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

Percentage Accurate: 84.1% → 99.3%

Time: 12.4s

Alternatives: 5

Speedup: 42.2×

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: 84.1% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+79}:\\ \;\;\;\;x + \frac{\frac{1}{y}}{e^{z}}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-26}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{y + z}\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.2e+79)
   (+ x (/ (/ 1.0 y) (exp z)))
   (if (<= y 8.5e-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 <= -1.2e+79) {
		tmp = x + ((1.0 / y) / exp(z));
	} else if (y <= 8.5e-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 <= (-1.2d+79)) then
        tmp = x + ((1.0d0 / y) / exp(z))
    else if (y <= 8.5d-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 <= -1.2e+79) {
		tmp = x + ((1.0 / y) / Math.exp(z));
	} else if (y <= 8.5e-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 <= -1.2e+79:
		tmp = x + ((1.0 / y) / math.exp(z))
	elif y <= 8.5e-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 <= -1.2e+79)
		tmp = Float64(x + Float64(Float64(1.0 / y) / exp(z)));
	elseif (y <= 8.5e-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 <= -1.2e+79)
		tmp = x + ((1.0 / y) / exp(z));
	elseif (y <= 8.5e-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, -1.2e+79], N[(x + N[(N[(1.0 / y), $MachinePrecision] / N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.5e-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 < -1.19999999999999993e79

   1. Initial program 77.5%

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

      1. *-commutative 77.5%

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

      2. exp-to-pow 77.5%

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

      3. +-commutative 77.5%

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

   3. Simplified 77.5%

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

   5. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. exp-neg 100.0%

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

      4. associate-/r/ 100.0%

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

      5. /-rgt-identity 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in y around 0 100.0%

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

       1. associate-/r* 100.0%

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

   12. Simplified 100.0%

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

   if -1.19999999999999993e79 < y < 8.50000000000000004e-26

   1. Initial program 89.1%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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. Add Preprocessing

   if 8.50000000000000004e-26 < y

   1. Initial program 82.9%

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

      1. *-commutative 82.9%

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

      2. exp-to-pow 82.9%

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

      3. +-commutative 82.9%

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

   3. Simplified 82.9%

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

   5. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+79}:\\ \;\;\;\;x + \frac{\frac{1}{y}}{e^{z}}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-26}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{y + z}\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \lor \neg \left(y \leq 8.5 \cdot 10^{-26}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.1) (not (<= y 8.5e-26)))
   (+ x (/ (exp (- z)) y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.1) || !(y <= 8.5e-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 <= (-1.1d0)) .or. (.not. (y <= 8.5d-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 <= -1.1) || !(y <= 8.5e-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 <= -1.1) or not (y <= 8.5e-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 <= -1.1) || !(y <= 8.5e-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 <= -1.1) || ~((y <= 8.5e-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, -1.1], N[Not[LessEqual[y, 8.5e-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 < -1.1000000000000001 or 8.50000000000000004e-26 < y

   1. Initial program 83.1%

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

      1. *-commutative 83.1%

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

      2. exp-to-pow 83.1%

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

      3. +-commutative 83.1%

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

   3. Simplified 83.1%

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

   5. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   if -1.1000000000000001 < y < 8.50000000000000004e-26

   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. Add Preprocessing

   5. Taylor expanded in z around 0 70.6%

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

      1. mul-1-neg 70.6%

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

      2. unsub-neg 70.6%

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

   7. Simplified 70.6%

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

   8. Taylor expanded in z around 0 100.0%

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

      1. +-commutative 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \lor \neg \left(y \leq 8.5 \cdot 10^{-26}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.96:\\ \;\;\;\;x + \frac{\frac{1}{y}}{e^{z}}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-28}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -0.96)
   (+ x (/ (/ 1.0 y) (exp z)))
   (if (<= y 5e-28) (+ x (/ 1.0 y)) (+ x (/ (exp (- z)) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.96) {
		tmp = x + ((1.0 / y) / exp(z));
	} else if (y <= 5e-28) {
		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 <= (-0.96d0)) then
        tmp = x + ((1.0d0 / y) / exp(z))
    else if (y <= 5d-28) 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 <= -0.96) {
		tmp = x + ((1.0 / y) / Math.exp(z));
	} else if (y <= 5e-28) {
		tmp = x + (1.0 / y);
	} else {
		tmp = x + (Math.exp(-z) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -0.96:
		tmp = x + ((1.0 / y) / math.exp(z))
	elif y <= 5e-28:
		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 <= -0.96)
		tmp = Float64(x + Float64(Float64(1.0 / y) / exp(z)));
	elseif (y <= 5e-28)
		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 <= -0.96)
		tmp = x + ((1.0 / y) / exp(z));
	elseif (y <= 5e-28)
		tmp = x + (1.0 / y);
	else
		tmp = x + (exp(-z) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -0.96], N[(x + N[(N[(1.0 / y), $MachinePrecision] / N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e-28], 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 < -0.95999999999999996

   1. Initial program 83.4%

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

      1. *-commutative 83.4%

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

      2. exp-to-pow 83.4%

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

      3. +-commutative 83.4%

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

   3. Simplified 83.4%

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

   5. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. exp-neg 100.0%

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

      4. associate-/r/ 100.0%

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

      5. /-rgt-identity 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in y around 0 100.0%

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

       1. associate-/r* 100.0%

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

   12. Simplified 100.0%

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

   if -0.95999999999999996 < y < 5.0000000000000002e-28

   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. Add Preprocessing

   5. Taylor expanded in z around 0 70.6%

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

      1. mul-1-neg 70.6%

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

      2. unsub-neg 70.6%

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

   7. Simplified 70.6%

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

   8. Taylor expanded in z around 0 100.0%

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

      1. +-commutative 100.0%

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

   10. Simplified 100.0%

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

   if 5.0000000000000002e-28 < y

   1. Initial program 82.9%

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

      1. *-commutative 82.9%

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

      2. exp-to-pow 82.9%

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

      3. +-commutative 82.9%

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

   3. Simplified 82.9%

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

   5. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.96:\\ \;\;\;\;x + \frac{\frac{1}{y}}{e^{z}}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-28}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.6% 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 84.8%

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

   1. *-commutative 84.8%

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

   2. exp-to-pow 84.8%

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

   3. +-commutative 84.8%

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

3. Simplified 84.8%

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

5. Taylor expanded in z around 0 67.5%

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

   1. mul-1-neg 67.5%

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

   2. unsub-neg 67.5%

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

7. Simplified 67.5%

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

8. Taylor expanded in z around 0 81.3%

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

   1. +-commutative 81.3%

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

10. Simplified 81.3%

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

11. Final simplification 81.3%

    \[\leadsto x + \frac{1}{y} \]
12. Add Preprocessing

Alternative 5: 50.3% 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 84.8%

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

   1. *-commutative 84.8%

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

   2. exp-to-pow 84.8%

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

   3. +-commutative 84.8%

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

3. Simplified 84.8%

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

5. Taylor expanded in z around 0 67.5%

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

   1. mul-1-neg 67.5%

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

   2. unsub-neg 67.5%

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

7. Simplified 67.5%

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

8. Taylor expanded in x around inf 48.4%

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

9. Final simplification 48.4%

   \[\leadsto x \]
10. Add Preprocessing

Developer target: 91.1% 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 2024024 
(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)))