2) - Math.sqrt(x)) + t_3;
	} else {
		tmp = (((1.0 - Math.sqrt(x)) + (1.0 - Math.sqrt(y))) + t_1) + t_3;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0)) - math.sqrt(z)
	t_2 = math.sqrt((x + 1.0))
	t_3 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_4 = math.sqrt((1.0 + y))
	t_5 = ((t_2 - math.sqrt(x)) + (t_4 - math.sqrt(y))) + t_1
	tmp = 0
	if t_5 <= 0.04:
		tmp = ((1.0 / (math.sqrt(x) + t_2)) + t_1) + t_3
	elif t_5 <= 1.9999999999999996:
		tmp = (((1.0 / (t_4 + math.sqrt(y))) + t_2) - math.sqrt(x)) + t_3
	else:
		tmp = (((1.0 - math.sqrt(x)) + (1.0 - math.sqrt(y))) + t_1) + t_3
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_4 = sqrt(Float64(1.0 + y))
	t_5 = Float64(Float64(Float64(t_2 - sqrt(x)) + Float64(t_4 - sqrt(y))) + t_1)
	tmp = 0.0
	if (t_5 <= 0.04)
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(x) + t_2)) + t_1) + t_3);
	elseif (t_5 <= 1.9999999999999996)
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(t_4 + sqrt(y))) + t_2) - sqrt(x)) + t_3);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - sqrt(x)) + Float64(1.0 - sqrt(y))) + t_1) + t_3);
	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((z + 1.0)) - sqrt(z);
	t_2 = sqrt((x + 1.0));
	t_3 = sqrt((t + 1.0)) - sqrt(t);
	t_4 = sqrt((1.0 + y));
	t_5 = ((t_2 - sqrt(x)) + (t_4 - sqrt(y))) + t_1;
	tmp = 0.0;
	if (t_5 <= 0.04)
		tmp = ((1.0 / (sqrt(x) + t_2)) + t_1) + t_3;
	elseif (t_5 <= 1.9999999999999996)
		tmp = (((1.0 / (t_4 + sqrt(y))) + t_2) - sqrt(x)) + t_3;
	else
		tmp = (((1.0 - sqrt(x)) + (1.0 - sqrt(y))) + t_1) + t_3;
	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[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$5, 0.04], N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision], If[LessEqual[t$95$5, 1.9999999999999996], N[(N[(N[(N[(1.0 / N[(t$95$4 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], N[(N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.0400000000000000008

   1. Initial program 75.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 77.2

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

   4. Applied rewrites 77.2%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in y around 0

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{1}}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 84.2%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{1}}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \color{blue}{\frac{1}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. frac-add N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{y} + \sqrt{y + 1}\right) + \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot 1}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y} + \sqrt{y + 1}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{y} + \sqrt{y + 1}\right) + \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot 1}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y} + \sqrt{y + 1}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Applied rewrites 84.3%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(\left(1 + x\right) - x, \sqrt{y + 1} + \sqrt{y}, \left(\sqrt{1 + x} + \sqrt{x}\right) \cdot 1\right)}{\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Taylor expanded in y around inf

       \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   11. 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. +-commutative N/A

          \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + x}} + \sqrt{x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. lower-sqrt.f64 77.6

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \color{blue}{\sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. Applied rewrites 77.6%

       \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.0400000000000000008 < (+.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))) < 1.9999999999999996

   1. Initial program 94.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{\sqrt{y}} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \sqrt{y} \cdot \color{blue}{\sqrt{y}}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\color{blue}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 95.0

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\color{blue}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 95.0%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   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. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{\sqrt{y} + \sqrt{1 + y}}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\left(\frac{1}{\color{blue}{\sqrt{1 + y} + \sqrt{y}}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\color{blue}{\sqrt{1 + y} + \sqrt{y}}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\color{blue}{\sqrt{1 + y}} + \sqrt{y}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\sqrt{1 + y} + \color{blue}{\sqrt{y}}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-sqrt.f64 43.6

          \[\leadsto \left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 43.6%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \sqrt{1 + x}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.9999999999999996 < (+.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 98.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(1 - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(1 - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 74.1

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(1 - \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 74.1%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(1 - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(1 - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(1 - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 51.5

         \[\leadsto \left(\left(\left(1 - \color{blue}{\sqrt{x}}\right) + \left(1 - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 51.5%

      \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(1 - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right) \leq 0.04:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{x + 1}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right) \leq 1.9999999999999996:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \sqrt{x + 1}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{x}\right) + \left(1 - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 93.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{t + 1} - \sqrt{t}\\ t_4 := \sqrt{1 + y}\\ t_5 := \left(\left(t\_2 - \sqrt{x}\right) + \left(t\_4 - \sqrt{y}\right)\right) + t\_1\\ \mathbf{if}\;t\_5 \leq 1.0001:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, -\sqrt{x}\right) + t\_2\\ \mathbf{elif}\;t\_5 \leq 2.0001:\\ \;\;\;\;\left(\left(\left(t\_4 - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right) + \frac{0.5}{\sqrt{z}}\right) + t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{x}\right) + \left(1 - \sqrt{y}\right)\right) + t\_1\right) + t\_3\\ \end{array} \end{array} \]

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 (+ z 1.0)) (sqrt z)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_4 (sqrt (+ 1.0 y)))
        (t_5 (+ (+ (- t_2 (sqrt x)) (- t_4 (sqrt y))) t_1)))
   (if (<= t_5 1.0001)
     (+ (fma 0.5 (+ (sqrt (/ 1.0 z)) (sqrt (/ 1.0 y))) (- (sqrt x))) t_2)
     (if (<= t_5 2.0001)
       (+ (+ (+ (- t_4 (+ (sqrt x) (sqrt y))) 1.0) (/ 0.5 (sqrt z))) t_3)
       (+ (+ (+ (- 1.0 (sqrt x)) (- 1.0 (sqrt y))) t_1) t_3)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = sqrt((x + 1.0));
	double t_3 = sqrt((t + 1.0)) - sqrt(t);
	double t_4 = sqrt((1.0 + y));
	double t_5 = ((t_2 - sqrt(x)) + (t_4 - sqrt(y))) + t_1;
	double tmp;
	if (t_5 <= 1.0001) {
		tmp = fma(0.5, (sqrt((1.0 / z)) + sqrt((1.0 / y))), -sqrt(x)) + t_2;
	} else if (t_5 <= 2.0001) {
		tmp = (((t_4 - (sqrt(x) + sqrt(y))) + 1.0) + (0.5 / sqrt(z))) + t_3;
	} else {
		tmp = (((1.0 - sqrt(x)) + (1.0 - sqrt(y))) + t_1) + t_3;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_4 = sqrt(Float64(1.0 + y))
	t_5 = Float64(Float64(Float64(t_2 - sqrt(x)) + Float64(t_4 - sqrt(y))) + t_1)
	tmp = 0.0
	if (t_5 <= 1.0001)
		tmp = Float64(fma(0.5, Float64(sqrt(Float64(1.0 / z)) + sqrt(Float64(1.0 / y))), Float64(-sqrt(x))) + t_2);
	elseif (t_5 <= 2.0001)
		tmp = Float64(Float64(Float64(Float64(t_4 - Float64(sqrt(x) + sqrt(y))) + 1.0) + Float64(0.5 / sqrt(z))) + t_3);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - sqrt(x)) + Float64(1.0 - sqrt(y))) + t_1) + t_3);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$5, 1.0001], N[(N[(0.5 * N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + (-N[Sqrt[x], $MachinePrecision])), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[t$95$5, 2.0001], N[(N[(N[(N[(t$95$4 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(0.5 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], N[(N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $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))) < 1.00009999999999999

   1. Initial program 90.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\color{blue}{\sqrt{z}} + \sqrt{y}\right) + \sqrt{x}\right) \]

      17. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \color{blue}{\sqrt{y}}\right) + \sqrt{x}\right) \]

      18. lower-sqrt.f64 3.1

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \color{blue}{\sqrt{x}}\right) \]

   5. Applied rewrites 3.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 20.9%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{1 + y}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right)} \]

   8. Taylor expanded in z around inf

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 21.3%

       \[\leadsto \sqrt{x + 1} + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   11. Taylor expanded in y around inf

       \[\leadsto \sqrt{x + 1} + \left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \sqrt{x}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 21.7%

       \[\leadsto \sqrt{x + 1} + \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, -\sqrt{x}\right) \]

   if 1.00009999999999999 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.00010000000000021

   1. Initial program 97.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{z}} \cdot \frac{1}{2}} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + x} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{z}}}, \frac{1}{2}, \sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{z}}}, \frac{1}{2}, \sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \color{blue}{\sqrt{1 + y} + \sqrt{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \color{blue}{\sqrt{1 + y} + \sqrt{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \color{blue}{\sqrt{1 + y}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{\color{blue}{1 + y}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y} + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y} + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y} + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y} + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. lower-sqrt.f64 39.9

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\sqrt{y} + \color{blue}{\sqrt{x}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 39.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(1 + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 38.8%

      \[\leadsto \left(1 + \color{blue}{\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 37.5%

       \[\leadsto \left(\frac{0.5}{\sqrt{z}} + \left(\left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right) + \color{blue}{1}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 2.00010000000000021 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 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 y around 0

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(1 - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(1 - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 91.3

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(1 - \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 91.3%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(1 - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(1 - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(1 - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 87.6

         \[\leadsto \left(\left(\left(1 - \color{blue}{\sqrt{x}}\right) + \left(1 - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 87.6%

      \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(1 - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right) \leq 1.0001:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, -\sqrt{x}\right) + \sqrt{x + 1}\\ \mathbf{elif}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right) \leq 2.0001:\\ \;\;\;\;\left(\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right) + \frac{0.5}{\sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{x}\right) + \left(1 - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 93.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{\frac{1}{z}}\\ t_4 := \sqrt{1 + y}\\ t_5 := \left(\left(t\_2 - \sqrt{x}\right) + \left(t\_4 - \sqrt{y}\right)\right) + t\_1\\ \mathbf{if}\;t\_5 \leq 1.0001:\\ \;\;\;\;\mathsf{fma}\left(0.5, t\_3 + \sqrt{\frac{1}{y}}, -\sqrt{x}\right) + t\_2\\ \mathbf{elif}\;t\_5 \leq 2.0001:\\ \;\;\;\;\left(\mathsf{fma}\left(t\_3, 0.5, t\_4\right) + t\_2\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{x}\right) + \left(1 - \sqrt{y}\right)\right) + t\_1\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \end{array} \]

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 (+ z 1.0)) (sqrt z)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (sqrt (/ 1.0 z)))
        (t_4 (sqrt (+ 1.0 y)))
        (t_5 (+ (+ (- t_2 (sqrt x)) (- t_4 (sqrt y))) t_1)))
   (if (<= t_5 1.0001)
     (+ (fma 0.5 (+ t_3 (sqrt (/ 1.0 y))) (- (sqrt x))) t_2)
     (if (<= t_5 2.0001)
       (- (+ (fma t_3 0.5 t_4) t_2) (+ (sqrt x) (sqrt y)))
       (+
        (+ (+ (- 1.0 (sqrt x)) (- 1.0 (sqrt y))) t_1)
        (- (sqrt (+ t 1.0)) (sqrt t)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = sqrt((x + 1.0));
	double t_3 = sqrt((1.0 / z));
	double t_4 = sqrt((1.0 + y));
	double t_5 = ((t_2 - sqrt(x)) + (t_4 - sqrt(y))) + t_1;
	double tmp;
	if (t_5 <= 1.0001) {
		tmp = fma(0.5, (t_3 + sqrt((1.0 / y))), -sqrt(x)) + t_2;
	} else if (t_5 <= 2.0001) {
		tmp = (fma(t_3, 0.5, t_4) + t_2) - (sqrt(x) + sqrt(y));
	} else {
		tmp = (((1.0 - sqrt(x)) + (1.0 - sqrt(y))) + t_1) + (sqrt((t + 1.0)) - sqrt(t));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = sqrt(Float64(1.0 / z))
	t_4 = sqrt(Float64(1.0 + y))
	t_5 = Float64(Float64(Float64(t_2 - sqrt(x)) + Float64(t_4 - sqrt(y))) + t_1)
	tmp = 0.0
	if (t_5 <= 1.0001)
		tmp = Float64(fma(0.5, Float64(t_3 + sqrt(Float64(1.0 / y))), Float64(-sqrt(x))) + t_2);
	elseif (t_5 <= 2.0001)
		tmp = Float64(Float64(fma(t_3, 0.5, t_4) + t_2) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - sqrt(x)) + Float64(1.0 - sqrt(y))) + t_1) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)));
	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[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$5, 1.0001], N[(N[(0.5 * N[(t$95$3 + N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + (-N[Sqrt[x], $MachinePrecision])), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[t$95$5, 2.0001], N[(N[(N[(t$95$3 * 0.5 + t$95$4), $MachinePrecision] + t$95$2), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $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))) < 1.00009999999999999

   1. Initial program 90.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\color{blue}{\sqrt{z}} + \sqrt{y}\right) + \sqrt{x}\right) \]

      17. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \color{blue}{\sqrt{y}}\right) + \sqrt{x}\right) \]

      18. lower-sqrt.f64 3.1

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \color{blue}{\sqrt{x}}\right) \]

   5. Applied rewrites 3.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 20.9%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{1 + y}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right)} \]

   8. Taylor expanded in z around inf

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 21.3%

       \[\leadsto \sqrt{x + 1} + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   11. Taylor expanded in y around inf

       \[\leadsto \sqrt{x + 1} + \left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \sqrt{x}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 21.7%

       \[\leadsto \sqrt{x + 1} + \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, -\sqrt{x}\right) \]

   if 1.00009999999999999 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.00010000000000021

   1. Initial program 97.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\color{blue}{\sqrt{z}} + \sqrt{y}\right) + \sqrt{x}\right) \]

      17. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \color{blue}{\sqrt{y}}\right) + \sqrt{x}\right) \]

      18. lower-sqrt.f64 8.8

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \color{blue}{\sqrt{x}}\right) \]

   5. Applied rewrites 8.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 25.4%

      \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} \]

   if 2.00010000000000021 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 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 y around 0

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(1 - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(1 - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 91.3

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(1 - \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 91.3%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(1 - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(1 - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(1 - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 87.6

         \[\leadsto \left(\left(\left(1 - \color{blue}{\sqrt{x}}\right) + \left(1 - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 87.6%

      \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(1 - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 32.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right) \leq 1.0001:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, -\sqrt{x}\right) + \sqrt{x + 1}\\ \mathbf{elif}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right) \leq 2.0001:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{x}\right) + \left(1 - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 88.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := \sqrt{x} + \sqrt{y}\\ t_3 := \sqrt{x + 1}\\ t_4 := \sqrt{\frac{1}{z}}\\ t_5 := \sqrt{1 + y}\\ t_6 := \left(\left(t\_3 - \sqrt{x}\right) + \left(t\_5 - \sqrt{y}\right)\right) + \left(t\_1 - \sqrt{z}\right)\\ \mathbf{if}\;t\_6 \leq 1.0001:\\ \;\;\;\;\mathsf{fma}\left(0.5, t\_4 + \sqrt{\frac{1}{y}}, -\sqrt{x}\right) + t\_3\\ \mathbf{elif}\;t\_6 \leq 2.0001:\\ \;\;\;\;\left(\mathsf{fma}\left(t\_4, 0.5, t\_5\right) + t\_3\right) - t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t\_5 + t\_1\right) + t\_3\right) - \sqrt{z}\right) - t\_2\\ \end{array} \end{array} \]

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 (+ z 1.0)))
        (t_2 (+ (sqrt x) (sqrt y)))
        (t_3 (sqrt (+ x 1.0)))
        (t_4 (sqrt (/ 1.0 z)))
        (t_5 (sqrt (+ 1.0 y)))
        (t_6 (+ (+ (- t_3 (sqrt x)) (- t_5 (sqrt y))) (- t_1 (sqrt z)))))
   (if (<= t_6 1.0001)
     (+ (fma 0.5 (+ t_4 (sqrt (/ 1.0 y))) (- (sqrt x))) t_3)
     (if (<= t_6 2.0001)
       (- (+ (fma t_4 0.5 t_5) t_3) t_2)
       (- (- (+ (+ t_5 t_1) t_3) (sqrt z)) t_2)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0));
	double t_2 = sqrt(x) + sqrt(y);
	double t_3 = sqrt((x + 1.0));
	double t_4 = sqrt((1.0 / z));
	double t_5 = sqrt((1.0 + y));
	double t_6 = ((t_3 - sqrt(x)) + (t_5 - sqrt(y))) + (t_1 - sqrt(z));
	double tmp;
	if (t_6 <= 1.0001) {
		tmp = fma(0.5, (t_4 + sqrt((1.0 / y))), -sqrt(x)) + t_3;
	} else if (t_6 <= 2.0001) {
		tmp = (fma(t_4, 0.5, t_5) + t_3) - t_2;
	} else {
		tmp = (((t_5 + t_1) + t_3) - sqrt(z)) - t_2;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	t_2 = Float64(sqrt(x) + sqrt(y))
	t_3 = sqrt(Float64(x + 1.0))
	t_4 = sqrt(Float64(1.0 / z))
	t_5 = sqrt(Float64(1.0 + y))
	t_6 = Float64(Float64(Float64(t_3 - sqrt(x)) + Float64(t_5 - sqrt(y))) + Float64(t_1 - sqrt(z)))
	tmp = 0.0
	if (t_6 <= 1.0001)
		tmp = Float64(fma(0.5, Float64(t_4 + sqrt(Float64(1.0 / y))), Float64(-sqrt(x))) + t_3);
	elseif (t_6 <= 2.0001)
		tmp = Float64(Float64(fma(t_4, 0.5, t_5) + t_3) - t_2);
	else
		tmp = Float64(Float64(Float64(Float64(t_5 + t_1) + t_3) - sqrt(z)) - t_2);
	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[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$5 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 1.0001], N[(N[(0.5 * N[(t$95$4 + N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + (-N[Sqrt[x], $MachinePrecision])), $MachinePrecision] + t$95$3), $MachinePrecision], If[LessEqual[t$95$6, 2.0001], N[(N[(N[(t$95$4 * 0.5 + t$95$5), $MachinePrecision] + t$95$3), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[(N[(N[(t$95$5 + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - t$95$2), $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))) < 1.00009999999999999

   1. Initial program 90.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\color{blue}{\sqrt{z}} + \sqrt{y}\right) + \sqrt{x}\right) \]

      17. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \color{blue}{\sqrt{y}}\right) + \sqrt{x}\right) \]

      18. lower-sqrt.f64 3.1

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \color{blue}{\sqrt{x}}\right) \]

   5. Applied rewrites 3.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 20.9%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{1 + y}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right)} \]

   8. Taylor expanded in z around inf

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 21.3%

       \[\leadsto \sqrt{x + 1} + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   11. Taylor expanded in y around inf

       \[\leadsto \sqrt{x + 1} + \left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \sqrt{x}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 21.7%

       \[\leadsto \sqrt{x + 1} + \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, -\sqrt{x}\right) \]

   if 1.00009999999999999 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.00010000000000021

   1. Initial program 97.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\color{blue}{\sqrt{z}} + \sqrt{y}\right) + \sqrt{x}\right) \]

      17. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \color{blue}{\sqrt{y}}\right) + \sqrt{x}\right) \]

      18. lower-sqrt.f64 8.8

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \color{blue}{\sqrt{x}}\right) \]

   5. Applied rewrites 8.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 25.4%

      \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} \]

   if 2.00010000000000021 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 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. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\color{blue}{\sqrt{z}} + \sqrt{y}\right) + \sqrt{x}\right) \]

      17. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \color{blue}{\sqrt{y}}\right) + \sqrt{x}\right) \]

      18. lower-sqrt.f64 62.6

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \color{blue}{\sqrt{x}}\right) \]

   5. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} + \left(\sqrt{z + 1} + \sqrt{1 + y}\right)\right) - \sqrt{z}\right) - \left(\sqrt{y} + \sqrt{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right) \leq 1.0001:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, -\sqrt{x}\right) + \sqrt{x + 1}\\ \mathbf{elif}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right) \leq 2.0001:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sqrt{1 + y} + \sqrt{z + 1}\right) + \sqrt{x + 1}\right) - \sqrt{z}\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 88.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{\frac{1}{z}}\\ t_4 := \sqrt{1 + y}\\ t_5 := \left(\left(t\_2 - \sqrt{x}\right) + \left(t\_4 - \sqrt{y}\right)\right) + \left(t\_1 - \sqrt{z}\right)\\ \mathbf{if}\;t\_5 \leq 1.0001:\\ \;\;\;\;\mathsf{fma}\left(0.5, t\_3 + \sqrt{\frac{1}{y}}, -\sqrt{x}\right) + t\_2\\ \mathbf{elif}\;t\_5 \leq 2.0001:\\ \;\;\;\;\left(\mathsf{fma}\left(t\_3, 0.5, t\_4\right) + t\_2\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_4 + 1\right) + t\_1\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\\ \end{array} \end{array} \]

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 (+ z 1.0)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (sqrt (/ 1.0 z)))
        (t_4 (sqrt (+ 1.0 y)))
        (t_5 (+ (+ (- t_2 (sqrt x)) (- t_4 (sqrt y))) (- t_1 (sqrt z)))))
   (if (<= t_5 1.0001)
     (+ (fma 0.5 (+ t_3 (sqrt (/ 1.0 y))) (- (sqrt x))) t_2)
     (if (<= t_5 2.0001)
       (- (+ (fma t_3 0.5 t_4) t_2) (+ (sqrt x) (sqrt y)))
       (- (+ (+ t_4 1.0) t_1) (+ (+ (sqrt z) (sqrt y)) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0));
	double t_2 = sqrt((x + 1.0));
	double t_3 = sqrt((1.0 / z));
	double t_4 = sqrt((1.0 + y));
	double t_5 = ((t_2 - sqrt(x)) + (t_4 - sqrt(y))) + (t_1 - sqrt(z));
	double tmp;
	if (t_5 <= 1.0001) {
		tmp = fma(0.5, (t_3 + sqrt((1.0 / y))), -sqrt(x)) + t_2;
	} else if (t_5 <= 2.0001) {
		tmp = (fma(t_3, 0.5, t_4) + t_2) - (sqrt(x) + sqrt(y));
	} else {
		tmp = ((t_4 + 1.0) + t_1) - ((sqrt(z) + sqrt(y)) + sqrt(x));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = sqrt(Float64(1.0 / z))
	t_4 = sqrt(Float64(1.0 + y))
	t_5 = Float64(Float64(Float64(t_2 - sqrt(x)) + Float64(t_4 - sqrt(y))) + Float64(t_1 - sqrt(z)))
	tmp = 0.0
	if (t_5 <= 1.0001)
		tmp = Float64(fma(0.5, Float64(t_3 + sqrt(Float64(1.0 / y))), Float64(-sqrt(x))) + t_2);
	elseif (t_5 <= 2.0001)
		tmp = Float64(Float64(fma(t_3, 0.5, t_4) + t_2) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(Float64(Float64(t_4 + 1.0) + t_1) - Float64(Float64(sqrt(z) + sqrt(y)) + sqrt(x)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 1.0001], N[(N[(0.5 * N[(t$95$3 + N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + (-N[Sqrt[x], $MachinePrecision])), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[t$95$5, 2.0001], N[(N[(N[(t$95$3 * 0.5 + t$95$4), $MachinePrecision] + t$95$2), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$4 + 1.0), $MachinePrecision] + t$95$1), $MachinePrecision] - N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $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))) < 1.00009999999999999

   1. Initial program 90.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\color{blue}{\sqrt{z}} + \sqrt{y}\right) + \sqrt{x}\right) \]

      17. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \color{blue}{\sqrt{y}}\right) + \sqrt{x}\right) \]

      18. lower-sqrt.f64 3.1

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \color{blue}{\sqrt{x}}\right) \]

   5. Applied rewrites 3.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 20.9%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{1 + y}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right)} \]

   8. Taylor expanded in z around inf

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 21.3%

       \[\leadsto \sqrt{x + 1} + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   11. Taylor expanded in y around inf

       \[\leadsto \sqrt{x + 1} + \left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \sqrt{x}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 21.7%

       \[\leadsto \sqrt{x + 1} + \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, -\sqrt{x}\right) \]

   if 1.00009999999999999 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.00010000000000021

   1. Initial program 97.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\color{blue}{\sqrt{z}} + \sqrt{y}\right) + \sqrt{x}\right) \]

      17. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \color{blue}{\sqrt{y}}\right) + \sqrt{x}\right) \]

      18. lower-sqrt.f64 8.8

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \color{blue}{\sqrt{x}}\right) \]

   5. Applied rewrites 8.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 25.4%

      \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} \]

   if 2.00010000000000021 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 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. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\color{blue}{\sqrt{z}} + \sqrt{y}\right) + \sqrt{x}\right) \]

      17. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \color{blue}{\sqrt{y}}\right) + \sqrt{x}\right) \]

      18. lower-sqrt.f64 62.6

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \color{blue}{\sqrt{x}}\right) \]

   5. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \sqrt{z} - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 2.0%

      \[\leadsto \sqrt{z} - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 58.9%

       \[\leadsto \left(\left(\sqrt{1 + y} + 1\right) + \sqrt{z + 1}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right) \leq 1.0001:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, -\sqrt{x}\right) + \sqrt{x + 1}\\ \mathbf{elif}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right) \leq 2.0001:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{1 + y} + 1\right) + \sqrt{z + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 95.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{x + 1}\\ t_3 := \left(\left(t\_2 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + t\_1\\ t_4 := \sqrt{t + 1} - \sqrt{t}\\ \mathbf{if}\;t\_3 \leq 0.04:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + t\_2} + t\_1\right) + t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_3 + t\_4\\ \end{array} \end{array} \]

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 (+ z 1.0)) (sqrt z)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (+ (+ (- t_2 (sqrt x)) (- (sqrt (+ 1.0 y)) (sqrt y))) t_1))
        (t_4 (- (sqrt (+ t 1.0)) (sqrt t))))
   (if (<= t_3 0.04) (+ (+ (/ 1.0 (+ (sqrt x) t_2)) t_1) t_4) (+ t_3 t_4))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = sqrt((x + 1.0));
	double t_3 = ((t_2 - sqrt(x)) + (sqrt((1.0 + y)) - sqrt(y))) + t_1;
	double t_4 = sqrt((t + 1.0)) - sqrt(t);
	double tmp;
	if (t_3 <= 0.04) {
		tmp = ((1.0 / (sqrt(x) + t_2)) + t_1) + t_4;
	} else {
		tmp = t_3 + t_4;
	}
	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((z + 1.0d0)) - sqrt(z)
    t_2 = sqrt((x + 1.0d0))
    t_3 = ((t_2 - sqrt(x)) + (sqrt((1.0d0 + y)) - sqrt(y))) + t_1
    t_4 = sqrt((t + 1.0d0)) - sqrt(t)
    if (t_3 <= 0.04d0) then
        tmp = ((1.0d0 / (sqrt(x) + t_2)) + t_1) + t_4
    else
        tmp = t_3 + t_4
    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((z + 1.0)) - Math.sqrt(z);
	double t_2 = Math.sqrt((x + 1.0));
	double t_3 = ((t_2 - Math.sqrt(x)) + (Math.sqrt((1.0 + y)) - Math.sqrt(y))) + t_1;
	double t_4 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double tmp;
	if (t_3 <= 0.04) {
		tmp = ((1.0 / (Math.sqrt(x) + t_2)) + t_1) + t_4;
	} else {
		tmp = t_3 + t_4;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0)) - math.sqrt(z)
	t_2 = math.sqrt((x + 1.0))
	t_3 = ((t_2 - math.sqrt(x)) + (math.sqrt((1.0 + y)) - math.sqrt(y))) + t_1
	t_4 = math.sqrt((t + 1.0)) - math.sqrt(t)
	tmp = 0
	if t_3 <= 0.04:
		tmp = ((1.0 / (math.sqrt(x) + t_2)) + t_1) + t_4
	else:
		tmp = t_3 + t_4
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = Float64(Float64(Float64(t_2 - sqrt(x)) + Float64(sqrt(Float64(1.0 + y)) - sqrt(y))) + t_1)
	t_4 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	tmp = 0.0
	if (t_3 <= 0.04)
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(x) + t_2)) + t_1) + t_4);
	else
		tmp = Float64(t_3 + t_4);
	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((z + 1.0)) - sqrt(z);
	t_2 = sqrt((x + 1.0));
	t_3 = ((t_2 - sqrt(x)) + (sqrt((1.0 + y)) - sqrt(y))) + t_1;
	t_4 = sqrt((t + 1.0)) - sqrt(t);
	tmp = 0.0;
	if (t_3 <= 0.04)
		tmp = ((1.0 / (sqrt(x) + t_2)) + t_1) + t_4;
	else
		tmp = t_3 + t_4;
	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[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(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] + t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.04], N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$4), $MachinePrecision], N[(t$95$3 + t$95$4), $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))) < 0.0400000000000000008

   1. Initial program 75.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{\sqrt{y}} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \sqrt{y} \cdot \color{blue}{\sqrt{y}}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\color{blue}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 77.2

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\color{blue}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 77.2%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in y around 0

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{1}}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 84.2%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{1}}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \color{blue}{\frac{1}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. frac-add N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{y} + \sqrt{y + 1}\right) + \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot 1}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y} + \sqrt{y + 1}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{y} + \sqrt{y + 1}\right) + \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot 1}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y} + \sqrt{y + 1}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Applied rewrites 84.3%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(\left(1 + x\right) - x, \sqrt{y + 1} + \sqrt{y}, \left(\sqrt{1 + x} + \sqrt{x}\right) \cdot 1\right)}{\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Taylor expanded in y around inf

       \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   11. 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. +-commutative N/A

          \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + x}} + \sqrt{x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. lower-sqrt.f64 77.6

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \color{blue}{\sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. Applied rewrites 77.6%

       \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.0400000000000000008 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 96.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right) \leq 0.04:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{x + 1}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 98.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1} - \sqrt{t}\\ t_2 := \sqrt{x + 1}\\ t_3 := t\_2 - \sqrt{x}\\ t_4 := \sqrt{z + 1} - \sqrt{z}\\ t_5 := \sqrt{1 + y}\\ \mathbf{if}\;t\_3 + \left(t\_5 - \sqrt{y}\right) \leq 0.04:\\ \;\;\;\;t\_1 + \left(t\_4 + \mathsf{fma}\left(\sqrt{\frac{1}{y}}, 0.5, \frac{1}{\sqrt{x} + t\_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{t\_5 + \sqrt{y}} + t\_3\right) + t\_4\right) + t\_1\\ \end{array} \end{array} \]

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 (+ t 1.0)) (sqrt t)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (- t_2 (sqrt x)))
        (t_4 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_5 (sqrt (+ 1.0 y))))
   (if (<= (+ t_3 (- t_5 (sqrt y))) 0.04)
     (+ t_1 (+ t_4 (fma (sqrt (/ 1.0 y)) 0.5 (/ 1.0 (+ (sqrt x) t_2)))))
     (+ (+ (+ (/ 1.0 (+ t_5 (sqrt y))) t_3) t_4) t_1))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((t + 1.0)) - sqrt(t);
	double t_2 = sqrt((x + 1.0));
	double t_3 = t_2 - sqrt(x);
	double t_4 = sqrt((z + 1.0)) - sqrt(z);
	double t_5 = sqrt((1.0 + y));
	double tmp;
	if ((t_3 + (t_5 - sqrt(y))) <= 0.04) {
		tmp = t_1 + (t_4 + fma(sqrt((1.0 / y)), 0.5, (1.0 / (sqrt(x) + t_2))));
	} else {
		tmp = (((1.0 / (t_5 + sqrt(y))) + t_3) + t_4) + t_1;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = Float64(t_2 - sqrt(x))
	t_4 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_5 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (Float64(t_3 + Float64(t_5 - sqrt(y))) <= 0.04)
		tmp = Float64(t_1 + Float64(t_4 + fma(sqrt(Float64(1.0 / y)), 0.5, Float64(1.0 / Float64(sqrt(x) + t_2)))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(t_5 + sqrt(y))) + t_3) + t_4) + 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[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$3 + N[(t$95$5 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.04], N[(t$95$1 + N[(t$95$4 + N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] * 0.5 + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 / N[(t$95$5 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) < 0.0400000000000000008

   1. Initial program 87.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{\sqrt{y}} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \sqrt{y} \cdot \color{blue}{\sqrt{y}}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\color{blue}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 88.2

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\color{blue}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 88.2%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in y around 0

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{1}}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 91.6%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{1}}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \color{blue}{\frac{1}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. frac-add N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{y} + \sqrt{y + 1}\right) + \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot 1}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y} + \sqrt{y + 1}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{y} + \sqrt{y + 1}\right) + \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot 1}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y} + \sqrt{y + 1}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Applied rewrites 91.9%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(\left(1 + x\right) - x, \sqrt{y + 1} + \sqrt{y}, \left(\sqrt{1 + x} + \sqrt{x}\right) \cdot 1\right)}{\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Taylor expanded in y around inf

       \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{y}} \cdot \frac{1}{2}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{y}}}, \frac{1}{2}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{y}}}, \frac{1}{2}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       8. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \frac{1}{\color{blue}{\sqrt{1 + x}} + \sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       9. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \frac{1}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       10. lower-sqrt.f64 92.9

           \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, 0.5, \frac{1}{\sqrt{1 + x} + \color{blue}{\sqrt{x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. Applied rewrites 92.9%

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{y}}, 0.5, \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.0400000000000000008 < (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)))

   1. Initial program 96.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{\sqrt{y}} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \sqrt{y} \cdot \color{blue}{\sqrt{y}}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\color{blue}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 96.7

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\color{blue}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 96.7%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in y around 0

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{1}}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 96.9%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{1}}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right) \leq 0.04:\\ \;\;\;\;\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \mathsf{fma}\left(\sqrt{\frac{1}{y}}, 0.5, \frac{1}{\sqrt{x} + \sqrt{x + 1}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 92.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{t + 1} - \sqrt{t}\\ t_4 := \sqrt{1 + y}\\ \mathbf{if}\;\left(\left(t\_2 - \sqrt{x}\right) + \left(t\_4 - \sqrt{y}\right)\right) + t\_1 \leq 1.9999999999999996:\\ \;\;\;\;\left(\left(\frac{1}{t\_4 + \sqrt{y}} + t\_2\right) - \sqrt{x}\right) + t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{x}\right) + \left(1 - \sqrt{y}\right)\right) + t\_1\right) + t\_3\\ \end{array} \end{array} \]

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 (+ z 1.0)) (sqrt z)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_4 (sqrt (+ 1.0 y))))
   (if (<= (+ (+ (- t_2 (sqrt x)) (- t_4 (sqrt y))) t_1) 1.9999999999999996)
     (+ (- (+ (/ 1.0 (+ t_4 (sqrt y))) t_2) (sqrt x)) t_3)
     (+ (+ (+ (- 1.0 (sqrt x)) (- 1.0 (sqrt y))) t_1) t_3))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = sqrt((x + 1.0));
	double t_3 = sqrt((t + 1.0)) - sqrt(t);
	double t_4 = sqrt((1.0 + y));
	double tmp;
	if ((((t_2 - sqrt(x)) + (t_4 - sqrt(y))) + t_1) <= 1.9999999999999996) {
		tmp = (((1.0 / (t_4 + sqrt(y))) + t_2) - sqrt(x)) + t_3;
	} else {
		tmp = (((1.0 - sqrt(x)) + (1.0 - sqrt(y))) + t_1) + t_3;
	}
	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((z + 1.0d0)) - sqrt(z)
    t_2 = sqrt((x + 1.0d0))
    t_3 = sqrt((t + 1.0d0)) - sqrt(t)
    t_4 = sqrt((1.0d0 + y))
    if ((((t_2 - sqrt(x)) + (t_4 - sqrt(y))) + t_1) <= 1.9999999999999996d0) then
        tmp = (((1.0d0 / (t_4 + sqrt(y))) + t_2) - sqrt(x)) + t_3
    else
        tmp = (((1.0d0 - sqrt(x)) + (1.0d0 - sqrt(y))) + t_1) + t_3
    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((z + 1.0)) - Math.sqrt(z);
	double t_2 = Math.sqrt((x + 1.0));
	double t_3 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_4 = Math.sqrt((1.0 + y));
	double tmp;
	if ((((t_2 - Math.sqrt(x)) + (t_4 - Math.sqrt(y))) + t_1) <= 1.9999999999999996) {
		tmp = (((1.0 / (t_4 + Math.sqrt(y))) + t_2) - Math.sqrt(x)) + t_3;
	} else {
		tmp = (((1.0 - Math.sqrt(x)) + (1.0 - Math.sqrt(y))) + t_1) + t_3;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0)) - math.sqrt(z)
	t_2 = math.sqrt((x + 1.0))
	t_3 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_4 = math.sqrt((1.0 + y))
	tmp = 0
	if (((t_2 - math.sqrt(x)) + (t_4 - math.sqrt(y))) + t_1) <= 1.9999999999999996:
		tmp = (((1.0 / (t_4 + math.sqrt(y))) + t_2) - math.sqrt(x)) + t_3
	else:
		tmp = (((1.0 - math.sqrt(x)) + (1.0 - math.sqrt(y))) + t_1) + t_3
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_4 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 - sqrt(x)) + Float64(t_4 - sqrt(y))) + t_1) <= 1.9999999999999996)
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(t_4 + sqrt(y))) + t_2) - sqrt(x)) + t_3);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - sqrt(x)) + Float64(1.0 - sqrt(y))) + t_1) + t_3);
	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((z + 1.0)) - sqrt(z);
	t_2 = sqrt((x + 1.0));
	t_3 = sqrt((t + 1.0)) - sqrt(t);
	t_4 = sqrt((1.0 + y));
	tmp = 0.0;
	if ((((t_2 - sqrt(x)) + (t_4 - sqrt(y))) + t_1) <= 1.9999999999999996)
		tmp = (((1.0 / (t_4 + sqrt(y))) + t_2) - sqrt(x)) + t_3;
	else
		tmp = (((1.0 - sqrt(x)) + (1.0 - sqrt(y))) + t_1) + t_3;
	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[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], 1.9999999999999996], N[(N[(N[(N[(1.0 / N[(t$95$4 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], N[(N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.9999999999999996

   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. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{\sqrt{y}} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \sqrt{y} \cdot \color{blue}{\sqrt{y}}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\color{blue}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 91.5

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\color{blue}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 91.5%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   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. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{\sqrt{y} + \sqrt{1 + y}}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\left(\frac{1}{\color{blue}{\sqrt{1 + y} + \sqrt{y}}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\color{blue}{\sqrt{1 + y} + \sqrt{y}}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\color{blue}{\sqrt{1 + y}} + \sqrt{y}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\sqrt{1 + y} + \color{blue}{\sqrt{y}}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-sqrt.f64 48.8

          \[\leadsto \left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 48.8%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \sqrt{1 + x}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.9999999999999996 < (+.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 98.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(1 - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(1 - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 74.1

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(1 - \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 74.1%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(1 - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(1 - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(1 - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 51.5

         \[\leadsto \left(\left(\left(1 - \color{blue}{\sqrt{x}}\right) + \left(1 - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 51.5%

      \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(1 - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right) \leq 1.9999999999999996:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \sqrt{x + 1}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{x}\right) + \left(1 - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 98.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{z + 1}\\ t_3 := \sqrt{1 + y} - \sqrt{y}\\ t_4 := \sqrt{t + 1} - \sqrt{t}\\ \mathbf{if}\;t\_3 \leq 0.0002:\\ \;\;\;\;t\_4 + \left(\left(t\_2 - \sqrt{z}\right) + \mathsf{fma}\left(\sqrt{\frac{1}{y}}, 0.5, \frac{1}{\sqrt{x} + t\_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(z + 1\right) - z}{\sqrt{z} + t\_2} + \left(\left(t\_1 - \sqrt{x}\right) + t\_3\right)\right) + t\_4\\ \end{array} \end{array} \]

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 1.0)))
        (t_2 (sqrt (+ z 1.0)))
        (t_3 (- (sqrt (+ 1.0 y)) (sqrt y)))
        (t_4 (- (sqrt (+ t 1.0)) (sqrt t))))
   (if (<= t_3 0.0002)
     (+
      t_4
      (+ (- t_2 (sqrt z)) (fma (sqrt (/ 1.0 y)) 0.5 (/ 1.0 (+ (sqrt x) t_1)))))
     (+
      (+ (/ (- (+ z 1.0) z) (+ (sqrt z) t_2)) (+ (- t_1 (sqrt x)) t_3))
      t_4))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double t_2 = sqrt((z + 1.0));
	double t_3 = sqrt((1.0 + y)) - sqrt(y);
	double t_4 = sqrt((t + 1.0)) - sqrt(t);
	double tmp;
	if (t_3 <= 0.0002) {
		tmp = t_4 + ((t_2 - sqrt(z)) + fma(sqrt((1.0 / y)), 0.5, (1.0 / (sqrt(x) + t_1))));
	} else {
		tmp = ((((z + 1.0) - z) / (sqrt(z) + t_2)) + ((t_1 - sqrt(x)) + t_3)) + t_4;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	t_2 = sqrt(Float64(z + 1.0))
	t_3 = Float64(sqrt(Float64(1.0 + y)) - sqrt(y))
	t_4 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	tmp = 0.0
	if (t_3 <= 0.0002)
		tmp = Float64(t_4 + Float64(Float64(t_2 - sqrt(z)) + fma(sqrt(Float64(1.0 / y)), 0.5, Float64(1.0 / Float64(sqrt(x) + t_1)))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(z + 1.0) - z) / Float64(sqrt(z) + t_2)) + Float64(Float64(t_1 - sqrt(x)) + t_3)) + t_4);
	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[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0002], N[(t$95$4 + N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] * 0.5 + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(z + 1.0), $MachinePrecision] - z), $MachinePrecision] / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)) < 2.0000000000000001e-4

   1. Initial program 91.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{\sqrt{y}} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \sqrt{y} \cdot \color{blue}{\sqrt{y}}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\color{blue}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 92.1

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\color{blue}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 92.1%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in y around 0

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{1}}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 94.1%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{1}}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \color{blue}{\frac{1}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. frac-add N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{y} + \sqrt{y + 1}\right) + \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot 1}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y} + \sqrt{y + 1}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{y} + \sqrt{y + 1}\right) + \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot 1}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y} + \sqrt{y + 1}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Applied rewrites 94.2%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(\left(1 + x\right) - x, \sqrt{y + 1} + \sqrt{y}, \left(\sqrt{1 + x} + \sqrt{x}\right) \cdot 1\right)}{\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Taylor expanded in y around inf

       \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{y}} \cdot \frac{1}{2}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{y}}}, \frac{1}{2}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{y}}}, \frac{1}{2}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       8. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \frac{1}{\color{blue}{\sqrt{1 + x}} + \sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       9. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \frac{1}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       10. lower-sqrt.f64 95.1

           \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, 0.5, \frac{1}{\sqrt{1 + x} + \color{blue}{\sqrt{x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. Applied rewrites 95.1%

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{y}}, 0.5, \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 2.0000000000000001e-4 < (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))

   1. Initial program 96.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\sqrt{z + 1}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{z + 1} \cdot \color{blue}{\sqrt{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{\sqrt{z}} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \sqrt{z} \cdot \color{blue}{\sqrt{z}}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 97.1

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 97.1%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + y} - \sqrt{y} \leq 0.0002:\\ \;\;\;\;\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \mathsf{fma}\left(\sqrt{\frac{1}{y}}, 0.5, \frac{1}{\sqrt{x} + \sqrt{x + 1}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 98.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{z + 1} - \sqrt{z}\\ \mathbf{if}\;y \leq 7500000:\\ \;\;\;\;\frac{\left(t + 1\right) - t}{\sqrt{t} + t\_1} + \left(\left(\left(t\_2 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 - \sqrt{t}\right) + \left(t\_3 + \mathsf{fma}\left(\sqrt{\frac{1}{y}}, 0.5, \frac{1}{\sqrt{x} + t\_2}\right)\right)\\ \end{array} \end{array} \]

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 (+ t 1.0)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (- (sqrt (+ z 1.0)) (sqrt z))))
   (if (<= y 7500000.0)
     (+
      (/ (- (+ t 1.0) t) (+ (sqrt t) t_1))
      (+ (+ (- t_2 (sqrt x)) (- (sqrt (+ 1.0 y)) (sqrt y))) t_3))
     (+
      (- t_1 (sqrt t))
      (+ t_3 (fma (sqrt (/ 1.0 y)) 0.5 (/ 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((t + 1.0));
	double t_2 = sqrt((x + 1.0));
	double t_3 = sqrt((z + 1.0)) - sqrt(z);
	double tmp;
	if (y <= 7500000.0) {
		tmp = (((t + 1.0) - t) / (sqrt(t) + t_1)) + (((t_2 - sqrt(x)) + (sqrt((1.0 + y)) - sqrt(y))) + t_3);
	} else {
		tmp = (t_1 - sqrt(t)) + (t_3 + fma(sqrt((1.0 / y)), 0.5, (1.0 / (sqrt(x) + t_2))));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(t + 1.0))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	tmp = 0.0
	if (y <= 7500000.0)
		tmp = Float64(Float64(Float64(Float64(t + 1.0) - t) / Float64(sqrt(t) + t_1)) + Float64(Float64(Float64(t_2 - sqrt(x)) + Float64(sqrt(Float64(1.0 + y)) - sqrt(y))) + t_3));
	else
		tmp = Float64(Float64(t_1 - sqrt(t)) + Float64(t_3 + fma(sqrt(Float64(1.0 / y)), 0.5, Float64(1.0 / Float64(sqrt(x) + t_2)))));
	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[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 7500000.0], N[(N[(N[(N[(t + 1.0), $MachinePrecision] - t), $MachinePrecision] / N[(N[Sqrt[t], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(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] + t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 + N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] * 0.5 + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y < 7.5e6

   1. Initial program 96.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\left(\sqrt{t + 1} - \sqrt{t}\right)} \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\sqrt{t + 1} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\sqrt{t + 1} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\sqrt{t + 1}} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\sqrt{t + 1} \cdot \color{blue}{\sqrt{t + 1}} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\left(t + 1\right)} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \color{blue}{\sqrt{t}} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \sqrt{t} \cdot \color{blue}{\sqrt{t}}}{\sqrt{t + 1} + \sqrt{t}} \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \color{blue}{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\left(t + 1\right) - t}}{\sqrt{t + 1} + \sqrt{t}} \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - t}{\color{blue}{\sqrt{t} + \sqrt{t + 1}}} \]

      12. lower-+.f64 97.5

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - t}{\color{blue}{\sqrt{t} + \sqrt{t + 1}}} \]

   4. Applied rewrites 97.5%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}}} \]

   if 7.5e6 < y

   1. Initial program 91.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{\sqrt{y}} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \sqrt{y} \cdot \color{blue}{\sqrt{y}}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\color{blue}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 92.1

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\color{blue}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 92.1%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in y around 0

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{1}}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 94.1%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{1}}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \color{blue}{\frac{1}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. frac-add N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{y} + \sqrt{y + 1}\right) + \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot 1}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y} + \sqrt{y + 1}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{y} + \sqrt{y + 1}\right) + \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot 1}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y} + \sqrt{y + 1}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Applied rewrites 94.2%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(\left(1 + x\right) - x, \sqrt{y + 1} + \sqrt{y}, \left(\sqrt{1 + x} + \sqrt{x}\right) \cdot 1\right)}{\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Taylor expanded in y around inf

       \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{y}} \cdot \frac{1}{2}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{y}}}, \frac{1}{2}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{y}}}, \frac{1}{2}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       8. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \frac{1}{\color{blue}{\sqrt{1 + x}} + \sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       9. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \frac{1}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       10. lower-sqrt.f64 95.1

           \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, 0.5, \frac{1}{\sqrt{1 + x} + \color{blue}{\sqrt{x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. Applied rewrites 95.1%

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{y}}, 0.5, \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7500000:\\ \;\;\;\;\frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}} + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \mathsf{fma}\left(\sqrt{\frac{1}{y}}, 0.5, \frac{1}{\sqrt{x} + \sqrt{x + 1}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 65.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ \mathbf{if}\;y \leq 6 \cdot 10^{+15}:\\ \;\;\;\;\left(\sqrt{1 + y} + t\_1\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{x}\right) + t\_1\\ \end{array} \end{array} \]

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 1.0))))
   (if (<= y 6e+15)
     (- (+ (sqrt (+ 1.0 y)) t_1) (+ (sqrt x) (sqrt y)))
     (+ (- (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((x + 1.0));
	double tmp;
	if (y <= 6e+15) {
		tmp = (sqrt((1.0 + y)) + t_1) - (sqrt(x) + sqrt(y));
	} else {
		tmp = -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) :: tmp
    t_1 = sqrt((x + 1.0d0))
    if (y <= 6d+15) then
        tmp = (sqrt((1.0d0 + y)) + t_1) - (sqrt(x) + sqrt(y))
    else
        tmp = -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((x + 1.0));
	double tmp;
	if (y <= 6e+15) {
		tmp = (Math.sqrt((1.0 + y)) + t_1) - (Math.sqrt(x) + Math.sqrt(y));
	} else {
		tmp = -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((x + 1.0))
	tmp = 0
	if y <= 6e+15:
		tmp = (math.sqrt((1.0 + y)) + t_1) - (math.sqrt(x) + math.sqrt(y))
	else:
		tmp = -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(x + 1.0))
	tmp = 0.0
	if (y <= 6e+15)
		tmp = Float64(Float64(sqrt(Float64(1.0 + y)) + t_1) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(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((x + 1.0));
	tmp = 0.0;
	if (y <= 6e+15)
		tmp = (sqrt((1.0 + y)) + t_1) - (sqrt(x) + sqrt(y));
	else
		tmp = -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[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 6e+15], N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[x], $MachinePrecision]) + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 6e15

   1. Initial program 96.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\color{blue}{\sqrt{z}} + \sqrt{y}\right) + \sqrt{x}\right) \]

      17. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \color{blue}{\sqrt{y}}\right) + \sqrt{x}\right) \]

      18. lower-sqrt.f64 23.3

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \color{blue}{\sqrt{x}}\right) \]

   5. Applied rewrites 23.3%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 24.9%

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} \]

   if 6e15 < y

   1. Initial program 91.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\color{blue}{\sqrt{z}} + \sqrt{y}\right) + \sqrt{x}\right) \]

      17. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \color{blue}{\sqrt{y}}\right) + \sqrt{x}\right) \]

      18. lower-sqrt.f64 3.1

          \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \color{blue}{\sqrt{x}}\right) \]

   5. Applied rewrites 3.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 23.3%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{1 + y}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \sqrt{x + 1} + -1 \cdot \color{blue}{\sqrt{x}} \]
   9. Step-by-step derivation

   10. Applied rewrites 23.2%

       \[\leadsto \sqrt{x + 1} + \left(-\sqrt{x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+15}:\\ \;\;\;\;\left(\sqrt{1 + y} + \sqrt{x + 1}\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{x}\right) + \sqrt{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 35.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(-\sqrt{x}\right) + \sqrt{x + 1} \end{array} \]

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)) (sqrt (+ x 1.0))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return -sqrt(x) + sqrt((x + 1.0));
}

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) + sqrt((x + 1.0d0))
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) + Math.sqrt((x + 1.0));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return -math.sqrt(x) + math.sqrt((x + 1.0))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(Float64(-sqrt(x)) + sqrt(Float64(x + 1.0)))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = -sqrt(x) + sqrt((x + 1.0));
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[x], $MachinePrecision]) + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.2%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
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. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   7. lower-+.f64 N/A

      \[\leadsto \left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   9. lower-+.f64 N/A

      \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   13. lower-+.f64 N/A

       \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   14. +-commutative N/A

       \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

   15. lower-+.f64 N/A

       \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

   16. lower-sqrt.f64 N/A

       \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\color{blue}{\sqrt{z}} + \sqrt{y}\right) + \sqrt{x}\right) \]

   17. lower-sqrt.f64 N/A

       \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \color{blue}{\sqrt{y}}\right) + \sqrt{x}\right) \]

   18. lower-sqrt.f64 13.5

       \[\leadsto \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \color{blue}{\sqrt{x}}\right) \]

5. Applied rewrites 13.5%

   \[\leadsto \color{blue}{\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]
6. Step-by-step derivation

7. Applied rewrites 25.5%

   \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{1 + y}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right)} \]

8. Taylor expanded in x around inf

   \[\leadsto \sqrt{x + 1} + -1 \cdot \color{blue}{\sqrt{x}} \]
9. Step-by-step derivation

10. Applied rewrites 17.3%

    \[\leadsto \sqrt{x + 1} + \left(-\sqrt{x}\right) \]

11. Final simplification 17.3%

    \[\leadsto \left(-\sqrt{x}\right) + \sqrt{x + 1} \]
12. 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 94.2%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
4. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

5. Applied rewrites 13.6%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\sqrt{\frac{1}{t}}, 0.5, \sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

6. Taylor expanded in t around 0

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{t}}} \]
7. Step-by-step derivation

8. Applied rewrites 7.3%

   \[\leadsto \sqrt{\frac{1}{t}} \cdot \color{blue}{0.5} \]
9. Step-by-step derivation

10. Applied rewrites 7.3%

    \[\leadsto \color{blue}{\frac{0.5}{\sqrt{t}}} \]
11. 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 2024275 
(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))))