Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Percentage Accurate: 84.8% → 95.8%

Time: 5.0s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.2%

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

   1. associate-*l/ 84.2%

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

   2. distribute-rgt-in 80.2%

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

   3. *-commutative 80.2%

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

   4. associate-/r/ 93.7%

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

   5. *-inverses 93.7%

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

   6. /-rgt-identity 93.7%

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

   7. associate-*r/ 94.3%

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

   8. *-commutative 94.3%

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

   9. associate-*r/ 97.7%

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

   10. fma-def 97.7%

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

3. Simplified 97.7%

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

   1. fma-udef 97.7%

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

5. Applied egg-rr 97.7%

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

6. Final simplification 97.7%

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

Alternative 2: 68.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{-77}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-156} \lor \neg \left(z \leq 6.3 \cdot 10^{-93}\right) \land z \leq 2.5 \cdot 10^{+73}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.65e-77)
   x
   (if (or (<= z 3.8e-156) (and (not (<= z 6.3e-93)) (<= z 2.5e+73)))
     (* x (/ y z))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.65e-77) {
		tmp = x;
	} else if ((z <= 3.8e-156) || (!(z <= 6.3e-93) && (z <= 2.5e+73))) {
		tmp = x * (y / z);
	} else {
		tmp = 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 <= (-1.65d-77)) then
        tmp = x
    else if ((z <= 3.8d-156) .or. (.not. (z <= 6.3d-93)) .and. (z <= 2.5d+73)) then
        tmp = x * (y / z)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.65e-77) {
		tmp = x;
	} else if ((z <= 3.8e-156) || (!(z <= 6.3e-93) && (z <= 2.5e+73))) {
		tmp = x * (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.65e-77:
		tmp = x
	elif (z <= 3.8e-156) or (not (z <= 6.3e-93) and (z <= 2.5e+73)):
		tmp = x * (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.65e-77)
		tmp = x;
	elseif ((z <= 3.8e-156) || (!(z <= 6.3e-93) && (z <= 2.5e+73)))
		tmp = Float64(x * Float64(y / z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.65e-77)
		tmp = x;
	elseif ((z <= 3.8e-156) || (~((z <= 6.3e-93)) && (z <= 2.5e+73)))
		tmp = x * (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.65e-77], x, If[Or[LessEqual[z, 3.8e-156], And[N[Not[LessEqual[z, 6.3e-93]], $MachinePrecision], LessEqual[z, 2.5e+73]]], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -1.64999999999999996e-77 or 3.80000000000000008e-156 < z < 6.30000000000000028e-93 or 2.49999999999999988e73 < z

   1. Initial program 80.5%

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

      1. associate-*l/ 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in z around inf 79.1%

      \[\leadsto \color{blue}{x} \]

   if -1.64999999999999996e-77 < z < 3.80000000000000008e-156 or 6.30000000000000028e-93 < z < 2.49999999999999988e73

   1. Initial program 95.1%

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

      1. associate-*l/ 89.3%

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

   3. Simplified 89.3%

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

   4. Taylor expanded in z around 0 78.3%

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

      1. *-commutative 78.3%

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

      2. associate-*r/ 77.0%

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

   6. Simplified 77.0%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{-77}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-156} \lor \neg \left(z \leq 6.3 \cdot 10^{-93}\right) \land z \leq 2.5 \cdot 10^{+73}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 68.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-76}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-156}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-91}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+72}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.8e-76)
   x
   (if (<= z 3.8e-156)
     (* x (/ y z))
     (if (<= z 2.9e-91) x (if (<= z 5.5e+72) (* y (/ x z)) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.8e-76) {
		tmp = x;
	} else if (z <= 3.8e-156) {
		tmp = x * (y / z);
	} else if (z <= 2.9e-91) {
		tmp = x;
	} else if (z <= 5.5e+72) {
		tmp = y * (x / z);
	} else {
		tmp = 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 <= (-4.8d-76)) then
        tmp = x
    else if (z <= 3.8d-156) then
        tmp = x * (y / z)
    else if (z <= 2.9d-91) then
        tmp = x
    else if (z <= 5.5d+72) then
        tmp = y * (x / z)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.8e-76) {
		tmp = x;
	} else if (z <= 3.8e-156) {
		tmp = x * (y / z);
	} else if (z <= 2.9e-91) {
		tmp = x;
	} else if (z <= 5.5e+72) {
		tmp = y * (x / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4.8e-76:
		tmp = x
	elif z <= 3.8e-156:
		tmp = x * (y / z)
	elif z <= 2.9e-91:
		tmp = x
	elif z <= 5.5e+72:
		tmp = y * (x / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.8e-76)
		tmp = x;
	elseif (z <= 3.8e-156)
		tmp = Float64(x * Float64(y / z));
	elseif (z <= 2.9e-91)
		tmp = x;
	elseif (z <= 5.5e+72)
		tmp = Float64(y * Float64(x / z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -4.8e-76)
		tmp = x;
	elseif (z <= 3.8e-156)
		tmp = x * (y / z);
	elseif (z <= 2.9e-91)
		tmp = x;
	elseif (z <= 5.5e+72)
		tmp = y * (x / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.8e-76], x, If[LessEqual[z, 3.8e-156], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.9e-91], x, If[LessEqual[z, 5.5e+72], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.80000000000000026e-76 or 3.80000000000000008e-156 < z < 2.9000000000000001e-91 or 5.5e72 < z

   1. Initial program 80.5%

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

      1. associate-*l/ 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in z around inf 79.1%

      \[\leadsto \color{blue}{x} \]

   if -4.80000000000000026e-76 < z < 3.80000000000000008e-156

   1. Initial program 95.5%

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

      1. associate-*l/ 89.1%

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

   3. Simplified 89.1%

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

   4. Taylor expanded in z around 0 86.0%

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

      1. *-commutative 86.0%

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

      2. associate-*r/ 82.3%

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

   6. Simplified 82.3%

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

   if 2.9000000000000001e-91 < z < 5.5e72

   1. Initial program 93.8%

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

      1. associate-*l/ 89.6%

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

   3. Simplified 89.6%

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

   4. Taylor expanded in z around 0 55.9%

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

      1. associate-*r/ 61.9%

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

   6. Simplified 61.9%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-76}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-156}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-91}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+72}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 68.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-83}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-156}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-92}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+72}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.8e-83)
   x
   (if (<= z 3.8e-156)
     (/ x (/ z y))
     (if (<= z 2.9e-92) x (if (<= z 5.5e+72) (* y (/ x z)) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.8e-83) {
		tmp = x;
	} else if (z <= 3.8e-156) {
		tmp = x / (z / y);
	} else if (z <= 2.9e-92) {
		tmp = x;
	} else if (z <= 5.5e+72) {
		tmp = y * (x / z);
	} else {
		tmp = 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 <= (-3.8d-83)) then
        tmp = x
    else if (z <= 3.8d-156) then
        tmp = x / (z / y)
    else if (z <= 2.9d-92) then
        tmp = x
    else if (z <= 5.5d+72) then
        tmp = y * (x / z)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.8e-83) {
		tmp = x;
	} else if (z <= 3.8e-156) {
		tmp = x / (z / y);
	} else if (z <= 2.9e-92) {
		tmp = x;
	} else if (z <= 5.5e+72) {
		tmp = y * (x / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.8e-83:
		tmp = x
	elif z <= 3.8e-156:
		tmp = x / (z / y)
	elif z <= 2.9e-92:
		tmp = x
	elif z <= 5.5e+72:
		tmp = y * (x / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.8e-83)
		tmp = x;
	elseif (z <= 3.8e-156)
		tmp = Float64(x / Float64(z / y));
	elseif (z <= 2.9e-92)
		tmp = x;
	elseif (z <= 5.5e+72)
		tmp = Float64(y * Float64(x / z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3.8e-83)
		tmp = x;
	elseif (z <= 3.8e-156)
		tmp = x / (z / y);
	elseif (z <= 2.9e-92)
		tmp = x;
	elseif (z <= 5.5e+72)
		tmp = y * (x / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.8e-83], x, If[LessEqual[z, 3.8e-156], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.9e-92], x, If[LessEqual[z, 5.5e+72], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.79999999999999977e-83 or 3.80000000000000008e-156 < z < 2.89999999999999985e-92 or 5.5e72 < z

   1. Initial program 80.5%

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

      1. associate-*l/ 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in z around inf 79.1%

      \[\leadsto \color{blue}{x} \]

   if -3.79999999999999977e-83 < z < 3.80000000000000008e-156

   1. Initial program 95.5%

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

      1. associate-*l/ 89.1%

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

   3. Simplified 89.1%

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

   4. Taylor expanded in z around 0 86.0%

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

      1. *-commutative 86.0%

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

      2. associate-*r/ 82.3%

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

   6. Simplified 82.3%

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

      1. clear-num 82.2%

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

      2. un-div-inv 82.5%

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

   8. Applied egg-rr 82.5%

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

   if 2.89999999999999985e-92 < z < 5.5e72

   1. Initial program 93.8%

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

      1. associate-*l/ 89.6%

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

   3. Simplified 89.6%

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

   4. Taylor expanded in z around 0 55.9%

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

      1. associate-*r/ 61.9%

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

   6. Simplified 61.9%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-83}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-156}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-92}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+72}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 70.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.65 \cdot 10^{-86}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-156}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-93}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+72}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.65e-86)
   x
   (if (<= z 3.8e-156)
     (/ (* x y) z)
     (if (<= z 5.2e-93) x (if (<= z 5.5e+72) (* y (/ x z)) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.65e-86) {
		tmp = x;
	} else if (z <= 3.8e-156) {
		tmp = (x * y) / z;
	} else if (z <= 5.2e-93) {
		tmp = x;
	} else if (z <= 5.5e+72) {
		tmp = y * (x / z);
	} else {
		tmp = 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 <= (-2.65d-86)) then
        tmp = x
    else if (z <= 3.8d-156) then
        tmp = (x * y) / z
    else if (z <= 5.2d-93) then
        tmp = x
    else if (z <= 5.5d+72) then
        tmp = y * (x / z)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.65e-86) {
		tmp = x;
	} else if (z <= 3.8e-156) {
		tmp = (x * y) / z;
	} else if (z <= 5.2e-93) {
		tmp = x;
	} else if (z <= 5.5e+72) {
		tmp = y * (x / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.65e-86:
		tmp = x
	elif z <= 3.8e-156:
		tmp = (x * y) / z
	elif z <= 5.2e-93:
		tmp = x
	elif z <= 5.5e+72:
		tmp = y * (x / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.65e-86)
		tmp = x;
	elseif (z <= 3.8e-156)
		tmp = Float64(Float64(x * y) / z);
	elseif (z <= 5.2e-93)
		tmp = x;
	elseif (z <= 5.5e+72)
		tmp = Float64(y * Float64(x / z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.65e-86)
		tmp = x;
	elseif (z <= 3.8e-156)
		tmp = (x * y) / z;
	elseif (z <= 5.2e-93)
		tmp = x;
	elseif (z <= 5.5e+72)
		tmp = y * (x / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.65e-86], x, If[LessEqual[z, 3.8e-156], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 5.2e-93], x, If[LessEqual[z, 5.5e+72], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.6499999999999998e-86 or 3.80000000000000008e-156 < z < 5.1999999999999997e-93 or 5.5e72 < z

   1. Initial program 80.5%

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

      1. associate-*l/ 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in z around inf 79.1%

      \[\leadsto \color{blue}{x} \]

   if -2.6499999999999998e-86 < z < 3.80000000000000008e-156

   1. Initial program 95.5%

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

      1. associate-*l/ 89.1%

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

   3. Simplified 89.1%

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

   4. Taylor expanded in z around 0 86.0%

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

   if 5.1999999999999997e-93 < z < 5.5e72

   1. Initial program 93.8%

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

      1. associate-*l/ 89.6%

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

   3. Simplified 89.6%

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

   4. Taylor expanded in z around 0 55.9%

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

      1. associate-*r/ 61.9%

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

   6. Simplified 61.9%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.65 \cdot 10^{-86}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-156}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-93}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+72}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 86.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{z} \cdot \left(y + z\right)\\ \mathbf{if}\;z \leq -1.45 \cdot 10^{-251}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10^{-218}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+93}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ x z) (+ y z))))
   (if (<= z -1.45e-251)
     t_0
     (if (<= z 1e-218) (/ (* x y) z) (if (<= z 7e+93) t_0 x)))))

C

double code(double x, double y, double z) {
	double t_0 = (x / z) * (y + z);
	double tmp;
	if (z <= -1.45e-251) {
		tmp = t_0;
	} else if (z <= 1e-218) {
		tmp = (x * y) / z;
	} else if (z <= 7e+93) {
		tmp = t_0;
	} else {
		tmp = 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) :: t_0
    real(8) :: tmp
    t_0 = (x / z) * (y + z)
    if (z <= (-1.45d-251)) then
        tmp = t_0
    else if (z <= 1d-218) then
        tmp = (x * y) / z
    else if (z <= 7d+93) then
        tmp = t_0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x / z) * (y + z);
	double tmp;
	if (z <= -1.45e-251) {
		tmp = t_0;
	} else if (z <= 1e-218) {
		tmp = (x * y) / z;
	} else if (z <= 7e+93) {
		tmp = t_0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x / z) * (y + z)
	tmp = 0
	if z <= -1.45e-251:
		tmp = t_0
	elif z <= 1e-218:
		tmp = (x * y) / z
	elif z <= 7e+93:
		tmp = t_0
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x / z) * Float64(y + z))
	tmp = 0.0
	if (z <= -1.45e-251)
		tmp = t_0;
	elseif (z <= 1e-218)
		tmp = Float64(Float64(x * y) / z);
	elseif (z <= 7e+93)
		tmp = t_0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x / z) * (y + z);
	tmp = 0.0;
	if (z <= -1.45e-251)
		tmp = t_0;
	elseif (z <= 1e-218)
		tmp = (x * y) / z;
	elseif (z <= 7e+93)
		tmp = t_0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / z), $MachinePrecision] * N[(y + z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.45e-251], t$95$0, If[LessEqual[z, 1e-218], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 7e+93], t$95$0, x]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.45e-251 or 1e-218 < z < 6.99999999999999996e93

   1. Initial program 87.3%

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

      1. associate-*l/ 89.9%

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

   3. Simplified 89.9%

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

   if -1.45e-251 < z < 1e-218

   1. Initial program 99.9%

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

      1. associate-*l/ 71.9%

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

   3. Simplified 71.9%

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

   4. Taylor expanded in z around 0 97.2%

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

   if 6.99999999999999996e93 < z

   1. Initial program 77.1%

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

      1. associate-*l/ 67.6%

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

   3. Simplified 67.6%

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

   4. Taylor expanded in z around inf 97.7%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-251}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + z\right)\\ \mathbf{elif}\;z \leq 10^{-218}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 50.7% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

function code(x, y, z)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 87.2%

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

   1. associate-*l/ 84.2%

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

3. Simplified 84.2%

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

4. Taylor expanded in z around inf 51.9%

   \[\leadsto \color{blue}{x} \]

5. Final simplification 51.9%

   \[\leadsto x \]

Developer target: 96.1% accurate, 1.0× speedup

Math

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

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(x / Float64(z / Float64(y + z)))
end

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023240 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (/ x (/ z (+ y z)))

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