Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2

Percentage Accurate: 82.7% → 99.3%

Time: 10.5s

Alternatives: 10

Speedup: 1.0×

• 26.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x y) (* (* z z) (+ z 1.0))))

C

double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * y) / ((z * z) * (z + 1.0d0))
end function

Java

public static double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}

Python

def code(x, y, z):
	return (x * y) / ((z * z) * (z + 1.0))

Julia

function code(x, y, z)
	return Float64(Float64(x * y) / Float64(Float64(z * z) * Float64(z + 1.0)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) / ((z * z) * (z + 1.0));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 82.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x y) (* (* z z) (+ z 1.0))))

C

double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * y) / ((z * z) * (z + 1.0d0))
end function

Java

public static double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}

Python

def code(x, y, z):
	return (x * y) / ((z * z) * (z + 1.0))

Julia

function code(x, y, z)
	return Float64(Float64(x * y) / Float64(Float64(z * z) * Float64(z + 1.0)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) / ((z * z) * (z + 1.0));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\sqrt{x\_m}}{z}\\ x\_s \cdot \left(t\_0 \cdot \left(t\_0 \cdot \frac{y}{z + 1}\right)\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (sqrt x_m) z))) (* x_s (* t_0 (* t_0 (/ y (+ z 1.0)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = sqrt(x_m) / z;
	return x_s * (t_0 * (t_0 * (y / (z + 1.0))));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    t_0 = sqrt(x_m) / z
    code = x_s * (t_0 * (t_0 * (y / (z + 1.0d0))))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = Math.sqrt(x_m) / z;
	return x_s * (t_0 * (t_0 * (y / (z + 1.0))));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = math.sqrt(x_m) / z
	return x_s * (t_0 * (t_0 * (y / (z + 1.0))))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(sqrt(x_m) / z)
	return Float64(x_s * Float64(t_0 * Float64(t_0 * Float64(y / Float64(z + 1.0)))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z)
	t_0 = sqrt(x_m) / z;
	tmp = x_s * (t_0 * (t_0 * (y / (z + 1.0))));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[Sqrt[x$95$m], $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * N[(t$95$0 * N[(t$95$0 * N[(y / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 80.1%

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

   1. times-frac 85.5%

      \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   2. add-sqr-sqrt 56.0%

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

   3. associate-*l* 56.0%

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

   4. sqrt-div 47.3%

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

   5. sqrt-prod 22.8%

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

   6. add-sqr-sqrt 32.6%

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

   7. sqrt-div 34.0%

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

   8. sqrt-prod 27.2%

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

   9. add-sqr-sqrt 55.6%

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

4. Applied egg-rr 55.6%

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

5. Final simplification 55.6%

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

Alternative 2: 95.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\frac{x\_m}{z}}{z \cdot \frac{z}{y}}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{-101}:\\ \;\;\;\;y \cdot \frac{\frac{x\_m}{z \cdot z}}{z + 1}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{y \cdot \frac{x\_m}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (/ x_m z) (* z (/ z y)))))
   (*
    x_s
    (if (<= z -2.5e+15)
      t_0
      (if (<= z -3.6e-101)
        (* y (/ (/ x_m (* z z)) (+ z 1.0)))
        (if (<= z 1.0) (/ (* y (/ x_m z)) z) t_0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m / z) / (z * (z / y));
	double tmp;
	if (z <= -2.5e+15) {
		tmp = t_0;
	} else if (z <= -3.6e-101) {
		tmp = y * ((x_m / (z * z)) / (z + 1.0));
	} else if (z <= 1.0) {
		tmp = (y * (x_m / z)) / z;
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_m / z) / (z * (z / y))
    if (z <= (-2.5d+15)) then
        tmp = t_0
    else if (z <= (-3.6d-101)) then
        tmp = y * ((x_m / (z * z)) / (z + 1.0d0))
    else if (z <= 1.0d0) then
        tmp = (y * (x_m / z)) / z
    else
        tmp = t_0
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m / z) / (z * (z / y));
	double tmp;
	if (z <= -2.5e+15) {
		tmp = t_0;
	} else if (z <= -3.6e-101) {
		tmp = y * ((x_m / (z * z)) / (z + 1.0));
	} else if (z <= 1.0) {
		tmp = (y * (x_m / z)) / z;
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = (x_m / z) / (z * (z / y))
	tmp = 0
	if z <= -2.5e+15:
		tmp = t_0
	elif z <= -3.6e-101:
		tmp = y * ((x_m / (z * z)) / (z + 1.0))
	elif z <= 1.0:
		tmp = (y * (x_m / z)) / z
	else:
		tmp = t_0
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(x_m / z) / Float64(z * Float64(z / y)))
	tmp = 0.0
	if (z <= -2.5e+15)
		tmp = t_0;
	elseif (z <= -3.6e-101)
		tmp = Float64(y * Float64(Float64(x_m / Float64(z * z)) / Float64(z + 1.0)));
	elseif (z <= 1.0)
		tmp = Float64(Float64(y * Float64(x_m / z)) / z);
	else
		tmp = t_0;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = (x_m / z) / (z * (z / y));
	tmp = 0.0;
	if (z <= -2.5e+15)
		tmp = t_0;
	elseif (z <= -3.6e-101)
		tmp = y * ((x_m / (z * z)) / (z + 1.0));
	elseif (z <= 1.0)
		tmp = (y * (x_m / z)) / z;
	else
		tmp = t_0;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(x$95$m / z), $MachinePrecision] / N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -2.5e+15], t$95$0, If[LessEqual[z, -3.6e-101], N[(y * N[(N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -2.5e15 or 1 < z

   1. Initial program 78.2%

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

      1. *-commutative 78.2%

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

      2. frac-times 93.5%

         \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}} \]

      3. associate-*l/ 92.0%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{z + 1}}{z \cdot z}} \]

      4. times-frac 97.5%

         \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{\frac{x}{z + 1}}{z}} \]

   4. Applied egg-rr 97.5%

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{\frac{x}{z + 1}}{z}} \]

   5. Taylor expanded in z around inf 96.8%

      \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{\frac{x}{z}}}{z} \]
   6. Step-by-step derivation

      1. clear-num 96.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{z}{y}}} \cdot \frac{\frac{x}{z}}{z} \]

      2. frac-times 96.0%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{z}}{\frac{z}{y} \cdot z}} \]

      3. *-un-lft-identity 96.0%

         \[\leadsto \frac{\color{blue}{\frac{x}{z}}}{\frac{z}{y} \cdot z} \]

   7. Applied egg-rr 96.0%

      \[\leadsto \color{blue}{\frac{\frac{x}{z}}{\frac{z}{y} \cdot z}} \]

   if -2.5e15 < z < -3.6e-101

   1. Initial program 90.5%

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

      1. *-commutative 90.5%

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

      2. associate-/l* 90.2%

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

      3. sqr-neg 90.2%

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

      4. associate-/r* 90.2%

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

      5. sqr-neg 90.2%

         \[\leadsto y \cdot \frac{\frac{x}{\color{blue}{z \cdot z}}}{z + 1} \]

   3. Simplified 90.2%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{x}{z \cdot z}}{z + 1}} \]
   4. Add Preprocessing

   if -3.6e-101 < z < 1

   1. Initial program 80.1%

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

      1. *-commutative 80.1%

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

      2. associate-/l* 77.2%

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

      3. sqr-neg 77.2%

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

      4. associate-/r* 77.2%

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

      5. sqr-neg 77.2%

         \[\leadsto y \cdot \frac{\frac{x}{\color{blue}{z \cdot z}}}{z + 1} \]

   3. Simplified 77.2%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{x}{z \cdot z}}{z + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 77.2%

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

      2. associate-*r/ 80.1%

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

      3. frac-times 79.6%

         \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}} \]

      4. frac-2neg 79.6%

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

      5. associate-/r* 92.8%

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

      6. frac-times 98.3%

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

      7. +-commutative 98.3%

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

      8. distribute-neg-in 98.3%

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

      9. metadata-eval 98.3%

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

   6. Applied egg-rr 98.3%

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

   7. Taylor expanded in z around 0 97.1%

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

      1. mul-1-neg 97.1%

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

   9. Simplified 97.1%

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

       1. *-commutative 97.1%

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

       2. associate-/l* 91.6%

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

       3. add-sqr-sqrt 41.0%

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

       4. sqrt-unprod 31.4%

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

       5. sqr-neg 31.4%

          \[\leadsto \sqrt{\color{blue}{x \cdot x}} \cdot \frac{\frac{y}{z}}{-z} \]

       6. sqrt-unprod 0.4%

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

       7. add-sqr-sqrt 2.2%

          \[\leadsto \color{blue}{x} \cdot \frac{\frac{y}{z}}{-z} \]

       8. add-sqr-sqrt 0.1%

          \[\leadsto x \cdot \frac{\frac{y}{z}}{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}} \]

       9. sqrt-unprod 54.1%

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

       10. sqr-neg 54.1%

           \[\leadsto x \cdot \frac{\frac{y}{z}}{\sqrt{\color{blue}{z \cdot z}}} \]

       11. sqrt-unprod 59.5%

           \[\leadsto x \cdot \frac{\frac{y}{z}}{\color{blue}{\sqrt{z} \cdot \sqrt{z}}} \]

       12. add-sqr-sqrt 91.6%

           \[\leadsto x \cdot \frac{\frac{y}{z}}{\color{blue}{z}} \]

   11. Applied egg-rr 91.6%

       \[\leadsto \color{blue}{x \cdot \frac{\frac{y}{z}}{z}} \]
   12. Step-by-step derivation

       1. associate-*r/ 97.1%

          \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{z}}{z}} \]

       2. frac-2neg 97.1%

          \[\leadsto \frac{x \cdot \color{blue}{\frac{-y}{-z}}}{z} \]

       3. associate-*r/ 87.8%

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

       4. distribute-neg-frac2 87.8%

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

       5. associate-*l/ 97.0%

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

       6. *-commutative 97.0%

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

       7. distribute-lft-neg-in 97.0%

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

       8. add-sqr-sqrt 52.9%

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

       9. sqrt-unprod 36.9%

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

       10. sqr-neg 36.9%

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

       11. sqrt-unprod 1.1%

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

       12. add-sqr-sqrt 2.2%

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

       13. add-sqr-sqrt 1.1%

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

       14. sqrt-unprod 36.1%

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

       15. sqr-neg 36.1%

           \[\leadsto \frac{\sqrt{\color{blue}{y \cdot y}} \cdot \frac{x}{z}}{z} \]

       16. sqrt-unprod 43.9%

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

       17. add-sqr-sqrt 97.0%

           \[\leadsto \frac{\color{blue}{y} \cdot \frac{x}{z}}{z} \]

   13. Applied egg-rr 97.0%

       \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{z}}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z}{y}}\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{-101}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z \cdot z}}{z + 1}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x\_m}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x\_m}{z}}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (or (<= z -1.0) (not (<= z 1.0)))
    (* (/ y z) (/ (/ x_m z) z))
    (/ (* y (/ x_m z)) z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = (y / z) * ((x_m / z) / z);
	} else {
		tmp = (y * (x_m / z)) / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = (y / z) * ((x_m / z) / z)
    else
        tmp = (y * (x_m / z)) / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = (y / z) * ((x_m / z) / z);
	} else {
		tmp = (y * (x_m / z)) / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.0):
		tmp = (y / z) * ((x_m / z) / z)
	else:
		tmp = (y * (x_m / z)) / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 1.0))
		tmp = Float64(Float64(y / z) * Float64(Float64(x_m / z) / z));
	else
		tmp = Float64(Float64(y * Float64(x_m / z)) / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 1.0)))
		tmp = (y / z) * ((x_m / z) / z);
	else
		tmp = (y * (x_m / z)) / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(y / z), $MachinePrecision] * N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 78.8%

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

      1. *-commutative 78.8%

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

      2. frac-times 93.7%

         \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}} \]

      3. associate-*l/ 92.2%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{z + 1}}{z \cdot z}} \]

      4. times-frac 97.6%

         \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{\frac{x}{z + 1}}{z}} \]

   4. Applied egg-rr 97.6%

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{\frac{x}{z + 1}}{z}} \]

   5. Taylor expanded in z around inf 95.6%

      \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{\frac{x}{z}}}{z} \]

   if -1 < z < 1

   1. Initial program 81.2%

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

      1. *-commutative 81.2%

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

      2. associate-/l* 78.6%

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

      3. sqr-neg 78.6%

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

      4. associate-/r* 78.6%

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

      5. sqr-neg 78.6%

         \[\leadsto y \cdot \frac{\frac{x}{\color{blue}{z \cdot z}}}{z + 1} \]

   3. Simplified 78.6%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{x}{z \cdot z}}{z + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 78.6%

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

      2. associate-*r/ 81.2%

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

      3. frac-times 82.1%

         \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}} \]

      4. frac-2neg 82.1%

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

      5. associate-/r* 93.6%

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

      6. frac-times 97.1%

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

      7. +-commutative 97.1%

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

      8. distribute-neg-in 97.1%

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

      9. metadata-eval 97.1%

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

   6. Applied egg-rr 97.1%

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

   7. Taylor expanded in z around 0 95.6%

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

      1. mul-1-neg 95.6%

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

   9. Simplified 95.6%

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

       1. *-commutative 95.6%

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

       2. associate-/l* 92.0%

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

       3. add-sqr-sqrt 40.3%

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

       4. sqrt-unprod 31.4%

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

       5. sqr-neg 31.4%

          \[\leadsto \sqrt{\color{blue}{x \cdot x}} \cdot \frac{\frac{y}{z}}{-z} \]

       6. sqrt-unprod 0.5%

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

       7. add-sqr-sqrt 2.2%

          \[\leadsto \color{blue}{x} \cdot \frac{\frac{y}{z}}{-z} \]

       8. add-sqr-sqrt 0.3%

          \[\leadsto x \cdot \frac{\frac{y}{z}}{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}} \]

       9. sqrt-unprod 47.7%

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

       10. sqr-neg 47.7%

           \[\leadsto x \cdot \frac{\frac{y}{z}}{\sqrt{\color{blue}{z \cdot z}}} \]

       11. sqrt-unprod 52.2%

           \[\leadsto x \cdot \frac{\frac{y}{z}}{\color{blue}{\sqrt{z} \cdot \sqrt{z}}} \]

       12. add-sqr-sqrt 92.0%

           \[\leadsto x \cdot \frac{\frac{y}{z}}{\color{blue}{z}} \]

   11. Applied egg-rr 92.0%

       \[\leadsto \color{blue}{x \cdot \frac{\frac{y}{z}}{z}} \]
   12. Step-by-step derivation

       1. associate-*r/ 95.6%

          \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{z}}{z}} \]

       2. frac-2neg 95.6%

          \[\leadsto \frac{x \cdot \color{blue}{\frac{-y}{-z}}}{z} \]

       3. associate-*r/ 87.4%

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

       4. distribute-neg-frac2 87.4%

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

       5. associate-*l/ 94.9%

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

       6. *-commutative 94.9%

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

       7. distribute-lft-neg-in 94.9%

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

       8. add-sqr-sqrt 50.0%

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

       9. sqrt-unprod 34.8%

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

       10. sqr-neg 34.8%

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

       11. sqrt-unprod 1.2%

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

       12. add-sqr-sqrt 2.2%

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

       13. add-sqr-sqrt 1.0%

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

       14. sqrt-unprod 36.4%

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

       15. sqr-neg 36.4%

           \[\leadsto \frac{\sqrt{\color{blue}{y \cdot y}} \cdot \frac{x}{z}}{z} \]

       16. sqrt-unprod 44.6%

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

       17. add-sqr-sqrt 94.9%

           \[\leadsto \frac{\color{blue}{y} \cdot \frac{x}{z}}{z} \]

   13. Applied egg-rr 94.9%

       \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{z}}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x\_m}{z}}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (or (<= z -1.0) (not (<= z 1.0)))
    (/ (/ x_m z) (* z (/ z y)))
    (/ (* y (/ x_m z)) z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = (x_m / z) / (z * (z / y));
	} else {
		tmp = (y * (x_m / z)) / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = (x_m / z) / (z * (z / y))
    else
        tmp = (y * (x_m / z)) / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = (x_m / z) / (z * (z / y));
	} else {
		tmp = (y * (x_m / z)) / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.0):
		tmp = (x_m / z) / (z * (z / y))
	else:
		tmp = (y * (x_m / z)) / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 1.0))
		tmp = Float64(Float64(x_m / z) / Float64(z * Float64(z / y)));
	else
		tmp = Float64(Float64(y * Float64(x_m / z)) / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 1.0)))
		tmp = (x_m / z) / (z * (z / y));
	else
		tmp = (y * (x_m / z)) / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(x$95$m / z), $MachinePrecision] / N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 78.8%

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

      1. *-commutative 78.8%

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

      2. frac-times 93.7%

         \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}} \]

      3. associate-*l/ 92.2%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{z + 1}}{z \cdot z}} \]

      4. times-frac 97.6%

         \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{\frac{x}{z + 1}}{z}} \]

   4. Applied egg-rr 97.6%

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{\frac{x}{z + 1}}{z}} \]

   5. Taylor expanded in z around inf 95.6%

      \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{\frac{x}{z}}}{z} \]
   6. Step-by-step derivation

      1. clear-num 95.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{z}{y}}} \cdot \frac{\frac{x}{z}}{z} \]

      2. frac-times 94.8%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{z}}{\frac{z}{y} \cdot z}} \]

      3. *-un-lft-identity 94.8%

         \[\leadsto \frac{\color{blue}{\frac{x}{z}}}{\frac{z}{y} \cdot z} \]

   7. Applied egg-rr 94.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{z}}{\frac{z}{y} \cdot z}} \]

   if -1 < z < 1

   1. Initial program 81.2%

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

      1. *-commutative 81.2%

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

      2. associate-/l* 78.6%

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

      3. sqr-neg 78.6%

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

      4. associate-/r* 78.6%

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

      5. sqr-neg 78.6%

         \[\leadsto y \cdot \frac{\frac{x}{\color{blue}{z \cdot z}}}{z + 1} \]

   3. Simplified 78.6%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{x}{z \cdot z}}{z + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 78.6%

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

      2. associate-*r/ 81.2%

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

      3. frac-times 82.1%

         \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}} \]

      4. frac-2neg 82.1%

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

      5. associate-/r* 93.6%

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

      6. frac-times 97.1%

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

      7. +-commutative 97.1%

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

      8. distribute-neg-in 97.1%

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

      9. metadata-eval 97.1%

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

   6. Applied egg-rr 97.1%

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

   7. Taylor expanded in z around 0 95.6%

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

      1. mul-1-neg 95.6%

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

   9. Simplified 95.6%

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

       1. *-commutative 95.6%

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

       2. associate-/l* 92.0%

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

       3. add-sqr-sqrt 40.3%

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

       4. sqrt-unprod 31.4%

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

       5. sqr-neg 31.4%

          \[\leadsto \sqrt{\color{blue}{x \cdot x}} \cdot \frac{\frac{y}{z}}{-z} \]

       6. sqrt-unprod 0.5%

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

       7. add-sqr-sqrt 2.2%

          \[\leadsto \color{blue}{x} \cdot \frac{\frac{y}{z}}{-z} \]

       8. add-sqr-sqrt 0.3%

          \[\leadsto x \cdot \frac{\frac{y}{z}}{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}} \]

       9. sqrt-unprod 47.7%

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

       10. sqr-neg 47.7%

           \[\leadsto x \cdot \frac{\frac{y}{z}}{\sqrt{\color{blue}{z \cdot z}}} \]

       11. sqrt-unprod 52.2%

           \[\leadsto x \cdot \frac{\frac{y}{z}}{\color{blue}{\sqrt{z} \cdot \sqrt{z}}} \]

       12. add-sqr-sqrt 92.0%

           \[\leadsto x \cdot \frac{\frac{y}{z}}{\color{blue}{z}} \]

   11. Applied egg-rr 92.0%

       \[\leadsto \color{blue}{x \cdot \frac{\frac{y}{z}}{z}} \]
   12. Step-by-step derivation

       1. associate-*r/ 95.6%

          \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{z}}{z}} \]

       2. frac-2neg 95.6%

          \[\leadsto \frac{x \cdot \color{blue}{\frac{-y}{-z}}}{z} \]

       3. associate-*r/ 87.4%

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

       4. distribute-neg-frac2 87.4%

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

       5. associate-*l/ 94.9%

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

       6. *-commutative 94.9%

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

       7. distribute-lft-neg-in 94.9%

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

       8. add-sqr-sqrt 50.0%

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

       9. sqrt-unprod 34.8%

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

       10. sqr-neg 34.8%

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

       11. sqrt-unprod 1.2%

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

       12. add-sqr-sqrt 2.2%

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

       13. add-sqr-sqrt 1.0%

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

       14. sqrt-unprod 36.4%

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

       15. sqr-neg 36.4%

           \[\leadsto \frac{\sqrt{\color{blue}{y \cdot y}} \cdot \frac{x}{z}}{z} \]

       16. sqrt-unprod 44.6%

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

       17. add-sqr-sqrt 94.9%

           \[\leadsto \frac{\color{blue}{y} \cdot \frac{x}{z}}{z} \]

   13. Applied egg-rr 94.9%

       \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{z}}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{y \cdot \frac{x\_m}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x\_m}{z}}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= z -1.0)
    (* (/ x_m z) (/ y (* z z)))
    (if (<= z 1.0) (/ (* y (/ x_m z)) z) (* (/ y z) (/ (/ x_m z) z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = (x_m / z) * (y / (z * z));
	} else if (z <= 1.0) {
		tmp = (y * (x_m / z)) / z;
	} else {
		tmp = (y / z) * ((x_m / z) / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.0d0)) then
        tmp = (x_m / z) * (y / (z * z))
    else if (z <= 1.0d0) then
        tmp = (y * (x_m / z)) / z
    else
        tmp = (y / z) * ((x_m / z) / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = (x_m / z) * (y / (z * z));
	} else if (z <= 1.0) {
		tmp = (y * (x_m / z)) / z;
	} else {
		tmp = (y / z) * ((x_m / z) / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = (x_m / z) * (y / (z * z))
	elif z <= 1.0:
		tmp = (y * (x_m / z)) / z
	else:
		tmp = (y / z) * ((x_m / z) / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(Float64(x_m / z) * Float64(y / Float64(z * z)));
	elseif (z <= 1.0)
		tmp = Float64(Float64(y * Float64(x_m / z)) / z);
	else
		tmp = Float64(Float64(y / z) * Float64(Float64(x_m / z) / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = (x_m / z) * (y / (z * z));
	elseif (z <= 1.0)
		tmp = (y * (x_m / z)) / z;
	else
		tmp = (y / z) * ((x_m / z) / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[z, -1.0], N[(N[(x$95$m / z), $MachinePrecision] * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 81.6%

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

      1. *-commutative 81.6%

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

      2. sqr-neg 81.6%

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

      3. times-frac 96.1%

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

      4. sqr-neg 96.1%

         \[\leadsto \frac{y}{\color{blue}{z \cdot z}} \cdot \frac{x}{z + 1} \]

   3. Simplified 96.1%

      \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 93.8%

      \[\leadsto \frac{y}{z \cdot z} \cdot \color{blue}{\frac{x}{z}} \]

   if -1 < z < 1

   1. Initial program 81.2%

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

      1. *-commutative 81.2%

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

      2. associate-/l* 78.6%

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

      3. sqr-neg 78.6%

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

      4. associate-/r* 78.6%

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

      5. sqr-neg 78.6%

         \[\leadsto y \cdot \frac{\frac{x}{\color{blue}{z \cdot z}}}{z + 1} \]

   3. Simplified 78.6%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{x}{z \cdot z}}{z + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 78.6%

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

      2. associate-*r/ 81.2%

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

      3. frac-times 82.1%

         \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}} \]

      4. frac-2neg 82.1%

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

      5. associate-/r* 93.6%

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

      6. frac-times 97.1%

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

      7. +-commutative 97.1%

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

      8. distribute-neg-in 97.1%

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

      9. metadata-eval 97.1%

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

   6. Applied egg-rr 97.1%

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

   7. Taylor expanded in z around 0 95.6%

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

      1. mul-1-neg 95.6%

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

   9. Simplified 95.6%

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

       1. *-commutative 95.6%

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

       2. associate-/l* 92.0%

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

       3. add-sqr-sqrt 40.3%

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

       4. sqrt-unprod 31.4%

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

       5. sqr-neg 31.4%

          \[\leadsto \sqrt{\color{blue}{x \cdot x}} \cdot \frac{\frac{y}{z}}{-z} \]

       6. sqrt-unprod 0.5%

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

       7. add-sqr-sqrt 2.2%

          \[\leadsto \color{blue}{x} \cdot \frac{\frac{y}{z}}{-z} \]

       8. add-sqr-sqrt 0.3%

          \[\leadsto x \cdot \frac{\frac{y}{z}}{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}} \]

       9. sqrt-unprod 47.7%

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

       10. sqr-neg 47.7%

           \[\leadsto x \cdot \frac{\frac{y}{z}}{\sqrt{\color{blue}{z \cdot z}}} \]

       11. sqrt-unprod 52.2%

           \[\leadsto x \cdot \frac{\frac{y}{z}}{\color{blue}{\sqrt{z} \cdot \sqrt{z}}} \]

       12. add-sqr-sqrt 92.0%

           \[\leadsto x \cdot \frac{\frac{y}{z}}{\color{blue}{z}} \]

   11. Applied egg-rr 92.0%

       \[\leadsto \color{blue}{x \cdot \frac{\frac{y}{z}}{z}} \]
   12. Step-by-step derivation

       1. associate-*r/ 95.6%

          \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{z}}{z}} \]

       2. frac-2neg 95.6%

          \[\leadsto \frac{x \cdot \color{blue}{\frac{-y}{-z}}}{z} \]

       3. associate-*r/ 87.4%

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

       4. distribute-neg-frac2 87.4%

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

       5. associate-*l/ 94.9%

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

       6. *-commutative 94.9%

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

       7. distribute-lft-neg-in 94.9%

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

       8. add-sqr-sqrt 50.0%

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

       9. sqrt-unprod 34.8%

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

       10. sqr-neg 34.8%

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

       11. sqrt-unprod 1.2%

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

       12. add-sqr-sqrt 2.2%

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

       13. add-sqr-sqrt 1.0%

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

       14. sqrt-unprod 36.4%

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

       15. sqr-neg 36.4%

           \[\leadsto \frac{\sqrt{\color{blue}{y \cdot y}} \cdot \frac{x}{z}}{z} \]

       16. sqrt-unprod 44.6%

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

       17. add-sqr-sqrt 94.9%

           \[\leadsto \frac{\color{blue}{y} \cdot \frac{x}{z}}{z} \]

   13. Applied egg-rr 94.9%

       \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{z}}{z}} \]

   if 1 < z

   1. Initial program 75.0%

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

      1. *-commutative 75.0%

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

      2. frac-times 90.5%

         \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}} \]

      3. associate-*l/ 90.4%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{z + 1}}{z \cdot z}} \]

      4. times-frac 99.6%

         \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{\frac{x}{z + 1}}{z}} \]

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{\frac{x}{z + 1}}{z}} \]

   5. Taylor expanded in z around inf 98.0%

      \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{\frac{x}{z}}}{z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\frac{y}{z} \cdot \frac{\frac{x\_m}{z + 1}}{z}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (* (/ y z) (/ (/ x_m (+ z 1.0)) z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	return x_s * ((y / z) * ((x_m / (z + 1.0)) / z));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x_s * ((y / z) * ((x_m / (z + 1.0d0)) / z))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	return x_s * ((y / z) * ((x_m / (z + 1.0)) / z));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	return x_s * ((y / z) * ((x_m / (z + 1.0)) / z))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(Float64(y / z) * Float64(Float64(x_m / Float64(z + 1.0)) / z)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * ((y / z) * ((x_m / (z + 1.0)) / z));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * N[(N[(y / z), $MachinePrecision] * N[(N[(x$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.1%

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

   1. *-commutative 80.1%

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

   2. frac-times 87.4%

      \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}} \]

   3. associate-*l/ 86.2%

      \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{z + 1}}{z \cdot z}} \]

   4. times-frac 97.0%

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{\frac{x}{z + 1}}{z}} \]

4. Applied egg-rr 97.0%

   \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{\frac{x}{z + 1}}{z}} \]

5. Final simplification 97.0%

   \[\leadsto \frac{y}{z} \cdot \frac{\frac{x}{z + 1}}{z} \]
6. Add Preprocessing

Alternative 7: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{\frac{y}{z + 1} \cdot \frac{x\_m}{z}}{z} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (/ (* (/ y (+ z 1.0)) (/ x_m z)) z)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	return x_s * (((y / (z + 1.0)) * (x_m / z)) / z);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x_s * (((y / (z + 1.0d0)) * (x_m / z)) / z)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	return x_s * (((y / (z + 1.0)) * (x_m / z)) / z);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	return x_s * (((y / (z + 1.0)) * (x_m / z)) / z)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(Float64(Float64(y / Float64(z + 1.0)) * Float64(x_m / z)) / z))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * (((y / (z + 1.0)) * (x_m / z)) / z);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * N[(N[(N[(y / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.1%

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

   1. *-commutative 80.1%

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

   2. associate-/l* 79.6%

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

   3. sqr-neg 79.6%

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

   4. associate-/r* 82.3%

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

   5. sqr-neg 82.3%

      \[\leadsto y \cdot \frac{\frac{x}{\color{blue}{z \cdot z}}}{z + 1} \]

3. Simplified 82.3%

   \[\leadsto \color{blue}{y \cdot \frac{\frac{x}{z \cdot z}}{z + 1}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r/ 84.1%

      \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{z \cdot z}}{z + 1}} \]

   2. *-commutative 84.1%

      \[\leadsto \frac{\color{blue}{\frac{x}{z \cdot z} \cdot y}}{z + 1} \]

   3. associate-*r/ 85.5%

      \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   4. associate-/r* 89.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{z}}{z}} \cdot \frac{y}{z + 1} \]

   5. associate-*l/ 96.3%

      \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot \frac{y}{z + 1}}{z}} \]

6. Applied egg-rr 96.3%

   \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot \frac{y}{z + 1}}{z}} \]

7. Final simplification 96.3%

   \[\leadsto \frac{\frac{y}{z + 1} \cdot \frac{x}{z}}{z} \]
8. Add Preprocessing

Alternative 8: 74.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(x\_m \cdot \frac{\frac{y}{z}}{z}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z) :precision binary64 (* x_s (* x_m (/ (/ y z) z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	return x_s * (x_m * ((y / z) / z));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x_s * (x_m * ((y / z) / z))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	return x_s * (x_m * ((y / z) / z));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	return x_s * (x_m * ((y / z) / z))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(x_m * Float64(Float64(y / z) / z)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * (x_m * ((y / z) / z));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * N[(x$95$m * N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.1%

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

   1. *-commutative 80.1%

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

   2. associate-/l* 79.6%

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

   3. sqr-neg 79.6%

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

   4. associate-/r* 82.3%

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

   5. sqr-neg 82.3%

      \[\leadsto y \cdot \frac{\frac{x}{\color{blue}{z \cdot z}}}{z + 1} \]

3. Simplified 82.3%

   \[\leadsto \color{blue}{y \cdot \frac{\frac{x}{z \cdot z}}{z + 1}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* 79.6%

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

   2. associate-*r/ 80.1%

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

   3. frac-times 87.4%

      \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}} \]

   4. frac-2neg 87.4%

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

   5. associate-/r* 95.1%

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

   6. frac-times 95.1%

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

   7. +-commutative 95.1%

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

   8. distribute-neg-in 95.1%

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

   9. metadata-eval 95.1%

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

6. Applied egg-rr 95.1%

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

7. Taylor expanded in z around 0 74.4%

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

   1. mul-1-neg 74.4%

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

9. Simplified 74.4%

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

    1. *-commutative 74.4%

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

    2. associate-/l* 74.3%

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

    3. add-sqr-sqrt 31.1%

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

    4. sqrt-unprod 36.2%

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

    5. sqr-neg 36.2%

       \[\leadsto \sqrt{\color{blue}{x \cdot x}} \cdot \frac{\frac{y}{z}}{-z} \]

    6. sqrt-unprod 15.2%

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

    7. add-sqr-sqrt 25.3%

       \[\leadsto \color{blue}{x} \cdot \frac{\frac{y}{z}}{-z} \]

    8. add-sqr-sqrt 14.7%

       \[\leadsto x \cdot \frac{\frac{y}{z}}{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}} \]

    9. sqrt-unprod 52.0%

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

    10. sqr-neg 52.0%

        \[\leadsto x \cdot \frac{\frac{y}{z}}{\sqrt{\color{blue}{z \cdot z}}} \]

    11. sqrt-unprod 39.1%

        \[\leadsto x \cdot \frac{\frac{y}{z}}{\color{blue}{\sqrt{z} \cdot \sqrt{z}}} \]

    12. add-sqr-sqrt 74.3%

        \[\leadsto x \cdot \frac{\frac{y}{z}}{\color{blue}{z}} \]

11. Applied egg-rr 74.3%

    \[\leadsto \color{blue}{x \cdot \frac{\frac{y}{z}}{z}} \]

12. Final simplification 74.3%

    \[\leadsto x \cdot \frac{\frac{y}{z}}{z} \]
13. Add Preprocessing

Alternative 9: 73.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y \cdot \frac{\frac{x\_m}{z}}{z}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z) :precision binary64 (* x_s (* y (/ (/ x_m z) z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	return x_s * (y * ((x_m / z) / z));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x_s * (y * ((x_m / z) / z))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	return x_s * (y * ((x_m / z) / z));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	return x_s * (y * ((x_m / z) / z))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(y * Float64(Float64(x_m / z) / z)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * (y * ((x_m / z) / z));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * N[(y * N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.1%

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

   1. *-commutative 80.1%

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

   2. associate-/l* 79.6%

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

   3. sqr-neg 79.6%

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

   4. associate-/r* 82.3%

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

   5. sqr-neg 82.3%

      \[\leadsto y \cdot \frac{\frac{x}{\color{blue}{z \cdot z}}}{z + 1} \]

3. Simplified 82.3%

   \[\leadsto \color{blue}{y \cdot \frac{\frac{x}{z \cdot z}}{z + 1}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative 82.3%

      \[\leadsto \color{blue}{\frac{\frac{x}{z \cdot z}}{z + 1} \cdot y} \]

   2. clear-num 82.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{z + 1}{\frac{x}{z \cdot z}}}} \cdot y \]

   3. associate-*l/ 82.2%

      \[\leadsto \color{blue}{\frac{1 \cdot y}{\frac{z + 1}{\frac{x}{z \cdot z}}}} \]

   4. associate-/r/ 82.8%

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

   5. frac-times 87.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{z + 1}{x}} \cdot \frac{y}{z \cdot z}} \]

   6. associate-/r* 94.7%

      \[\leadsto \frac{1}{\frac{z + 1}{x}} \cdot \color{blue}{\frac{\frac{y}{z}}{z}} \]

   7. frac-times 97.2%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{y}{z}}{\frac{z + 1}{x} \cdot z}} \]

   8. *-un-lft-identity 97.2%

      \[\leadsto \frac{\color{blue}{\frac{y}{z}}}{\frac{z + 1}{x} \cdot z} \]

6. Applied egg-rr 97.2%

   \[\leadsto \color{blue}{\frac{\frac{y}{z}}{\frac{z + 1}{x} \cdot z}} \]

7. Taylor expanded in z around 0 74.5%

   \[\leadsto \frac{\frac{y}{z}}{\color{blue}{\frac{z}{x}}} \]
8. Step-by-step derivation

   1. associate-/l/ 68.7%

      \[\leadsto \color{blue}{\frac{y}{\frac{z}{x} \cdot z}} \]

   2. add-sqr-sqrt 34.3%

      \[\leadsto \frac{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}{\frac{z}{x} \cdot z} \]

   3. sqrt-unprod 39.8%

      \[\leadsto \frac{\color{blue}{\sqrt{y \cdot y}}}{\frac{z}{x} \cdot z} \]

   4. sqr-neg 39.8%

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

   5. sqrt-unprod 11.4%

      \[\leadsto \frac{\color{blue}{\sqrt{-y} \cdot \sqrt{-y}}}{\frac{z}{x} \cdot z} \]

   6. add-sqr-sqrt 24.0%

      \[\leadsto \frac{\color{blue}{-y}}{\frac{z}{x} \cdot z} \]

   7. *-un-lft-identity 24.0%

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

   8. frac-times 23.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{z}{x}} \cdot \frac{-y}{z}} \]

   9. clear-num 23.7%

      \[\leadsto \color{blue}{\frac{x}{z}} \cdot \frac{-y}{z} \]

   10. associate-/l* 22.6%

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

   11. *-commutative 22.6%

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

   12. *-un-lft-identity 22.6%

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

   13. times-frac 24.0%

       \[\leadsto \color{blue}{\frac{-y}{1} \cdot \frac{\frac{x}{z}}{z}} \]

   14. add-sqr-sqrt 11.4%

       \[\leadsto \frac{\color{blue}{\sqrt{-y} \cdot \sqrt{-y}}}{1} \cdot \frac{\frac{x}{z}}{z} \]

   15. sqrt-unprod 39.5%

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

   16. sqr-neg 39.5%

       \[\leadsto \frac{\sqrt{\color{blue}{y \cdot y}}}{1} \cdot \frac{\frac{x}{z}}{z} \]

   17. sqrt-unprod 33.5%

       \[\leadsto \frac{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}{1} \cdot \frac{\frac{x}{z}}{z} \]

   18. add-sqr-sqrt 67.6%

       \[\leadsto \frac{\color{blue}{y}}{1} \cdot \frac{\frac{x}{z}}{z} \]

   19. /-rgt-identity 67.6%

       \[\leadsto \color{blue}{y} \cdot \frac{\frac{x}{z}}{z} \]

9. Applied egg-rr 67.6%

   \[\leadsto \color{blue}{y \cdot \frac{\frac{x}{z}}{z}} \]

10. Final simplification 67.6%

    \[\leadsto y \cdot \frac{\frac{x}{z}}{z} \]
11. Add Preprocessing

Alternative 10: 74.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{y}{z \cdot \frac{z}{x\_m}} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z) :precision binary64 (* x_s (/ y (* z (/ z x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	return x_s * (y / (z * (z / x_m)));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x_s * (y / (z * (z / x_m)))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	return x_s * (y / (z * (z / x_m)));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	return x_s * (y / (z * (z / x_m)))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(y / Float64(z * Float64(z / x_m))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * (y / (z * (z / x_m)));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * N[(y / N[(z * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.1%

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

   1. *-commutative 80.1%

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

   2. frac-times 87.4%

      \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}} \]

   3. associate-*l/ 86.2%

      \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{z + 1}}{z \cdot z}} \]

   4. times-frac 97.0%

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{\frac{x}{z + 1}}{z}} \]

4. Applied egg-rr 97.0%

   \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{\frac{x}{z + 1}}{z}} \]
5. Step-by-step derivation

   1. *-commutative 97.0%

      \[\leadsto \color{blue}{\frac{\frac{x}{z + 1}}{z} \cdot \frac{y}{z}} \]

   2. clear-num 96.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{z}{\frac{x}{z + 1}}}} \cdot \frac{y}{z} \]

   3. frac-times 86.3%

      \[\leadsto \color{blue}{\frac{1 \cdot y}{\frac{z}{\frac{x}{z + 1}} \cdot z}} \]

   4. *-un-lft-identity 86.3%

      \[\leadsto \frac{\color{blue}{y}}{\frac{z}{\frac{x}{z + 1}} \cdot z} \]

   5. div-inv 86.3%

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

   6. clear-num 86.3%

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

6. Applied egg-rr 86.3%

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

7. Taylor expanded in z around 0 68.7%

   \[\leadsto \frac{y}{\color{blue}{\frac{z}{x}} \cdot z} \]

8. Final simplification 68.7%

   \[\leadsto \frac{y}{z \cdot \frac{z}{x}} \]
9. Add Preprocessing

Developer target: 95.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< z 249.6182814532307)
   (/ (* y (/ x z)) (+ z (* z z)))
   (/ (* (/ (/ y z) (+ 1.0 z)) x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z < 249.6182814532307) {
		tmp = (y * (x / z)) / (z + (z * z));
	} else {
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z < 249.6182814532307d0) then
        tmp = (y * (x / z)) / (z + (z * z))
    else
        tmp = (((y / z) / (1.0d0 + z)) * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z < 249.6182814532307) {
		tmp = (y * (x / z)) / (z + (z * z));
	} else {
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z < 249.6182814532307:
		tmp = (y * (x / z)) / (z + (z * z))
	else:
		tmp = (((y / z) / (1.0 + z)) * x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z < 249.6182814532307)
		tmp = Float64(Float64(y * Float64(x / z)) / Float64(z + Float64(z * z)));
	else
		tmp = Float64(Float64(Float64(Float64(y / z) / Float64(1.0 + z)) * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z < 249.6182814532307)
		tmp = (y * (x / z)) / (z + (z * z));
	else
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[z, 249.6182814532307], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / N[(z + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y / z), $MachinePrecision] / N[(1.0 + z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision]]

Reproduce

herbie shell --seed 2024078 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1.0 z)) x) z))

  (/ (* x y) (* (* z z) (+ z 1.0))))