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

Percentage Accurate: 84.7% → 99.3%

Time: 8.1s

Alternatives: 9

Speedup: 14.0×

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: 84.7% 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: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+98} \lor \neg \left(y \leq 0.0005\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{y + z}\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5e+98) (not (<= y 0.0005)))
   (+ x (/ (exp (- z)) y))
   (+ x (/ (pow (exp y) (log (/ y (+ y z)))) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5e+98) || !(y <= 0.0005)) {
		tmp = x + (exp(-z) / y);
	} else {
		tmp = x + (pow(exp(y), log((y / (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 ((y <= (-5d+98)) .or. (.not. (y <= 0.0005d0))) then
        tmp = x + (exp(-z) / y)
    else
        tmp = x + ((exp(y) ** log((y / (y + z)))) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5e+98) || !(y <= 0.0005)) {
		tmp = x + (Math.exp(-z) / y);
	} else {
		tmp = x + (Math.pow(Math.exp(y), Math.log((y / (y + z)))) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -5e+98) or not (y <= 0.0005):
		tmp = x + (math.exp(-z) / y)
	else:
		tmp = x + (math.pow(math.exp(y), math.log((y / (y + z)))) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5e+98) || !(y <= 0.0005))
		tmp = Float64(x + Float64(exp(Float64(-z)) / y));
	else
		tmp = Float64(x + Float64((exp(y) ^ log(Float64(y / Float64(y + z)))) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -5e+98) || ~((y <= 0.0005)))
		tmp = x + (exp(-z) / y);
	else
		tmp = x + ((exp(y) ^ log((y / (y + z)))) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5e+98], N[Not[LessEqual[y, 0.0005]], $MachinePrecision]], N[(x + N[(N[Exp[(-z)], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Power[N[Exp[y], $MachinePrecision], N[Log[N[(y / N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.9999999999999998e98 or 5.0000000000000001e-4 < y

   1. Initial program 87.2%

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

      1. *-commutative 87.2%

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

      2. exp-to-pow 87.2%

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

      3. +-commutative 87.2%

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

   3. Simplified 87.2%

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

   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 -4.9999999999999998e98 < y < 5.0000000000000001e-4

   1. Initial program 83.7%

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

      1. exp-prod 99.7%

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

      2. +-commutative 99.7%

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

   3. Simplified 99.7%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+98} \lor \neg \left(y \leq 0.0005\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{y + z}\right)}}{y}\\ \end{array} \]

Alternative 2: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \lor \neg \left(y \leq 0.082\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -7.8) (not (<= y 0.082)))
   (+ x (/ (exp (- z)) y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7.8) || !(y <= 0.082)) {
		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 <= (-7.8d0)) .or. (.not. (y <= 0.082d0))) 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 <= -7.8) || !(y <= 0.082)) {
		tmp = x + (Math.exp(-z) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -7.8) or not (y <= 0.082):
		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 <= -7.8) || !(y <= 0.082))
		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 <= -7.8) || ~((y <= 0.082)))
		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, -7.8], N[Not[LessEqual[y, 0.082]], $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 < -7.79999999999999982 or 0.0820000000000000034 < y

   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%

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

   if -7.79999999999999982 < y < 0.0820000000000000034

   1. Initial program 81.1%

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

      1. exp-prod 99.7%

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

      2. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 98.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \lor \neg \left(y \leq 0.082\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 3: 86.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+76}:\\ \;\;\;\;\frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.6e+76) (/ (exp (- z)) y) (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.6e+76) {
		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 <= (-2.6d+76)) 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 <= -2.6e+76) {
		tmp = Math.exp(-z) / y;
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.6e+76:
		tmp = math.exp(-z) / y
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.6e+76)
		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 <= -2.6e+76)
		tmp = exp(-z) / y;
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.6e+76], 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 < -2.5999999999999999e76

   1. Initial program 44.9%

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

      1. *-commutative 44.9%

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

      2. exp-to-pow 44.9%

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

      3. +-commutative 44.9%

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

   3. Simplified 44.9%

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

   4. Taylor expanded in y around inf 66.8%

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

      1. mul-1-neg 66.8%

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

   6. Simplified 66.8%

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

   7. Taylor expanded in x around 0 66.8%

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

   if -2.5999999999999999e76 < z

   1. Initial program 93.2%

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

      1. exp-prod 96.8%

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

      2. +-commutative 96.8%

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

   3. Simplified 96.8%

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

   4. Taylor expanded in y around 0 93.7%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+76}:\\ \;\;\;\;\frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 4: 87.1% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8:\\ \;\;\;\;x + \frac{\left(1 - z\right) + z \cdot \left(z \cdot 0.5\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.8) (+ x (/ (+ (- 1.0 z) (* z (* z 0.5))) y)) (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.8) {
		tmp = x + (((1.0 - z) + (z * (z * 0.5))) / 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 <= (-7.8d0)) then
        tmp = x + (((1.0d0 - z) + (z * (z * 0.5d0))) / 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 <= -7.8) {
		tmp = x + (((1.0 - z) + (z * (z * 0.5))) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -7.8:
		tmp = x + (((1.0 - z) + (z * (z * 0.5))) / y)
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7.8)
		tmp = Float64(x + Float64(Float64(Float64(1.0 - z) + Float64(z * Float64(z * 0.5))) / 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 <= -7.8)
		tmp = x + (((1.0 - z) + (z * (z * 0.5))) / y);
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7.8], N[(x + N[(N[(N[(1.0 - z), $MachinePrecision] + N[(z * N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.79999999999999982

   1. Initial program 87.1%

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

      1. exp-prod 87.1%

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

      2. +-commutative 87.1%

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

   3. Simplified 87.1%

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

   4. Taylor expanded in y around inf 66.5%

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

      1. associate-+r+ 66.5%

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

      2. +-commutative 66.5%

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

      3. associate-*r/ 66.5%

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

      4. distribute-rgt-out 66.5%

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

      5. metadata-eval 66.5%

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

      6. *-rgt-identity 66.5%

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

      7. associate-*l/ 66.5%

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

      8. metadata-eval 66.5%

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

      9. associate-*r/ 66.5%

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

      10. distribute-rgt-in 82.9%

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

      11. mul-1-neg 82.9%

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

      12. unsub-neg 82.9%

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

      13. unpow2 82.9%

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

      14. associate-*r/ 82.9%

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

      15. metadata-eval 82.9%

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

   6. Simplified 82.9%

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

   7. Taylor expanded in y around inf 82.9%

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

      1. unpow2 82.9%

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

      2. *-commutative 82.9%

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

      3. associate-*l* 82.9%

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

   9. Simplified 82.9%

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

   if -7.79999999999999982 < y

   1. Initial program 85.0%

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

      1. exp-prod 95.1%

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

      2. +-commutative 95.1%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in y around 0 91.3%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8:\\ \;\;\;\;x + \frac{\left(1 - z\right) + z \cdot \left(z \cdot 0.5\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 5: 83.9% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+150}:\\ \;\;\;\;x + \frac{z \cdot \left(z \cdot 0.5\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.8e+150) (+ x (/ (* z (* z 0.5)) y)) (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.8e+150) {
		tmp = x + ((z * (z * 0.5)) / 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 <= (-3.8d+150)) then
        tmp = x + ((z * (z * 0.5d0)) / 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 <= -3.8e+150) {
		tmp = x + ((z * (z * 0.5)) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.8e+150:
		tmp = x + ((z * (z * 0.5)) / y)
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.8e+150)
		tmp = Float64(x + Float64(Float64(z * Float64(z * 0.5)) / 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 <= -3.8e+150)
		tmp = x + ((z * (z * 0.5)) / y);
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.8e+150], N[(x + N[(N[(z * N[(z * 0.5), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.79999999999999989e150

   1. Initial program 82.9%

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

      1. exp-prod 82.9%

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

      2. +-commutative 82.9%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in y around inf 61.5%

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

      1. associate-+r+ 61.5%

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

      2. +-commutative 61.5%

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

      3. associate-*r/ 61.5%

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

      4. distribute-rgt-out 61.5%

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

      5. metadata-eval 61.5%

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

      6. *-rgt-identity 61.5%

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

      7. associate-*l/ 61.5%

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

      8. metadata-eval 61.5%

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

      9. associate-*r/ 61.5%

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

      10. distribute-rgt-in 81.1%

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

      11. mul-1-neg 81.1%

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

      12. unsub-neg 81.1%

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

      13. unpow2 81.1%

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

      14. associate-*r/ 81.1%

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

      15. metadata-eval 81.1%

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

   6. Simplified 81.1%

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

   7. Taylor expanded in y around inf 81.1%

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

      1. unpow2 81.1%

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

      2. *-commutative 81.1%

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

      3. associate-*l* 81.1%

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

   9. Simplified 81.1%

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

   10. Taylor expanded in z around inf 79.2%

       \[\leadsto x + \frac{\color{blue}{0.5 \cdot {z}^{2}}}{y} \]
   11. Step-by-step derivation

       1. unpow2 79.2%

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

       2. *-commutative 79.2%

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

       3. associate-*r* 79.2%

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

   12. Simplified 79.2%

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

   if -3.79999999999999989e150 < y

   1. Initial program 86.4%

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

      1. exp-prod 95.1%

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

      2. +-commutative 95.1%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in y around 0 88.9%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+150}:\\ \;\;\;\;x + \frac{z \cdot \left(z \cdot 0.5\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 6: 84.6% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+229}:\\ \;\;\;\;\frac{0.5}{y} \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.1e+229) (* (/ 0.5 y) (* z z)) (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.1e+229) {
		tmp = (0.5 / y) * (z * z);
	} 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 <= (-3.1d+229)) then
        tmp = (0.5d0 / y) * (z * z)
    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 <= -3.1e+229) {
		tmp = (0.5 / y) * (z * z);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.1e+229:
		tmp = (0.5 / y) * (z * z)
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.1e+229)
		tmp = Float64(Float64(0.5 / y) * Float64(z * z));
	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 <= -3.1e+229)
		tmp = (0.5 / y) * (z * z);
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.1e+229], N[(N[(0.5 / y), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.10000000000000014e229

   1. Initial program 58.8%

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

      1. exp-prod 85.9%

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

      2. +-commutative 85.9%

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

   3. Simplified 85.9%

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

   4. Taylor expanded in y around inf 0.0%

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

      1. associate-+r+ 0.0%

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

      2. +-commutative 0.0%

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

      3. associate-*r/ 0.0%

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

      4. distribute-rgt-out 7.1%

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

      5. metadata-eval 7.1%

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

      6. *-rgt-identity 7.1%

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

      7. associate-*l/ 7.1%

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

      8. metadata-eval 7.1%

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

      9. associate-*r/ 7.1%

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

      10. distribute-rgt-in 71.5%

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

      11. mul-1-neg 71.5%

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

      12. unsub-neg 71.5%

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

      13. unpow2 71.5%

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

      14. associate-*r/ 71.5%

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

      15. metadata-eval 71.5%

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

   6. Simplified 71.5%

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

   7. Taylor expanded in z around inf 71.5%

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

      1. unpow2 71.5%

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

      2. *-commutative 71.5%

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

      3. associate-*r/ 71.5%

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

      4. metadata-eval 71.5%

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

      5. +-commutative 71.5%

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

      6. associate-*r/ 71.5%

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

      7. metadata-eval 71.5%

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

      8. unpow2 71.5%

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

   9. Simplified 71.5%

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

   10. Taylor expanded in y around inf 72.9%

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

   if -3.10000000000000014e229 < z

   1. Initial program 87.2%

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

      1. exp-prod 93.0%

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

      2. +-commutative 93.0%

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

   3. Simplified 93.0%

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

   4. Taylor expanded in y around 0 87.8%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+229}:\\ \;\;\;\;\frac{0.5}{y} \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 7: 66.5% accurate, 29.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+97}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.2e+97) x (if (<= y 1.0) (/ 1.0 y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.2e+97) {
		tmp = x;
	} else if (y <= 1.0) {
		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 <= (-3.2d+97)) then
        tmp = x
    else if (y <= 1.0d0) 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 <= -3.2e+97) {
		tmp = x;
	} else if (y <= 1.0) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.2e+97:
		tmp = x
	elif y <= 1.0:
		tmp = 1.0 / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.2e+97)
		tmp = x;
	elseif (y <= 1.0)
		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 <= -3.2e+97)
		tmp = x;
	elseif (y <= 1.0)
		tmp = 1.0 / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.2e+97], x, If[LessEqual[y, 1.0], N[(1.0 / y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -3.20000000000000016e97 or 1 < y

   1. Initial program 87.3%

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

      1. *-commutative 87.3%

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

      2. exp-to-pow 87.3%

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

      3. +-commutative 87.3%

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

   3. Simplified 87.3%

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

   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 x around inf 69.8%

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

   if -3.20000000000000016e97 < y < 1

   1. Initial program 83.5%

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

      1. *-commutative 83.5%

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

      2. exp-to-pow 83.5%

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

      3. +-commutative 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in y around inf 65.3%

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

      1. mul-1-neg 65.3%

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

   6. Simplified 65.3%

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

   7. Taylor expanded in x around 0 48.1%

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

   8. Taylor expanded in z around 0 72.0%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+97}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 84.2% 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 85.7%

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

   1. exp-prod 92.6%

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

   2. +-commutative 92.6%

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

3. Simplified 92.6%

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

4. Taylor expanded in y around 0 84.7%

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

5. Final simplification 84.7%

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

Alternative 9: 49.3% 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 85.7%

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

   1. *-commutative 85.7%

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

   2. exp-to-pow 85.7%

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

   3. +-commutative 85.7%

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

3. Simplified 85.7%

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

4. Taylor expanded in y around inf 85.1%

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

   1. mul-1-neg 85.1%

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

6. Simplified 85.1%

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

7. Taylor expanded in x around inf 50.1%

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

8. Final simplification 50.1%

   \[\leadsto x \]

Developer target: 91.3% 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 2023290 
(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)))