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

Percentage Accurate: 83.4% → 99.3%

Time: 11.7s

Alternatives: 9

Speedup: 1.0×

• 25.7% 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: 83.4% 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

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

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ (sqrt y_m) z))) (* y_s (* t_0 (* t_0 (/ x (+ z 1.0)))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = sqrt(y_m) / z;
	return y_s * (t_0 * (t_0 * (x / (z + 1.0))));
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    t_0 = sqrt(y_m) / z
    code = y_s * (t_0 * (t_0 * (x / (z + 1.0d0))))
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = Math.sqrt(y_m) / z;
	return y_s * (t_0 * (t_0 * (x / (z + 1.0))));
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	t_0 = math.sqrt(y_m) / z
	return y_s * (t_0 * (t_0 * (x / (z + 1.0))))

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(sqrt(y_m) / z)
	return Float64(y_s * Float64(t_0 * Float64(t_0 * Float64(x / Float64(z + 1.0)))))
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m, z)
	t_0 = sqrt(y_m) / z;
	tmp = y_s * (t_0 * (t_0 * (x / (z + 1.0))));
end

Wolfram

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

Derivation

1. Initial program 84.0%

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

   1. *-commutative 84.0%

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

   2. frac-times 87.2%

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

   3. add-sqr-sqrt 57.5%

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

   4. associate-*l* 57.4%

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

   5. sqrt-div 48.5%

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

   6. sqrt-prod 23.4%

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

   7. add-sqr-sqrt 34.6%

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

   8. sqrt-div 34.5%

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

   9. sqrt-prod 27.6%

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

   10. add-sqr-sqrt 55.7%

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

4. Applied egg-rr 55.7%

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

5. Final simplification 55.7%

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

Alternative 2: 95.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.75\right):\\ \;\;\;\;\frac{y_m}{z} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m}{z} \cdot \left(\frac{x}{z} - x\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (or (<= z -1.0) (not (<= z 0.75)))
    (* (/ y_m z) (/ (/ x z) z))
    (* (/ y_m z) (- (/ x z) x)))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 0.75)) {
		tmp = (y_m / z) * ((x / z) / z);
	} else {
		tmp = (y_m / z) * ((x / z) - x);
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 0.75d0))) then
        tmp = (y_m / z) * ((x / z) / z)
    else
        tmp = (y_m / z) * ((x / z) - x)
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 0.75)) {
		tmp = (y_m / z) * ((x / z) / z);
	} else {
		tmp = (y_m / z) * ((x / z) - x);
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 0.75):
		tmp = (y_m / z) * ((x / z) / z)
	else:
		tmp = (y_m / z) * ((x / z) - x)
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 0.75))
		tmp = Float64(Float64(y_m / z) * Float64(Float64(x / z) / z));
	else
		tmp = Float64(Float64(y_m / z) * Float64(Float64(x / z) - x));
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 0.75)))
		tmp = (y_m / z) * ((x / z) / z);
	else
		tmp = (y_m / z) * ((x / z) - x);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 0.75]], $MachinePrecision]], N[(N[(y$95$m / z), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / z), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 0.75 < z

   1. Initial program 87.9%

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

      1. *-commutative 87.9%

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

      2. frac-times 92.8%

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

      3. associate-*l/ 94.4%

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

      4. times-frac 97.9%

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

   4. Applied egg-rr 97.9%

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

   5. Taylor expanded in z around inf 97.0%

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

   if -1 < z < 0.75

   1. Initial program 80.0%

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

      1. *-commutative 80.0%

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

      2. frac-times 81.4%

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

      3. associate-*l/ 80.0%

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

      4. times-frac 98.2%

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

   4. Applied egg-rr 98.2%

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

   5. Taylor expanded in z around 0 97.3%

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

      1. neg-mul-1 97.3%

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

      2. +-commutative 97.3%

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

      3. unsub-neg 97.3%

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

   7. Simplified 97.3%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.75\right):\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{x}{z} - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-85}:\\ \;\;\;\;\frac{y_m}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y_m}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= x 2.8e-85) (* (/ y_m z) (/ x z)) (* x (/ y_m (* z z))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 2.8e-85) {
		tmp = (y_m / z) * (x / z);
	} else {
		tmp = x * (y_m / (z * z));
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 2.8d-85) then
        tmp = (y_m / z) * (x / z)
    else
        tmp = x * (y_m / (z * z))
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 2.8e-85) {
		tmp = (y_m / z) * (x / z);
	} else {
		tmp = x * (y_m / (z * z));
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 2.8e-85:
		tmp = (y_m / z) * (x / z)
	else:
		tmp = x * (y_m / (z * z))
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 2.8e-85)
		tmp = Float64(Float64(y_m / z) * Float64(x / z));
	else
		tmp = Float64(x * Float64(y_m / Float64(z * z)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 2.8e-85)
		tmp = (y_m / z) * (x / z);
	else
		tmp = x * (y_m / (z * z));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 2.8e-85], N[(N[(y$95$m / z), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.80000000000000017e-85

   1. Initial program 82.8%

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

      1. *-commutative 82.8%

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

      2. frac-times 86.0%

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

      3. associate-*l/ 85.4%

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

      4. times-frac 98.7%

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in z around 0 78.7%

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

   if 2.80000000000000017e-85 < x

   1. Initial program 86.2%

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

      1. sqr-neg 86.2%

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

      2. *-commutative 86.2%

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

      3. times-frac 89.5%

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

      4. sqr-neg 89.5%

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

   3. Simplified 89.5%

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

   5. Taylor expanded in z around 0 63.3%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-85}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.9% accurate, 1.0× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (* (/ y_m z) (/ x (* z (+ z 1.0))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	return y_s * ((y_m / z) * (x / (z * (z + 1.0))));
}

Fortran

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

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	return y_s * ((y_m / z) * (x / (z * (z + 1.0))));
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	return y_s * ((y_m / z) * (x / (z * (z + 1.0))))

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(Float64(y_m / z) * Float64(x / Float64(z * Float64(z + 1.0)))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

   1. *-commutative 84.0%

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

   2. frac-times 87.2%

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

   3. associate-*l/ 87.4%

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

   4. times-frac 98.1%

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

4. Applied egg-rr 98.1%

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

5. Taylor expanded in x around 0 96.0%

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

   1. +-commutative 96.0%

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

7. Simplified 96.0%

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

8. Final simplification 96.0%

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

Alternative 5: 96.1% accurate, 1.0× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (* (/ y_m z) (/ (/ x (+ z 1.0)) z))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	return y_s * ((y_m / z) * ((x / (z + 1.0)) / z));
}

Fortran

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

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	return y_s * ((y_m / z) * ((x / (z + 1.0)) / z));
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	return y_s * ((y_m / z) * ((x / (z + 1.0)) / z))

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(Float64(y_m / z) * Float64(Float64(x / Float64(z + 1.0)) / z)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

   1. *-commutative 84.0%

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

   2. frac-times 87.2%

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

   3. associate-*l/ 87.4%

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

   4. times-frac 98.1%

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

4. Applied egg-rr 98.1%

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

5. Final simplification 98.1%

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

Alternative 6: 97.3% accurate, 1.0× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (/ (* (/ x (+ z 1.0)) (/ y_m z)) z)))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	return y_s * (((x / (z + 1.0)) * (y_m / z)) / z);
}

Fortran

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

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	return y_s * (((x / (z + 1.0)) * (y_m / z)) / z);
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	return y_s * (((x / (z + 1.0)) * (y_m / z)) / z)

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(Float64(Float64(x / Float64(z + 1.0)) * Float64(y_m / z)) / z))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

   1. *-commutative 84.0%

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

   2. associate-*l/ 85.4%

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

   3. *-commutative 85.4%

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

   4. sqr-neg 85.4%

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

   5. *-commutative 85.4%

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

   6. distribute-rgt1-in 73.7%

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

   7. sqr-neg 73.7%

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

   8. fma-def 85.4%

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

   9. sqr-neg 85.4%

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

   10. cube-unmult 85.5%

       \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{{z}^{3}}\right)} \]

3. Simplified 85.5%

   \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-def 73.7%

      \[\leadsto x \cdot \frac{y}{\color{blue}{z \cdot z + {z}^{3}}} \]

   2. associate-*r/ 73.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{z \cdot z + {z}^{3}}} \]

   3. *-commutative 73.1%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{z \cdot z + {z}^{3}} \]

   4. cube-mult 73.1%

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

   5. distribute-rgt1-in 84.0%

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

   6. *-commutative 84.0%

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

   7. frac-times 87.2%

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

   8. associate-/r* 94.6%

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

   9. associate-*l/ 97.6%

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

6. Applied egg-rr 97.6%

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

7. Final simplification 97.6%

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

Alternative 7: 74.9% accurate, 1.6× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z) :precision binary64 (* y_s (* (/ y_m z) (/ x z))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	return y_s * ((y_m / z) * (x / z));
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * ((y_m / z) * (x / z))
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	return y_s * ((y_m / z) * (x / z));
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	return y_s * ((y_m / z) * (x / z))

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(Float64(y_m / z) * Float64(x / z)))
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m, z)
	tmp = y_s * ((y_m / z) * (x / z));
end

Wolfram

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

Derivation

1. Initial program 84.0%

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

   1. *-commutative 84.0%

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

   2. frac-times 87.2%

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

   3. associate-*l/ 87.4%

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

   4. times-frac 98.1%

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

4. Applied egg-rr 98.1%

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

5. Taylor expanded in z around 0 69.9%

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

6. Final simplification 69.9%

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

Alternative 8: 75.0% accurate, 1.6× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z) :precision binary64 (* y_s (/ y_m (* z (/ z x)))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	return y_s * (y_m / (z * (z / x)));
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (y_m / (z * (z / x)))
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	return y_s * (y_m / (z * (z / x)));
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	return y_s * (y_m / (z * (z / x)))

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(y_m / Float64(z * Float64(z / x))))
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m, z)
	tmp = y_s * (y_m / (z * (z / x)));
end

Wolfram

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

Derivation

1. Initial program 84.0%

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

   1. *-commutative 84.0%

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

   2. frac-times 87.2%

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

   3. associate-*l/ 87.4%

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

   4. times-frac 98.1%

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

4. Applied egg-rr 98.1%

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

5. Taylor expanded in z around 0 69.9%

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

   1. *-commutative 69.9%

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

   2. clear-num 70.2%

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

   3. frac-times 70.9%

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

   4. *-un-lft-identity 70.9%

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

7. Applied egg-rr 70.9%

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

8. Final simplification 70.9%

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

Alternative 9: 30.6% accurate, 1.8× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z) :precision binary64 (* y_s (* y_m (/ (- x) z))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	return y_s * (y_m * (-x / z));
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (y_m * (-x / z))
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	return y_s * (y_m * (-x / z));
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	return y_s * (y_m * (-x / z))

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(y_m * Float64(Float64(-x) / z)))
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m, z)
	tmp = y_s * (y_m * (-x / z));
end

Wolfram

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

Derivation

1. Initial program 84.0%

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

   1. *-commutative 84.0%

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

   2. frac-times 87.2%

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

   3. associate-*l/ 87.4%

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

   4. times-frac 98.1%

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

4. Applied egg-rr 98.1%

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

5. Taylor expanded in z around 0 63.2%

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

   1. neg-mul-1 63.2%

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

   2. +-commutative 63.2%

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

   3. unsub-neg 63.2%

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

7. Simplified 63.2%

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

8. Taylor expanded in z around inf 21.6%

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

   1. mul-1-neg 21.6%

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

   2. distribute-frac-neg 21.6%

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

   3. distribute-lft-neg-out 21.6%

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

   4. *-commutative 21.6%

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

   5. associate-*r/ 26.1%

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

10. Simplified 26.1%

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

11. Final simplification 26.1%

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

Developer target: 95.4% 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 2024020 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1.0 z)) x) z))

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