Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G

Percentage Accurate: 85.6% → 99.6%

Time: 11.6s

Alternatives: 9

Speedup: 12.4×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

def code(x, y, z):
	return x + (math.exp((y * math.log((y / (z + y))))) / y)

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x, y, z):
	return x + (math.exp((y * math.log((y / (z + y))))) / y)

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -10000000:\\ \;\;\;\;x - \frac{\frac{-1}{y}}{e^{z}}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-26}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{y + z}\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -10000000.0)
   (- x (/ (/ -1.0 y) (exp z)))
   (if (<= y 2.25e-26)
     (+ x (/ (pow (exp y) (log (/ y (+ y z)))) y))
     (+ x (/ (exp (- z)) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -10000000.0) {
		tmp = x - ((-1.0 / y) / exp(z));
	} else if (y <= 2.25e-26) {
		tmp = x + (pow(exp(y), log((y / (y + z)))) / y);
	} else {
		tmp = x + (exp(-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 <= (-10000000.0d0)) then
        tmp = x - (((-1.0d0) / y) / exp(z))
    else if (y <= 2.25d-26) then
        tmp = x + ((exp(y) ** log((y / (y + z)))) / y)
    else
        tmp = x + (exp(-z) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -10000000.0) {
		tmp = x - ((-1.0 / y) / Math.exp(z));
	} else if (y <= 2.25e-26) {
		tmp = x + (Math.pow(Math.exp(y), Math.log((y / (y + z)))) / y);
	} else {
		tmp = x + (Math.exp(-z) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -10000000.0:
		tmp = x - ((-1.0 / y) / math.exp(z))
	elif y <= 2.25e-26:
		tmp = x + (math.pow(math.exp(y), math.log((y / (y + z)))) / y)
	else:
		tmp = x + (math.exp(-z) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -10000000.0)
		tmp = Float64(x - Float64(Float64(-1.0 / y) / exp(z)));
	elseif (y <= 2.25e-26)
		tmp = Float64(x + Float64((exp(y) ^ log(Float64(y / Float64(y + z)))) / y));
	else
		tmp = Float64(x + Float64(exp(Float64(-z)) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -10000000.0)
		tmp = x - ((-1.0 / y) / exp(z));
	elseif (y <= 2.25e-26)
		tmp = x + ((exp(y) ^ log((y / (y + z)))) / y);
	else
		tmp = x + (exp(-z) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -10000000.0], N[(x - N[(N[(-1.0 / y), $MachinePrecision] / N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.25e-26], N[(x + N[(N[Power[N[Exp[y], $MachinePrecision], N[Log[N[(y / N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Exp[(-z)], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1e7

   1. Initial program 88.1%

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

      1. *-commutative 88.1%

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

      2. exp-to-pow 88.1%

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

      3. +-commutative 88.1%

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

   3. Simplified 88.1%

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

   4. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

      1. frac-2neg 100.0%

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

      2. neg-sub0 100.0%

         \[\leadsto x + \frac{\color{blue}{0 - e^{-z}}}{-y} \]

      3. div-sub 100.0%

         \[\leadsto x + \color{blue}{\left(\frac{0}{-y} - \frac{e^{-z}}{-y}\right)} \]

      4. frac-2neg 100.0%

         \[\leadsto x + \left(\frac{0}{-y} - \color{blue}{\frac{-e^{-z}}{-\left(-y\right)}}\right) \]

      5. exp-neg 100.0%

         \[\leadsto x + \left(\frac{0}{-y} - \frac{-\color{blue}{\frac{1}{e^{z}}}}{-\left(-y\right)}\right) \]

      6. distribute-neg-frac 100.0%

         \[\leadsto x + \left(\frac{0}{-y} - \frac{\color{blue}{\frac{-1}{e^{z}}}}{-\left(-y\right)}\right) \]

      7. metadata-eval 100.0%

         \[\leadsto x + \left(\frac{0}{-y} - \frac{\frac{\color{blue}{-1}}{e^{z}}}{-\left(-y\right)}\right) \]

      8. remove-double-neg 100.0%

         \[\leadsto x + \left(\frac{0}{-y} - \frac{\frac{-1}{e^{z}}}{\color{blue}{y}}\right) \]

   8. Applied egg-rr 100.0%

      \[\leadsto x + \color{blue}{\left(\frac{0}{-y} - \frac{\frac{-1}{e^{z}}}{y}\right)} \]
   9. Step-by-step derivation

      1. div0 100.0%

         \[\leadsto x + \left(\color{blue}{0} - \frac{\frac{-1}{e^{z}}}{y}\right) \]

      2. neg-sub0 100.0%

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

      3. distribute-neg-frac 100.0%

         \[\leadsto x + \color{blue}{\frac{-\frac{-1}{e^{z}}}{y}} \]

      4. mul-1-neg 100.0%

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

      5. associate-*l/ 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l/ 100.0%

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

      8. mul-1-neg 100.0%

         \[\leadsto x + \frac{\color{blue}{-\frac{-1}{y}}}{e^{z}} \]

   10. Simplified 100.0%

       \[\leadsto x + \color{blue}{\frac{-\frac{-1}{y}}{e^{z}}} \]

   if -1e7 < y < 2.2499999999999999e-26

   1. Initial program 82.4%

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

      1. exp-prod 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 2.2499999999999999e-26 < y

   1. Initial program 90.1%

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

      1. *-commutative 90.1%

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

      2. exp-to-pow 90.1%

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

      3. +-commutative 90.1%

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

   3. Simplified 90.1%

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

   4. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -10000000:\\ \;\;\;\;x - \frac{\frac{-1}{y}}{e^{z}}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-26}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{y + z}\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array} \]

Alternative 2: 99.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -520 \lor \neg \left(y \leq 2.25 \cdot 10^{-26}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -520.0) (not (<= y 2.25e-26)))
   (+ x (/ (exp (- z)) y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -520.0) || !(y <= 2.25e-26)) {
		tmp = x + (exp(-z) / y);
	} else {
		tmp = x + (1.0 / 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 <= (-520.0d0)) .or. (.not. (y <= 2.25d-26))) then
        tmp = x + (exp(-z) / y)
    else
        tmp = x + (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -520.0) || !(y <= 2.25e-26)) {
		tmp = x + (Math.exp(-z) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -520.0) or not (y <= 2.25e-26):
		tmp = x + (math.exp(-z) / y)
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -520.0) || !(y <= 2.25e-26))
		tmp = Float64(x + Float64(exp(Float64(-z)) / y));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -520.0) || ~((y <= 2.25e-26)))
		tmp = x + (exp(-z) / y);
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -520.0], N[Not[LessEqual[y, 2.25e-26]], $MachinePrecision]], N[(x + N[(N[Exp[(-z)], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -520 or 2.2499999999999999e-26 < y

   1. Initial program 89.4%

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

      1. *-commutative 89.4%

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

      2. exp-to-pow 89.4%

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

      3. +-commutative 89.4%

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

   3. Simplified 89.4%

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

   4. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   if -520 < y < 2.2499999999999999e-26

   1. Initial program 82.0%

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

      1. exp-prod 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{y} + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -520 \lor \neg \left(y \leq 2.25 \cdot 10^{-26}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 3: 99.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -520:\\ \;\;\;\;x - \frac{\frac{-1}{y}}{e^{z}}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-26}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -520.0)
   (- x (/ (/ -1.0 y) (exp z)))
   (if (<= y 2.25e-26) (+ x (/ 1.0 y)) (+ x (/ (exp (- z)) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -520.0) {
		tmp = x - ((-1.0 / y) / exp(z));
	} else if (y <= 2.25e-26) {
		tmp = x + (1.0 / y);
	} else {
		tmp = x + (exp(-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 <= (-520.0d0)) then
        tmp = x - (((-1.0d0) / y) / exp(z))
    else if (y <= 2.25d-26) then
        tmp = x + (1.0d0 / y)
    else
        tmp = x + (exp(-z) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -520.0) {
		tmp = x - ((-1.0 / y) / Math.exp(z));
	} else if (y <= 2.25e-26) {
		tmp = x + (1.0 / y);
	} else {
		tmp = x + (Math.exp(-z) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -520.0:
		tmp = x - ((-1.0 / y) / math.exp(z))
	elif y <= 2.25e-26:
		tmp = x + (1.0 / y)
	else:
		tmp = x + (math.exp(-z) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -520.0)
		tmp = Float64(x - Float64(Float64(-1.0 / y) / exp(z)));
	elseif (y <= 2.25e-26)
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = Float64(x + Float64(exp(Float64(-z)) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -520.0)
		tmp = x - ((-1.0 / y) / exp(z));
	elseif (y <= 2.25e-26)
		tmp = x + (1.0 / y);
	else
		tmp = x + (exp(-z) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -520.0], N[(x - N[(N[(-1.0 / y), $MachinePrecision] / N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.25e-26], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Exp[(-z)], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -520

   1. Initial program 88.4%

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

      1. *-commutative 88.4%

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

      2. exp-to-pow 88.4%

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

      3. +-commutative 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

      1. frac-2neg 100.0%

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

      2. neg-sub0 100.0%

         \[\leadsto x + \frac{\color{blue}{0 - e^{-z}}}{-y} \]

      3. div-sub 100.0%

         \[\leadsto x + \color{blue}{\left(\frac{0}{-y} - \frac{e^{-z}}{-y}\right)} \]

      4. frac-2neg 100.0%

         \[\leadsto x + \left(\frac{0}{-y} - \color{blue}{\frac{-e^{-z}}{-\left(-y\right)}}\right) \]

      5. exp-neg 100.0%

         \[\leadsto x + \left(\frac{0}{-y} - \frac{-\color{blue}{\frac{1}{e^{z}}}}{-\left(-y\right)}\right) \]

      6. distribute-neg-frac 100.0%

         \[\leadsto x + \left(\frac{0}{-y} - \frac{\color{blue}{\frac{-1}{e^{z}}}}{-\left(-y\right)}\right) \]

      7. metadata-eval 100.0%

         \[\leadsto x + \left(\frac{0}{-y} - \frac{\frac{\color{blue}{-1}}{e^{z}}}{-\left(-y\right)}\right) \]

      8. remove-double-neg 100.0%

         \[\leadsto x + \left(\frac{0}{-y} - \frac{\frac{-1}{e^{z}}}{\color{blue}{y}}\right) \]

   8. Applied egg-rr 100.0%

      \[\leadsto x + \color{blue}{\left(\frac{0}{-y} - \frac{\frac{-1}{e^{z}}}{y}\right)} \]
   9. Step-by-step derivation

      1. div0 100.0%

         \[\leadsto x + \left(\color{blue}{0} - \frac{\frac{-1}{e^{z}}}{y}\right) \]

      2. neg-sub0 100.0%

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

      3. distribute-neg-frac 100.0%

         \[\leadsto x + \color{blue}{\frac{-\frac{-1}{e^{z}}}{y}} \]

      4. mul-1-neg 100.0%

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

      5. associate-*l/ 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l/ 100.0%

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

      8. mul-1-neg 100.0%

         \[\leadsto x + \frac{\color{blue}{-\frac{-1}{y}}}{e^{z}} \]

   10. Simplified 100.0%

       \[\leadsto x + \color{blue}{\frac{-\frac{-1}{y}}{e^{z}}} \]

   if -520 < y < 2.2499999999999999e-26

   1. Initial program 82.0%

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

      1. exp-prod 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   if 2.2499999999999999e-26 < y

   1. Initial program 90.1%

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

      1. *-commutative 90.1%

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

      2. exp-to-pow 90.1%

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

      3. +-commutative 90.1%

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

   3. Simplified 90.1%

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

   4. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -520:\\ \;\;\;\;x - \frac{\frac{-1}{y}}{e^{z}}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-26}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array} \]

Alternative 4: 89.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -980:\\ \;\;\;\;\frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -980.0) (/ (exp (- z)) y) (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -980.0) {
		tmp = exp(-z) / y;
	} else {
		tmp = x + (1.0 / 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 <= (-980.0d0)) then
        tmp = exp(-z) / y
    else
        tmp = x + (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -980.0) {
		tmp = Math.exp(-z) / y;
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -980.0:
		tmp = math.exp(-z) / y
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -980.0)
		tmp = Float64(exp(Float64(-z)) / y);
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -980.0)
		tmp = exp(-z) / y;
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -980.0], N[(N[Exp[(-z)], $MachinePrecision] / y), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -980

   1. Initial program 41.5%

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

      1. *-commutative 41.5%

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

      2. exp-to-pow 41.5%

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

      3. +-commutative 41.5%

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

   3. Simplified 41.5%

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

   4. Taylor expanded in y around inf 60.1%

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

      1. mul-1-neg 60.1%

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

   6. Simplified 60.1%

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

   7. Taylor expanded in x around 0 60.1%

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

   if -980 < z

   1. Initial program 94.7%

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

      1. exp-prod 98.3%

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

      2. +-commutative 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in y around inf 95.7%

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

      1. +-commutative 95.7%

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

   6. Simplified 95.7%

      \[\leadsto \color{blue}{\frac{1}{y} + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -980:\\ \;\;\;\;\frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 5: 87.5% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1000:\\ \;\;\;\;x + \left(0.5 \cdot \frac{z \cdot z}{y} + \frac{1 - z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1000.0)
   (+ x (+ (* 0.5 (/ (* z z) y)) (/ (- 1.0 z) y)))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1000.0) {
		tmp = x + ((0.5 * ((z * z) / y)) + ((1.0 - z) / y));
	} else {
		tmp = x + (1.0 / 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 <= (-1000.0d0)) then
        tmp = x + ((0.5d0 * ((z * z) / y)) + ((1.0d0 - z) / y))
    else
        tmp = x + (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1000.0) {
		tmp = x + ((0.5 * ((z * z) / y)) + ((1.0 - z) / y));
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1000.0:
		tmp = x + ((0.5 * ((z * z) / y)) + ((1.0 - z) / y))
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1000.0)
		tmp = Float64(x + Float64(Float64(0.5 * Float64(Float64(z * z) / y)) + Float64(Float64(1.0 - z) / y)));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1000.0)
		tmp = x + ((0.5 * ((z * z) / y)) + ((1.0 - z) / y));
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1000.0], N[(x + N[(N[(0.5 * N[(N[(z * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1e3

   1. Initial program 88.4%

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

      1. *-commutative 88.4%

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

      2. exp-to-pow 88.4%

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

      3. +-commutative 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in z around 0 85.3%

      \[\leadsto \color{blue}{x + \left(-1 \cdot \frac{z}{y} + \left(0.5 \cdot \frac{{z}^{2}}{y} + \frac{1}{y}\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative 85.3%

         \[\leadsto x + \color{blue}{\left(\left(0.5 \cdot \frac{{z}^{2}}{y} + \frac{1}{y}\right) + -1 \cdot \frac{z}{y}\right)} \]

      2. mul-1-neg 85.3%

         \[\leadsto x + \left(\left(0.5 \cdot \frac{{z}^{2}}{y} + \frac{1}{y}\right) + \color{blue}{\left(-\frac{z}{y}\right)}\right) \]

      3. associate-+l+ 85.3%

         \[\leadsto x + \color{blue}{\left(0.5 \cdot \frac{{z}^{2}}{y} + \left(\frac{1}{y} + \left(-\frac{z}{y}\right)\right)\right)} \]

      4. unpow2 85.3%

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

      5. sub-neg 85.3%

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

      6. div-sub 85.2%

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

   9. Simplified 85.2%

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

   if -1e3 < y

   1. Initial program 86.1%

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

      1. exp-prod 95.0%

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

      2. +-commutative 95.0%

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

   3. Simplified 95.0%

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

   4. Taylor expanded in y around inf 92.4%

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

      1. +-commutative 92.4%

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

   6. Simplified 92.4%

      \[\leadsto \color{blue}{\frac{1}{y} + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1000:\\ \;\;\;\;x + \left(0.5 \cdot \frac{z \cdot z}{y} + \frac{1 - z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 6: 87.5% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -520:\\ \;\;\;\;\frac{1}{y} + \left(x + \frac{z \cdot \left(z \cdot 0.5\right) - z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -520.0)
   (+ (/ 1.0 y) (+ x (/ (- (* z (* z 0.5)) z) y)))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -520.0) {
		tmp = (1.0 / y) + (x + (((z * (z * 0.5)) - z) / y));
	} else {
		tmp = x + (1.0 / 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 <= (-520.0d0)) then
        tmp = (1.0d0 / y) + (x + (((z * (z * 0.5d0)) - z) / y))
    else
        tmp = x + (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -520.0) {
		tmp = (1.0 / y) + (x + (((z * (z * 0.5)) - z) / y));
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -520.0:
		tmp = (1.0 / y) + (x + (((z * (z * 0.5)) - z) / y))
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -520.0)
		tmp = Float64(Float64(1.0 / y) + Float64(x + Float64(Float64(Float64(z * Float64(z * 0.5)) - z) / y)));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -520.0)
		tmp = (1.0 / y) + (x + (((z * (z * 0.5)) - z) / y));
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -520.0], N[(N[(1.0 / y), $MachinePrecision] + N[(x + N[(N[(N[(z * N[(z * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -520

   1. Initial program 88.4%

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

      1. *-commutative 88.4%

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

      2. exp-to-pow 88.4%

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

      3. +-commutative 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in z around 0 85.3%

      \[\leadsto \color{blue}{x + \left(-1 \cdot \frac{z}{y} + \left(0.5 \cdot \frac{{z}^{2}}{y} + \frac{1}{y}\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative 85.3%

         \[\leadsto x + \color{blue}{\left(\left(0.5 \cdot \frac{{z}^{2}}{y} + \frac{1}{y}\right) + -1 \cdot \frac{z}{y}\right)} \]

      2. mul-1-neg 85.3%

         \[\leadsto x + \left(\left(0.5 \cdot \frac{{z}^{2}}{y} + \frac{1}{y}\right) + \color{blue}{\left(-\frac{z}{y}\right)}\right) \]

      3. associate-+l+ 85.3%

         \[\leadsto x + \color{blue}{\left(0.5 \cdot \frac{{z}^{2}}{y} + \left(\frac{1}{y} + \left(-\frac{z}{y}\right)\right)\right)} \]

      4. unpow2 85.3%

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

      5. sub-neg 85.3%

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

      6. div-sub 85.2%

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

   9. Simplified 85.2%

      \[\leadsto \color{blue}{x + \left(0.5 \cdot \frac{z \cdot z}{y} + \frac{1 - z}{y}\right)} \]
   10. Step-by-step derivation

       1. +-commutative 85.2%

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

       2. +-commutative 85.2%

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

       3. div-sub 85.3%

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

       4. associate-+l- 85.3%

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

       5. associate-+l- 85.3%

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

       6. associate-*r/ 85.3%

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

       7. sub-div 85.3%

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

       8. *-commutative 85.3%

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

       9. associate-*l* 85.3%

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

   11. Applied egg-rr 85.3%

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

   if -520 < y

   1. Initial program 86.1%

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

      1. exp-prod 95.0%

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

      2. +-commutative 95.0%

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

   3. Simplified 95.0%

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

   4. Taylor expanded in y around inf 92.4%

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

      1. +-commutative 92.4%

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

   6. Simplified 92.4%

      \[\leadsto \color{blue}{\frac{1}{y} + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -520:\\ \;\;\;\;\frac{1}{y} + \left(x + \frac{z \cdot \left(z \cdot 0.5\right) - z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 7: 67.6% accurate, 29.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-88}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -8.5e-88) x (if (<= y 1.1e-27) (/ 1.0 y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.5e-88) {
		tmp = x;
	} else if (y <= 1.1e-27) {
		tmp = 1.0 / y;
	} 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 (y <= (-8.5d-88)) then
        tmp = x
    else if (y <= 1.1d-27) then
        tmp = 1.0d0 / y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.5e-88) {
		tmp = x;
	} else if (y <= 1.1e-27) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -8.5e-88:
		tmp = x
	elif y <= 1.1e-27:
		tmp = 1.0 / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -8.5e-88)
		tmp = x;
	elseif (y <= 1.1e-27)
		tmp = Float64(1.0 / y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -8.5e-88)
		tmp = x;
	elseif (y <= 1.1e-27)
		tmp = 1.0 / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -8.5e-88], x, If[LessEqual[y, 1.1e-27], N[(1.0 / y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -8.4999999999999996e-88 or 1.09999999999999993e-27 < y

   1. Initial program 90.4%

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

      1. exp-prod 89.9%

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

      2. +-commutative 89.9%

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

   3. Simplified 89.9%

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

   4. Taylor expanded in x around inf 69.2%

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

   if -8.4999999999999996e-88 < y < 1.09999999999999993e-27

   1. Initial program 77.6%

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

      1. exp-prod 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 80.1%

      \[\leadsto \color{blue}{\frac{1}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-88}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 85.3% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

   1. exp-prod 92.8%

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

   2. +-commutative 92.8%

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

3. Simplified 92.8%

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

4. Taylor expanded in y around inf 87.9%

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

   1. +-commutative 87.9%

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

6. Simplified 87.9%

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

7. Final simplification 87.9%

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

Alternative 9: 49.8% accurate, 211.0× 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 86.8%

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

   1. exp-prod 92.8%

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

   2. +-commutative 92.8%

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

3. Simplified 92.8%

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

4. Taylor expanded in x around inf 55.4%

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

5. Final simplification 55.4%

   \[\leadsto x \]

Developer target: 92.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y}{z + y} < 7.11541576 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< (/ y (+ z y)) 7.11541576e-315)
   (+ x (/ (exp (/ -1.0 z)) y))
   (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y / (z + y)) < 7.11541576e-315) {
		tmp = x + (exp((-1.0 / z)) / y);
	} else {
		tmp = x + (exp(log(pow((y / (y + z)), y))) / 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 / (z + y)) < 7.11541576d-315) then
        tmp = x + (exp(((-1.0d0) / z)) / y)
    else
        tmp = x + (exp(log(((y / (y + z)) ** y))) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y / (z + y)) < 7.11541576e-315) {
		tmp = x + (Math.exp((-1.0 / z)) / y);
	} else {
		tmp = x + (Math.exp(Math.log(Math.pow((y / (y + z)), y))) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y / (z + y)) < 7.11541576e-315:
		tmp = x + (math.exp((-1.0 / z)) / y)
	else:
		tmp = x + (math.exp(math.log(math.pow((y / (y + z)), y))) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(y / Float64(z + y)) < 7.11541576e-315)
		tmp = Float64(x + Float64(exp(Float64(-1.0 / z)) / y));
	else
		tmp = Float64(x + Float64(exp(log((Float64(y / Float64(y + z)) ^ y))) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y / (z + y)) < 7.11541576e-315)
		tmp = x + (exp((-1.0 / z)) / y);
	else
		tmp = x + (exp(log(((y / (y + z)) ^ y))) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[N[(y / N[(z + y), $MachinePrecision]), $MachinePrecision], 7.11541576e-315], N[(x + N[(N[Exp[N[(-1.0 / z), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Exp[N[Log[N[Power[N[(y / N[(y + z), $MachinePrecision]), $MachinePrecision], y], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2023297 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G"
  :precision binary64

  :herbie-target
  (if (< (/ y (+ z y)) 7.11541576e-315) (+ x (/ (exp (/ -1.0 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y)))

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