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

Percentage Accurate: 82.8% → 96.6%

Time: 10.8s

Alternatives: 14

Speedup: 1.0×

• 25.3% 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: 82.8% 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: 96.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return ((x / z) * (y / 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 / z) * (y / z)) / (z + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.3%

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

   1. associate-*l* 81.3%

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

   2. times-frac 95.0%

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

   3. distribute-lft-in 95.0%

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

   4. fma-def 95.0%

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

   5. *-rgt-identity 95.0%

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

3. Simplified 95.0%

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

   1. fma-udef 95.0%

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

   2. *-rgt-identity 95.0%

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

   3. distribute-lft-in 95.0%

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

   4. times-frac 81.3%

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

   5. associate-*l* 81.3%

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

   6. associate-/r* 84.4%

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

   7. times-frac 98.1%

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

5. Applied egg-rr 98.1%

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

6. Final simplification 98.1%

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

Alternative 2: 94.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ \mathbf{if}\;t_0 \leq -10000000 \lor \neg \left(t_0 \leq 10^{-18}\right):\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ z 1.0) (* z z))))
   (if (or (<= t_0 -10000000.0) (not (<= t_0 1e-18)))
     (* (/ y z) (/ (/ x z) z))
     (* (/ x z) (- (/ y z) y)))))

C

double code(double x, double y, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double tmp;
	if ((t_0 <= -10000000.0) || !(t_0 <= 1e-18)) {
		tmp = (y / z) * ((x / z) / z);
	} else {
		tmp = (x / z) * ((y / z) - y);
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (z + 1.0d0) * (z * z)
    if ((t_0 <= (-10000000.0d0)) .or. (.not. (t_0 <= 1d-18))) then
        tmp = (y / z) * ((x / z) / z)
    else
        tmp = (x / z) * ((y / z) - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double tmp;
	if ((t_0 <= -10000000.0) || !(t_0 <= 1e-18)) {
		tmp = (y / z) * ((x / z) / z);
	} else {
		tmp = (x / z) * ((y / z) - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z + 1.0) * (z * z)
	tmp = 0
	if (t_0 <= -10000000.0) or not (t_0 <= 1e-18):
		tmp = (y / z) * ((x / z) / z)
	else:
		tmp = (x / z) * ((y / z) - y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if ((t_0 <= -10000000.0) || !(t_0 <= 1e-18))
		tmp = Float64(Float64(y / z) * Float64(Float64(x / z) / z));
	else
		tmp = Float64(Float64(x / z) * Float64(Float64(y / z) - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z + 1.0) * (z * z);
	tmp = 0.0;
	if ((t_0 <= -10000000.0) || ~((t_0 <= 1e-18)))
		tmp = (y / z) * ((x / z) / z);
	else
		tmp = (x / z) * ((y / z) - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -10000000.0], N[Not[LessEqual[t$95$0, 1e-18]], $MachinePrecision]], N[(N[(y / z), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z 1)) < -1e7 or 1.0000000000000001e-18 < (*.f64 (*.f64 z z) (+.f64 z 1))

   1. Initial program 82.1%

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

      1. /-rgt-identity 82.1%

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

      2. associate-/l* 82.1%

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

      3. associate-/l/ 83.8%

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

      4. associate-*l* 88.7%

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

      5. associate-*r/ 88.7%

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

      6. *-rgt-identity 88.7%

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

      7. associate-*l* 91.4%

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

      8. associate-*r/ 88.7%

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

      9. distribute-lft-in 88.7%

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

      10. fma-def 88.7%

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

      11. *-rgt-identity 88.7%

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

   3. Simplified 88.7%

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

   4. Taylor expanded in z around inf 87.2%

      \[\leadsto \frac{x}{z \cdot \color{blue}{\frac{{z}^{2}}{y}}} \]
   5. Step-by-step derivation

      1. unpow2 87.2%

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

   6. Simplified 87.2%

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

      1. associate-/r* 90.5%

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

      2. associate-/l* 95.8%

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

      3. associate-/r/ 97.2%

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

   8. Applied egg-rr 97.2%

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

   if -1e7 < (*.f64 (*.f64 z z) (+.f64 z 1)) < 1.0000000000000001e-18

   1. Initial program 80.3%

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

      1. associate-*l* 80.3%

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

      2. times-frac 98.0%

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

      3. distribute-lft-in 98.0%

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

      4. fma-def 98.0%

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

      5. *-rgt-identity 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in z around 0 98.0%

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

      1. neg-mul-1 98.0%

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

      2. +-commutative 98.0%

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

      3. unsub-neg 98.0%

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

   6. Simplified 98.0%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -10000000 \lor \neg \left(\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 10^{-18}\right):\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \end{array} \]

Alternative 3: 93.1% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 0.75)))
   (* (/ x z) (/ y (* z z)))
   (* (/ x z) (- (/ y z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 0.75)) {
		tmp = (x / z) * (y / (z * z));
	} else {
		tmp = (x / z) * ((y / z) - y);
	}
	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 <= (-1.0d0)) .or. (.not. (z <= 0.75d0))) then
        tmp = (x / z) * (y / (z * z))
    else
        tmp = (x / z) * ((y / z) - y)
    end if
    code = tmp
end function

Java

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

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 0.75):
		tmp = (x / z) * (y / (z * z))
	else:
		tmp = (x / z) * ((y / z) - y)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 0.75]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 0.75 < z

   1. Initial program 82.1%

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

      1. associate-*l* 82.1%

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

      2. times-frac 92.6%

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

      3. distribute-lft-in 92.6%

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

      4. fma-def 92.6%

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

      5. *-rgt-identity 92.6%

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

   3. Simplified 92.6%

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

   4. Taylor expanded in z around inf 91.1%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{y}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 91.1%

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

   6. Simplified 91.1%

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

   if -1 < z < 0.75

   1. Initial program 80.3%

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

      1. associate-*l* 80.3%

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

      2. times-frac 98.0%

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

      3. distribute-lft-in 98.0%

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

      4. fma-def 98.0%

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

      5. *-rgt-identity 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in z around 0 98.0%

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

      1. neg-mul-1 98.0%

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

      2. +-commutative 98.0%

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

      3. unsub-neg 98.0%

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

   6. Simplified 98.0%

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

4. Final simplification 94.2%

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

Alternative 4: 92.9% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 0.75)))
   (* (/ y z) (/ x (* z z)))
   (* (/ x z) (- (/ y z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 0.75)) {
		tmp = (y / z) * (x / (z * z));
	} else {
		tmp = (x / z) * ((y / z) - y);
	}
	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 <= (-1.0d0)) .or. (.not. (z <= 0.75d0))) then
        tmp = (y / z) * (x / (z * z))
    else
        tmp = (x / z) * ((y / z) - y)
    end if
    code = tmp
end function

Java

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

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 0.75):
		tmp = (y / z) * (x / (z * z))
	else:
		tmp = (x / z) * ((y / z) - y)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 0.75]], $MachinePrecision]], N[(N[(y / z), $MachinePrecision] * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 0.75 < z

   1. Initial program 82.1%

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

      1. /-rgt-identity 82.1%

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

      2. associate-/l* 82.1%

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

      3. associate-/l/ 83.8%

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

      4. associate-*l* 88.7%

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

      5. associate-*r/ 88.7%

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

      6. *-rgt-identity 88.7%

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

      7. associate-*l* 91.4%

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

      8. associate-*r/ 88.7%

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

      9. distribute-lft-in 88.7%

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

      10. fma-def 88.7%

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

      11. *-rgt-identity 88.7%

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

   3. Simplified 88.7%

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

   4. Taylor expanded in z around inf 87.2%

      \[\leadsto \frac{x}{z \cdot \color{blue}{\frac{{z}^{2}}{y}}} \]
   5. Step-by-step derivation

      1. unpow2 87.2%

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

   6. Simplified 87.2%

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

      1. associate-/l* 89.9%

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

      2. add-cbrt-cube 82.3%

         \[\leadsto \frac{x}{z \cdot \frac{\color{blue}{\sqrt[3]{\left(z \cdot z\right) \cdot z}}}{\frac{y}{z}}} \]

      3. cbrt-prod 87.1%

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

      4. *-un-lft-identity 87.1%

         \[\leadsto \frac{x}{z \cdot \frac{\sqrt[3]{z \cdot z} \cdot \sqrt[3]{z}}{\color{blue}{1 \cdot \frac{y}{z}}}} \]

      5. times-frac 87.1%

         \[\leadsto \frac{x}{z \cdot \color{blue}{\left(\frac{\sqrt[3]{z \cdot z}}{1} \cdot \frac{\sqrt[3]{z}}{\frac{y}{z}}\right)}} \]

      6. cbrt-prod 89.7%

         \[\leadsto \frac{x}{z \cdot \left(\frac{\color{blue}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{1} \cdot \frac{\sqrt[3]{z}}{\frac{y}{z}}\right)} \]

      7. pow2 89.7%

         \[\leadsto \frac{x}{z \cdot \left(\frac{\color{blue}{{\left(\sqrt[3]{z}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{z}}{\frac{y}{z}}\right)} \]

   8. Applied egg-rr 89.7%

      \[\leadsto \frac{x}{z \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{z}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{z}}{\frac{y}{z}}\right)}} \]
   9. Step-by-step derivation

      1. /-rgt-identity 89.7%

         \[\leadsto \frac{x}{z \cdot \left(\color{blue}{{\left(\sqrt[3]{z}\right)}^{2}} \cdot \frac{\sqrt[3]{z}}{\frac{y}{z}}\right)} \]

      2. associate-*r/ 89.7%

         \[\leadsto \frac{x}{z \cdot \color{blue}{\frac{{\left(\sqrt[3]{z}\right)}^{2} \cdot \sqrt[3]{z}}{\frac{y}{z}}}} \]

      3. unpow2 89.7%

         \[\leadsto \frac{x}{z \cdot \frac{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)} \cdot \sqrt[3]{z}}{\frac{y}{z}}} \]

      4. rem-3cbrt-lft 89.9%

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

   10. Simplified 89.9%

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

       1. associate-/r* 95.8%

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

       2. *-un-lft-identity 95.8%

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

       3. associate-/r/ 95.8%

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

       4. times-frac 97.0%

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

       5. clear-num 97.2%

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

       6. associate-/l/ 91.9%

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

   12. Applied egg-rr 91.9%

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

   if -1 < z < 0.75

   1. Initial program 80.3%

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

      1. associate-*l* 80.3%

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

      2. times-frac 98.0%

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

      3. distribute-lft-in 98.0%

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

      4. fma-def 98.0%

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

      5. *-rgt-identity 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in z around 0 98.0%

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

      1. neg-mul-1 98.0%

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

      2. +-commutative 98.0%

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

      3. unsub-neg 98.0%

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

   6. Simplified 98.0%

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

4. Final simplification 94.6%

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

Alternative 5: 40.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \frac{y}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (/ x (/ z y))
   (if (<= z -5e-310) (* x (/ y (- z))) (/ y (/ z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = x / (z / y);
	} else if (z <= -5e-310) {
		tmp = x * (y / -z);
	} else {
		tmp = y / (z / x);
	}
	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 <= (-1.0d0)) then
        tmp = x / (z / y)
    else if (z <= (-5d-310)) then
        tmp = x * (y / -z)
    else
        tmp = y / (z / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = x / (z / y);
	} else if (z <= -5e-310) {
		tmp = x * (y / -z);
	} else {
		tmp = y / (z / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = x / (z / y)
	elif z <= -5e-310:
		tmp = x * (y / -z)
	else:
		tmp = y / (z / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(x / Float64(z / y));
	elseif (z <= -5e-310)
		tmp = Float64(x * Float64(y / Float64(-z)));
	else
		tmp = Float64(y / Float64(z / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = x / (z / y);
	elseif (z <= -5e-310)
		tmp = x * (y / -z);
	else
		tmp = y / (z / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.0], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -5e-310], N[(x * N[(y / (-z)), $MachinePrecision]), $MachinePrecision], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 79.0%

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

      1. associate-*l* 79.0%

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

      2. times-frac 93.0%

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

      3. distribute-lft-in 93.0%

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

      4. fma-def 93.0%

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

      5. *-rgt-identity 93.0%

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

   3. Simplified 93.0%

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

   4. Taylor expanded in z around 0 28.6%

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

      1. neg-mul-1 28.6%

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

      2. +-commutative 28.6%

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

      3. unsub-neg 28.6%

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

   6. Simplified 28.6%

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

   7. Taylor expanded in z around inf 24.1%

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

      1. associate-*r/ 24.1%

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

      2. *-commutative 24.1%

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

      3. neg-mul-1 24.1%

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

      4. distribute-lft-neg-in 24.1%

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

      5. associate-*r/ 30.7%

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

      6. distribute-lft-neg-out 30.7%

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

      7. distribute-rgt-neg-in 30.7%

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

      8. *-lft-identity 30.7%

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

      9. metadata-eval 30.7%

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

      10. times-frac 30.7%

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

      11. neg-mul-1 30.7%

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

      12. neg-mul-1 30.7%

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

      13. distribute-frac-neg 30.7%

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

      14. remove-double-neg 30.7%

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

   9. Simplified 30.7%

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

       1. associate-*r/ 24.1%

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

       2. associate-/l* 34.0%

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

       3. add-sqr-sqrt 34.0%

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

       4. sqrt-unprod 52.3%

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

       5. sqr-neg 52.3%

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

       6. sqrt-prod 0.0%

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

       7. add-sqr-sqrt 36.9%

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

   11. Applied egg-rr 36.9%

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

   if -1 < z < -4.999999999999985e-310

   1. Initial program 79.2%

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

      1. associate-*l* 79.2%

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

      2. times-frac 98.3%

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

      3. distribute-lft-in 98.3%

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

      4. fma-def 98.3%

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

      5. *-rgt-identity 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in z around 0 98.3%

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

      1. neg-mul-1 98.3%

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

      2. +-commutative 98.3%

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

      3. unsub-neg 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in z around inf 30.0%

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

      1. associate-*r/ 30.0%

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

      2. *-commutative 30.0%

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

      3. neg-mul-1 30.0%

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

      4. distribute-lft-neg-in 30.0%

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

      5. associate-*r/ 38.7%

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

      6. distribute-lft-neg-out 38.7%

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

      7. distribute-rgt-neg-in 38.7%

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

      8. *-lft-identity 38.7%

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

      9. metadata-eval 38.7%

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

      10. times-frac 38.7%

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

      11. neg-mul-1 38.7%

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

      12. neg-mul-1 38.7%

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

      13. distribute-frac-neg 38.7%

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

      14. remove-double-neg 38.7%

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

   9. Simplified 38.7%

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

   if -4.999999999999985e-310 < z

   1. Initial program 84.3%

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

      1. associate-*l* 84.3%

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

      2. times-frac 94.7%

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

      3. distribute-lft-in 94.7%

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

      4. fma-def 94.7%

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

      5. *-rgt-identity 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in z around 0 66.6%

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

      1. neg-mul-1 66.6%

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

      2. +-commutative 66.6%

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

      3. unsub-neg 66.6%

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

   6. Simplified 66.6%

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

   7. Taylor expanded in z around inf 17.8%

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

      1. associate-*r/ 17.8%

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

      2. *-commutative 17.8%

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

      3. neg-mul-1 17.8%

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

      4. distribute-lft-neg-in 17.8%

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

      5. associate-*r/ 20.4%

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

      6. distribute-lft-neg-out 20.4%

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

      7. distribute-rgt-neg-in 20.4%

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

      8. *-lft-identity 20.4%

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

      9. metadata-eval 20.4%

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

      10. times-frac 20.4%

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

      11. neg-mul-1 20.4%

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

      12. neg-mul-1 20.4%

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

      13. distribute-frac-neg 20.4%

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

      14. remove-double-neg 20.4%

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

   9. Simplified 20.4%

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

       1. *-commutative 20.4%

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

       2. associate-*l/ 17.8%

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

       3. associate-/l* 22.3%

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

       4. add-sqr-sqrt 0.0%

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

       5. sqrt-unprod 51.9%

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

       6. sqr-neg 51.9%

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

       7. sqrt-prod 37.7%

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

       8. add-sqr-sqrt 37.7%

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

   11. Applied egg-rr 37.7%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \frac{y}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 6: 75.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+157}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -9.5e+157) (* x (/ y (* z z))) (* (/ x z) (/ y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -9.5e+157) {
		tmp = x * (y / (z * z));
	} else {
		tmp = (x / z) * (y / 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 <= (-9.5d+157)) then
        tmp = x * (y / (z * z))
    else
        tmp = (x / z) * (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -9.5e+157) {
		tmp = x * (y / (z * z));
	} else {
		tmp = (x / z) * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -9.5e+157:
		tmp = x * (y / (z * z))
	else:
		tmp = (x / z) * (y / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -9.5e+157)
		tmp = Float64(x * Float64(y / Float64(z * z)));
	else
		tmp = Float64(Float64(x / z) * Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -9.5e+157)
		tmp = x * (y / (z * z));
	else
		tmp = (x / z) * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -9.5e+157], N[(x * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.4999999999999996e157

   1. Initial program 80.9%

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

      1. associate-*l* 80.9%

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

      2. times-frac 84.2%

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

      3. distribute-lft-in 84.2%

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

      4. fma-def 84.2%

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

      5. *-rgt-identity 84.2%

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

   3. Simplified 84.2%

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

   4. Taylor expanded in z around 0 80.9%

      \[\leadsto \color{blue}{\frac{y \cdot x}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 80.9%

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

      2. associate-/l* 84.2%

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

   6. Simplified 84.2%

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

      1. associate-/r/ 84.2%

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

   8. Applied egg-rr 84.2%

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

   if -9.4999999999999996e157 < z

   1. Initial program 81.4%

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

      1. associate-*l* 81.4%

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

      2. times-frac 96.8%

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

      3. distribute-lft-in 96.8%

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

      4. fma-def 96.8%

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

      5. *-rgt-identity 96.8%

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

   3. Simplified 96.8%

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

   4. Taylor expanded in z around 0 63.4%

      \[\leadsto \color{blue}{\frac{y \cdot x}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 63.4%

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

      2. associate-/l/ 66.7%

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

      3. associate-*r/ 70.0%

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

      4. associate-*l/ 72.7%

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

   6. Simplified 72.7%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+157}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \]

Alternative 7: 76.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1e-7) (* x (/ y (* z z))) (* y (/ (/ x z) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1e-7) {
		tmp = x * (y / (z * z));
	} else {
		tmp = y * ((x / z) / 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 (x <= (-1d-7)) then
        tmp = x * (y / (z * z))
    else
        tmp = y * ((x / z) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1e-7) {
		tmp = x * (y / (z * z));
	} else {
		tmp = y * ((x / z) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1e-7:
		tmp = x * (y / (z * z))
	else:
		tmp = y * ((x / z) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1e-7)
		tmp = Float64(x * Float64(y / Float64(z * z)));
	else
		tmp = Float64(y * Float64(Float64(x / z) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1e-7)
		tmp = x * (y / (z * z));
	else
		tmp = y * ((x / z) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1e-7], N[(x * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.9999999999999995e-8

   1. Initial program 71.9%

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

      1. associate-*l* 71.9%

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

      2. times-frac 87.2%

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

      3. distribute-lft-in 87.2%

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

      4. fma-def 87.2%

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

      5. *-rgt-identity 87.2%

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

   3. Simplified 87.2%

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

   4. Taylor expanded in z around 0 55.8%

      \[\leadsto \color{blue}{\frac{y \cdot x}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 55.8%

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

      2. associate-/l* 51.3%

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

   6. Simplified 51.3%

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

      1. associate-/r/ 57.5%

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

   8. Applied egg-rr 57.5%

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

   if -9.9999999999999995e-8 < x

   1. Initial program 85.7%

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

      1. associate-*l* 85.7%

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

      2. times-frac 98.6%

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

      3. distribute-lft-in 98.6%

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

      4. fma-def 98.6%

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

      5. *-rgt-identity 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in z around 0 70.5%

      \[\leadsto \color{blue}{\frac{y \cdot x}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 70.5%

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

      2. associate-/l* 74.2%

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

   6. Simplified 74.2%

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

      1. clear-num 74.4%

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

      2. associate-/r/ 73.8%

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

      3. clear-num 73.8%

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

      4. associate-/r* 77.8%

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

   8. Applied egg-rr 77.8%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 8: 76.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.05e-24) (/ x (* z (/ z y))) (* y (/ (/ x z) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.05e-24) {
		tmp = x / (z * (z / y));
	} else {
		tmp = y * ((x / z) / 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 (x <= (-1.05d-24)) then
        tmp = x / (z * (z / y))
    else
        tmp = y * ((x / z) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.05e-24) {
		tmp = x / (z * (z / y));
	} else {
		tmp = y * ((x / z) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.05e-24:
		tmp = x / (z * (z / y))
	else:
		tmp = y * ((x / z) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.05e-24)
		tmp = Float64(x / Float64(z * Float64(z / y)));
	else
		tmp = Float64(y * Float64(Float64(x / z) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.05e-24)
		tmp = x / (z * (z / y));
	else
		tmp = y * ((x / z) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.05e-24], N[(x / N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05e-24

   1. Initial program 72.4%

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

      1. /-rgt-identity 72.4%

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

      2. associate-/l* 72.4%

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

      3. associate-/l/ 73.8%

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

      4. associate-*l* 80.4%

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

      5. associate-*r/ 80.4%

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

      6. *-rgt-identity 80.4%

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

      7. associate-*l* 88.0%

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

      8. associate-*r/ 85.7%

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

      9. distribute-lft-in 85.7%

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

      10. fma-def 85.7%

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

      11. *-rgt-identity 85.7%

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

   3. Simplified 85.7%

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

   4. Taylor expanded in z around 0 62.9%

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

   if -1.05e-24 < x

   1. Initial program 85.8%

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

      1. associate-*l* 85.8%

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

      2. times-frac 98.6%

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

      3. distribute-lft-in 98.6%

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

      4. fma-def 98.6%

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

      5. *-rgt-identity 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in z around 0 70.2%

      \[\leadsto \color{blue}{\frac{y \cdot x}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 70.2%

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

      2. associate-/l* 74.0%

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

   6. Simplified 74.0%

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

      1. clear-num 74.2%

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

      2. associate-/r/ 73.6%

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

      3. clear-num 73.6%

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

      4. associate-/r* 77.8%

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

   8. Applied egg-rr 77.8%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 9: 76.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z}}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3e-25) (/ x (/ z (/ y z))) (* y (/ (/ x z) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3e-25) {
		tmp = x / (z / (y / z));
	} else {
		tmp = y * ((x / z) / 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 (x <= (-3d-25)) then
        tmp = x / (z / (y / z))
    else
        tmp = y * ((x / z) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3e-25) {
		tmp = x / (z / (y / z));
	} else {
		tmp = y * ((x / z) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3e-25:
		tmp = x / (z / (y / z))
	else:
		tmp = y * ((x / z) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3e-25)
		tmp = Float64(x / Float64(z / Float64(y / z)));
	else
		tmp = Float64(y * Float64(Float64(x / z) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3e-25)
		tmp = x / (z / (y / z));
	else
		tmp = y * ((x / z) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3e-25], N[(x / N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.9999999999999998e-25

   1. Initial program 72.4%

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

      1. associate-*l* 72.4%

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

      2. times-frac 87.9%

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

      3. distribute-lft-in 87.9%

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

      4. fma-def 87.9%

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

      5. *-rgt-identity 87.9%

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

   3. Simplified 87.9%

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

   4. Taylor expanded in z around 0 57.2%

      \[\leadsto \color{blue}{\frac{y \cdot x}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 57.2%

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

      2. associate-/l* 53.0%

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

   6. Simplified 53.0%

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

      1. clear-num 53.0%

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

      2. associate-/r/ 52.0%

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

      3. clear-num 52.0%

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

      4. associate-/r* 49.4%

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

   8. Applied egg-rr 49.4%

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

      1. associate-/l/ 52.0%

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

      2. associate-/r/ 58.9%

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

      3. associate-/l* 62.9%

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

   10. Applied egg-rr 62.9%

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

   if -2.9999999999999998e-25 < x

   1. Initial program 85.8%

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

      1. associate-*l* 85.8%

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

      2. times-frac 98.6%

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

      3. distribute-lft-in 98.6%

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

      4. fma-def 98.6%

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

      5. *-rgt-identity 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in z around 0 70.2%

      \[\leadsto \color{blue}{\frac{y \cdot x}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 70.2%

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

      2. associate-/l* 74.0%

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

   6. Simplified 74.0%

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

      1. clear-num 74.2%

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

      2. associate-/r/ 73.6%

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

      3. clear-num 73.6%

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

      4. associate-/r* 77.8%

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

   8. Applied egg-rr 77.8%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z}}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 10: 31.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{+163}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.1e+163) (* x (/ y z)) (* (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.1e+163) {
		tmp = x * (y / z);
	} else {
		tmp = (x / z) * y;
	}
	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 (x <= (-5.1d+163)) then
        tmp = x * (y / z)
    else
        tmp = (x / z) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.1e+163) {
		tmp = x * (y / z);
	} else {
		tmp = (x / z) * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.1e+163:
		tmp = x * (y / z)
	else:
		tmp = (x / z) * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.1e+163)
		tmp = Float64(x * Float64(y / z));
	else
		tmp = Float64(Float64(x / z) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -5.1e+163)
		tmp = x * (y / z);
	else
		tmp = (x / z) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.1e+163], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.1000000000000002e163

   1. Initial program 76.4%

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

      1. associate-*l* 76.4%

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

      2. times-frac 77.1%

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

      3. distribute-lft-in 77.1%

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

      4. fma-def 77.1%

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

      5. *-rgt-identity 77.1%

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

   3. Simplified 77.1%

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

   4. Taylor expanded in z around 0 41.3%

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

      1. neg-mul-1 41.3%

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

      2. +-commutative 41.3%

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

      3. unsub-neg 41.3%

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

   6. Simplified 41.3%

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

   7. Taylor expanded in z around inf 16.7%

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

      1. associate-*r/ 16.7%

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

      2. *-commutative 16.7%

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

      3. neg-mul-1 16.7%

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

      4. distribute-lft-neg-in 16.7%

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

      5. associate-*r/ 31.1%

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

      6. distribute-lft-neg-out 31.1%

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

      7. distribute-rgt-neg-in 31.1%

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

      8. *-lft-identity 31.1%

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

      9. metadata-eval 31.1%

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

      10. times-frac 31.1%

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

      11. neg-mul-1 31.1%

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

      12. neg-mul-1 31.1%

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

      13. distribute-frac-neg 31.1%

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

      14. remove-double-neg 31.1%

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

   9. Simplified 31.1%

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

       1. expm1-log1p-u 23.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{y}{-z}\right)\right)} \]

       2. expm1-udef 23.5%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{y}{-z}\right)} - 1} \]

       3. associate-*r/ 23.5%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot y}{-z}}\right)} - 1 \]

       4. add-sqr-sqrt 22.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x \cdot y}{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}}\right)} - 1 \]

       5. sqrt-unprod 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x \cdot y}{\color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}}\right)} - 1 \]

       6. sqr-neg 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x \cdot y}{\sqrt{\color{blue}{z \cdot z}}}\right)} - 1 \]

       7. sqrt-prod 6.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x \cdot y}{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}\right)} - 1 \]

       8. add-sqr-sqrt 20.8%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x \cdot y}{\color{blue}{z}}\right)} - 1 \]

   11. Applied egg-rr 20.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot y}{z}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 6.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot y}{z}\right)\right)} \]

       2. expm1-log1p 17.4%

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

       3. *-commutative 17.4%

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

       4. associate-*l/ 31.8%

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

   13. Simplified 31.8%

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

   if -5.1000000000000002e163 < x

   1. Initial program 81.9%

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

      1. associate-*l* 81.9%

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

      2. times-frac 97.2%

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

      3. distribute-lft-in 97.2%

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

      4. fma-def 97.2%

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

      5. *-rgt-identity 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in z around 0 64.4%

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

      1. neg-mul-1 64.4%

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

      2. +-commutative 64.4%

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

      3. unsub-neg 64.4%

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

   6. Simplified 64.4%

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

   7. Taylor expanded in z around inf 23.7%

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

      1. associate-*r/ 23.7%

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

      2. *-commutative 23.7%

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

      3. neg-mul-1 23.7%

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

      4. distribute-lft-neg-in 23.7%

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

      5. associate-*r/ 28.1%

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

      6. distribute-lft-neg-out 28.1%

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

      7. distribute-rgt-neg-in 28.1%

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

      8. *-lft-identity 28.1%

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

      9. metadata-eval 28.1%

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

      10. times-frac 28.1%

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

      11. neg-mul-1 28.1%

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

      12. neg-mul-1 28.1%

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

      13. distribute-frac-neg 28.1%

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

      14. remove-double-neg 28.1%

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

   9. Simplified 28.1%

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

       1. expm1-log1p-u 23.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{y}{-z}\right)\right)} \]

       2. expm1-udef 38.1%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{y}{-z}\right)} - 1} \]

       3. associate-*r/ 37.7%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot y}{-z}}\right)} - 1 \]

       4. add-sqr-sqrt 22.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x \cdot y}{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}}\right)} - 1 \]

       5. sqrt-unprod 43.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x \cdot y}{\color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}}\right)} - 1 \]

       6. sqr-neg 43.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x \cdot y}{\sqrt{\color{blue}{z \cdot z}}}\right)} - 1 \]

       7. sqrt-prod 17.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x \cdot y}{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}\right)} - 1 \]

       8. add-sqr-sqrt 36.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x \cdot y}{\color{blue}{z}}\right)} - 1 \]

   11. Applied egg-rr 36.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot y}{z}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 20.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot y}{z}\right)\right)} \]

       2. expm1-log1p 23.9%

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

       3. *-commutative 23.9%

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

       4. associate-*r/ 27.7%

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

   13. Simplified 27.7%

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

4. Final simplification 28.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{+163}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \]

Alternative 11: 31.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+80}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1e+80) (/ x (/ z y)) (* (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e+80) {
		tmp = x / (z / y);
	} else {
		tmp = (x / z) * y;
	}
	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 (y <= 1d+80) then
        tmp = x / (z / y)
    else
        tmp = (x / z) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e+80) {
		tmp = x / (z / y);
	} else {
		tmp = (x / z) * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1e+80:
		tmp = x / (z / y)
	else:
		tmp = (x / z) * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1e+80)
		tmp = Float64(x / Float64(z / y));
	else
		tmp = Float64(Float64(x / z) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1e+80)
		tmp = x / (z / y);
	else
		tmp = (x / z) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1e+80], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1e80

   1. Initial program 81.4%

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

      1. associate-*l* 81.4%

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

      2. times-frac 94.8%

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

      3. distribute-lft-in 94.8%

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

      4. fma-def 94.8%

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

      5. *-rgt-identity 94.8%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in z around 0 62.6%

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

      1. neg-mul-1 62.6%

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

      2. +-commutative 62.6%

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

      3. unsub-neg 62.6%

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

   6. Simplified 62.6%

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

   7. Taylor expanded in z around inf 22.9%

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

      1. associate-*r/ 22.9%

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

      2. *-commutative 22.9%

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

      3. neg-mul-1 22.9%

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

      4. distribute-lft-neg-in 22.9%

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

      5. associate-*r/ 27.8%

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

      6. distribute-lft-neg-out 27.8%

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

      7. distribute-rgt-neg-in 27.8%

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

      8. *-lft-identity 27.8%

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

      9. metadata-eval 27.8%

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

      10. times-frac 27.8%

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

      11. neg-mul-1 27.8%

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

      12. neg-mul-1 27.8%

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

      13. distribute-frac-neg 27.8%

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

      14. remove-double-neg 27.8%

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

   9. Simplified 27.8%

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

       1. associate-*r/ 22.9%

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

       2. associate-/l* 29.6%

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

       3. add-sqr-sqrt 19.3%

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

       4. sqrt-unprod 48.6%

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

       5. sqr-neg 48.6%

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

       6. sqrt-prod 17.2%

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

       7. add-sqr-sqrt 32.2%

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

   11. Applied egg-rr 32.2%

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

   if 1e80 < y

   1. Initial program 80.9%

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

      1. associate-*l* 80.9%

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

      2. times-frac 95.8%

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

      3. distribute-lft-in 95.9%

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

      4. fma-def 95.9%

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

      5. *-rgt-identity 95.9%

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

   3. Simplified 95.9%

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

   4. Taylor expanded in z around 0 58.9%

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

      1. neg-mul-1 58.9%

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

      2. +-commutative 58.9%

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

      3. unsub-neg 58.9%

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

   6. Simplified 58.9%

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

   7. Taylor expanded in z around inf 23.3%

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

      1. associate-*r/ 23.3%

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

      2. *-commutative 23.3%

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

      3. neg-mul-1 23.3%

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

      4. distribute-lft-neg-in 23.3%

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

      5. associate-*r/ 31.0%

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

      6. distribute-lft-neg-out 31.0%

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

      7. distribute-rgt-neg-in 31.0%

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

      8. *-lft-identity 31.0%

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

      9. metadata-eval 31.0%

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

      10. times-frac 31.0%

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

      11. neg-mul-1 31.0%

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

      12. neg-mul-1 31.0%

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

      13. distribute-frac-neg 31.0%

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

      14. remove-double-neg 31.0%

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

   9. Simplified 31.0%

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

       1. expm1-log1p-u 13.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{y}{-z}\right)\right)} \]

       2. expm1-udef 32.9%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{y}{-z}\right)} - 1} \]

       3. associate-*r/ 32.9%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot y}{-z}}\right)} - 1 \]

       4. add-sqr-sqrt 20.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x \cdot y}{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}}\right)} - 1 \]

       5. sqrt-unprod 35.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x \cdot y}{\color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}}\right)} - 1 \]

       6. sqr-neg 35.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x \cdot y}{\sqrt{\color{blue}{z \cdot z}}}\right)} - 1 \]

       7. sqrt-prod 14.8%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x \cdot y}{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}\right)} - 1 \]

       8. add-sqr-sqrt 27.4%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x \cdot y}{\color{blue}{z}}\right)} - 1 \]

   11. Applied egg-rr 27.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot y}{z}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 8.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot y}{z}\right)\right)} \]

       2. expm1-log1p 15.6%

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

       3. *-commutative 15.6%

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

       4. associate-*r/ 31.5%

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

   13. Simplified 31.5%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+80}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \]

Alternative 12: 31.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{+79}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.4e+79) (/ x (/ z y)) (/ y (/ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.4e+79) {
		tmp = x / (z / y);
	} else {
		tmp = y / (z / x);
	}
	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 (y <= 4.4d+79) then
        tmp = x / (z / y)
    else
        tmp = y / (z / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.4e+79) {
		tmp = x / (z / y);
	} else {
		tmp = y / (z / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.4e+79:
		tmp = x / (z / y)
	else:
		tmp = y / (z / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.4e+79)
		tmp = Float64(x / Float64(z / y));
	else
		tmp = Float64(y / Float64(z / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.4e+79)
		tmp = x / (z / y);
	else
		tmp = y / (z / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.4e+79], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.3999999999999998e79

   1. Initial program 81.4%

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

      1. associate-*l* 81.4%

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

      2. times-frac 94.8%

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

      3. distribute-lft-in 94.8%

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

      4. fma-def 94.8%

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

      5. *-rgt-identity 94.8%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in z around 0 62.6%

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

      1. neg-mul-1 62.6%

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

      2. +-commutative 62.6%

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

      3. unsub-neg 62.6%

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

   6. Simplified 62.6%

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

   7. Taylor expanded in z around inf 22.9%

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

      1. associate-*r/ 22.9%

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

      2. *-commutative 22.9%

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

      3. neg-mul-1 22.9%

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

      4. distribute-lft-neg-in 22.9%

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

      5. associate-*r/ 27.8%

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

      6. distribute-lft-neg-out 27.8%

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

      7. distribute-rgt-neg-in 27.8%

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

      8. *-lft-identity 27.8%

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

      9. metadata-eval 27.8%

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

      10. times-frac 27.8%

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

      11. neg-mul-1 27.8%

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

      12. neg-mul-1 27.8%

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

      13. distribute-frac-neg 27.8%

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

      14. remove-double-neg 27.8%

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

   9. Simplified 27.8%

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

       1. associate-*r/ 22.9%

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

       2. associate-/l* 29.6%

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

       3. add-sqr-sqrt 19.3%

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

       4. sqrt-unprod 48.6%

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

       5. sqr-neg 48.6%

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

       6. sqrt-prod 17.2%

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

       7. add-sqr-sqrt 32.2%

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

   11. Applied egg-rr 32.2%

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

   if 4.3999999999999998e79 < y

   1. Initial program 80.9%

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

      1. associate-*l* 80.9%

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

      2. times-frac 95.8%

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

      3. distribute-lft-in 95.9%

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

      4. fma-def 95.9%

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

      5. *-rgt-identity 95.9%

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

   3. Simplified 95.9%

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

   4. Taylor expanded in z around 0 58.9%

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

      1. neg-mul-1 58.9%

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

      2. +-commutative 58.9%

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

      3. unsub-neg 58.9%

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

   6. Simplified 58.9%

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

   7. Taylor expanded in z around inf 23.3%

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

      1. associate-*r/ 23.3%

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

      2. *-commutative 23.3%

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

      3. neg-mul-1 23.3%

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

      4. distribute-lft-neg-in 23.3%

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

      5. associate-*r/ 31.0%

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

      6. distribute-lft-neg-out 31.0%

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

      7. distribute-rgt-neg-in 31.0%

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

      8. *-lft-identity 31.0%

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

      9. metadata-eval 31.0%

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

      10. times-frac 31.0%

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

      11. neg-mul-1 31.0%

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

      12. neg-mul-1 31.0%

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

      13. distribute-frac-neg 31.0%

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

      14. remove-double-neg 31.0%

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

   9. Simplified 31.0%

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

       1. *-commutative 31.0%

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

       2. associate-*l/ 23.3%

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

       3. associate-/l* 39.1%

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

       4. add-sqr-sqrt 26.5%

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

       5. sqrt-unprod 52.0%

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

       6. sqr-neg 52.0%

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

       7. sqrt-prod 21.3%

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

       8. add-sqr-sqrt 31.5%

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

   11. Applied egg-rr 31.5%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{+79}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 13: 73.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{x}{z} \cdot \frac{y}{z} \end{array} \]

FPCore

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

C

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

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 / z) * (y / z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.3%

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

   1. associate-*l* 81.3%

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

   2. times-frac 95.0%

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

   3. distribute-lft-in 95.0%

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

   4. fma-def 95.0%

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

   5. *-rgt-identity 95.0%

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

3. Simplified 95.0%

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

4. Taylor expanded in z around 0 65.9%

   \[\leadsto \color{blue}{\frac{y \cdot x}{{z}^{2}}} \]
5. Step-by-step derivation

   1. unpow2 65.9%

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

   2. associate-/l/ 66.4%

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

   3. associate-*r/ 69.6%

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

   4. associate-*l/ 72.0%

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

6. Simplified 72.0%

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

7. Final simplification 72.0%

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

Alternative 14: 30.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{x}{z} \cdot y \end{array} \]

FPCore

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

C

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

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 / z) * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.3%

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

   1. associate-*l* 81.3%

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

   2. times-frac 95.0%

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

   3. distribute-lft-in 95.0%

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

   4. fma-def 95.0%

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

   5. *-rgt-identity 95.0%

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

3. Simplified 95.0%

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

4. Taylor expanded in z around 0 61.9%

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

   1. neg-mul-1 61.9%

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

   2. +-commutative 61.9%

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

   3. unsub-neg 61.9%

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

6. Simplified 61.9%

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

7. Taylor expanded in z around inf 23.0%

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

   1. associate-*r/ 23.0%

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

   2. *-commutative 23.0%

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

   3. neg-mul-1 23.0%

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

   4. distribute-lft-neg-in 23.0%

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

   5. associate-*r/ 28.4%

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

   6. distribute-lft-neg-out 28.4%

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

   7. distribute-rgt-neg-in 28.4%

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

   8. *-lft-identity 28.4%

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

   9. metadata-eval 28.4%

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

   10. times-frac 28.4%

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

   11. neg-mul-1 28.4%

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

   12. neg-mul-1 28.4%

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

   13. distribute-frac-neg 28.4%

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

   14. remove-double-neg 28.4%

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

9. Simplified 28.4%

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

    1. expm1-log1p-u 23.8%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{y}{-z}\right)\right)} \]

    2. expm1-udef 36.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{y}{-z}\right)} - 1} \]

    3. associate-*r/ 36.1%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot y}{-z}}\right)} - 1 \]

    4. add-sqr-sqrt 22.3%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{x \cdot y}{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}}\right)} - 1 \]

    5. sqrt-unprod 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{x \cdot y}{\color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}}\right)} - 1 \]

    6. sqr-neg 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{x \cdot y}{\sqrt{\color{blue}{z \cdot z}}}\right)} - 1 \]

    7. sqrt-prod 15.9%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{x \cdot y}{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}\right)} - 1 \]

    8. add-sqr-sqrt 34.5%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{x \cdot y}{\color{blue}{z}}\right)} - 1 \]

11. Applied egg-rr 34.5%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot y}{z}\right)} - 1} \]
12. Step-by-step derivation

    1. expm1-def 18.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot y}{z}\right)\right)} \]

    2. expm1-log1p 23.2%

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

    3. *-commutative 23.2%

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

    4. associate-*r/ 26.9%

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

13. Simplified 26.9%

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

14. Final simplification 26.9%

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

Developer target: 95.0% accurate, 0.8× 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 2023238 
(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))))