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

Percentage Accurate: 84.3% → 98.9%

Time: 8.0s

Alternatives: 7

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: 84.3% 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: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+130} \lor \neg \left(y \leq 8.5 \cdot 10^{-12}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{y + z}\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5e+130) (not (<= y 8.5e-12)))
   (+ 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+130) || !(y <= 8.5e-12)) {
		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+130)) .or. (.not. (y <= 8.5d-12))) 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+130) || !(y <= 8.5e-12)) {
		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+130) or not (y <= 8.5e-12):
		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+130) || !(y <= 8.5e-12))
		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+130) || ~((y <= 8.5e-12)))
		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+130], N[Not[LessEqual[y, 8.5e-12]], $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 < -4.9999999999999996e130 or 8.4999999999999997e-12 < y

   1. Initial program 83.2%

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

      1. *-commutative 83.2%

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

      2. exp-to-pow 83.2%

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

      3. +-commutative 83.2%

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

   3. Simplified 83.2%

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

   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 -4.9999999999999996e130 < y < 8.4999999999999997e-12

   1. Initial program 89.5%

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

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+130} \lor \neg \left(y \leq 8.5 \cdot 10^{-12}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{y + z}\right)}}{y}\\ \end{array} \]

Alternative 2: 99.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \lor \neg \left(y \leq 8.5 \cdot 10^{-12}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.7) (not (<= y 8.5e-12)))
   (+ x (/ (exp (- z)) y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.7) || !(y <= 8.5e-12)) {
		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 <= (-1.7d0)) .or. (.not. (y <= 8.5d-12))) 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 <= -1.7) || !(y <= 8.5e-12)) {
		tmp = x + (Math.exp(-z) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.7) or not (y <= 8.5e-12):
		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 <= -1.7) || !(y <= 8.5e-12))
		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 <= -1.7) || ~((y <= 8.5e-12)))
		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, -1.7], N[Not[LessEqual[y, 8.5e-12]], $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 < -1.69999999999999996 or 8.4999999999999997e-12 < y

   1. Initial program 86.9%

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

      1. *-commutative 86.9%

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

      2. exp-to-pow 86.9%

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

      3. +-commutative 86.9%

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

   3. Simplified 86.9%

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

   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 -1.69999999999999996 < y < 8.4999999999999997e-12

   1. Initial program 86.7%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 0 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \lor \neg \left(y \leq 8.5 \cdot 10^{-12}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 3: 86.9% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.85:\\ \;\;\;\;x + \frac{\left(1 - z\right) + z \cdot \left(z \cdot 0.5\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -0.85) (+ 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 <= -0.85) {
		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 <= (-0.85d0)) 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 <= -0.85) {
		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 <= -0.85:
		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 <= -0.85)
		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 <= -0.85)
		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, -0.85], 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 < -0.849999999999999978

   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 89.1%

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

      2. +-commutative 89.1%

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

   3. Simplified 89.1%

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

   4. Taylor expanded in y around inf 68.5%

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

      1. associate-+r+ 68.5%

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

      2. +-commutative 68.5%

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

      3. associate-*r/ 68.5%

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

      4. distribute-rgt-out 68.5%

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

      5. metadata-eval 68.5%

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

      6. *-rgt-identity 68.5%

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

      7. associate-*l/ 68.5%

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

      8. metadata-eval 68.5%

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

      9. associate-*r/ 68.5%

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

      10. distribute-rgt-in 80.8%

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

      11. mul-1-neg 80.8%

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

      12. unsub-neg 80.8%

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

      13. unpow2 80.8%

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

      14. associate-*r/ 80.8%

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

      15. metadata-eval 80.8%

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

   6. Simplified 80.8%

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

   7. Taylor expanded in y around inf 80.8%

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

      1. unpow2 80.8%

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

      2. *-commutative 80.8%

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

      3. associate-*r* 80.8%

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

   9. Simplified 80.8%

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

   if -0.849999999999999978 < y

   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 94.5%

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

      2. +-commutative 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in y around 0 92.8%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.85:\\ \;\;\;\;x + \frac{\left(1 - z\right) + z \cdot \left(z \cdot 0.5\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 4: 84.9% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.02 \cdot 10^{+130}:\\ \;\;\;\;\frac{0.5}{y} \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.02e+130) (* (/ 0.5 y) (* z z)) (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.02e+130) {
		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 <= (-2.02d+130)) 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 <= -2.02e+130) {
		tmp = (0.5 / y) * (z * z);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.02e+130:
		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 <= -2.02e+130)
		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 <= -2.02e+130)
		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, -2.02e+130], 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 < -2.0199999999999999e130

   1. Initial program 54.4%

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

      1. exp-prod 79.2%

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

      2. +-commutative 79.2%

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

   3. Simplified 79.2%

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

   4. Taylor expanded in y around inf 0.5%

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

      1. associate-+r+ 0.5%

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

      2. +-commutative 0.5%

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

      3. associate-*r/ 0.5%

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

      4. distribute-rgt-out 0.5%

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

      5. metadata-eval 0.5%

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

      6. *-rgt-identity 0.5%

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

      7. associate-*l/ 0.5%

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

      8. metadata-eval 0.5%

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

      9. associate-*r/ 0.5%

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

      10. distribute-rgt-in 53.5%

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

      11. mul-1-neg 53.5%

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

      12. unsub-neg 53.5%

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

      13. unpow2 53.5%

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

      14. associate-*r/ 53.5%

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

      15. metadata-eval 53.5%

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

   6. Simplified 53.5%

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

   7. Taylor expanded in z around inf 53.7%

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

      1. unpow2 53.7%

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

      2. *-commutative 53.7%

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

      3. associate-*r/ 53.7%

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

      4. metadata-eval 53.7%

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

      5. +-commutative 53.7%

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

      6. associate-*r/ 53.7%

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

      7. metadata-eval 53.7%

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

      8. unpow2 53.7%

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

   9. Simplified 53.7%

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

   10. Taylor expanded in y around inf 55.1%

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

   if -2.0199999999999999e130 < z

   1. Initial program 89.4%

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

      1. exp-prod 93.9%

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

      2. +-commutative 93.9%

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

   3. Simplified 93.9%

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

   4. Taylor expanded in y around 0 90.6%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.02 \cdot 10^{+130}:\\ \;\;\;\;\frac{0.5}{y} \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 5: 68.1% accurate, 29.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{-18}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{-50}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -8.6e-18) x (if (<= y 4.1e-50) (/ 1.0 y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.6e-18) {
		tmp = x;
	} else if (y <= 4.1e-50) {
		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 <= (-8.6d-18)) then
        tmp = x
    else if (y <= 4.1d-50) 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 <= -8.6e-18) {
		tmp = x;
	} else if (y <= 4.1e-50) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -8.6e-18:
		tmp = x
	elif y <= 4.1e-50:
		tmp = 1.0 / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -8.6e-18)
		tmp = x;
	elseif (y <= 4.1e-50)
		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 <= -8.6e-18)
		tmp = x;
	elseif (y <= 4.1e-50)
		tmp = 1.0 / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -8.6e-18], x, If[LessEqual[y, 4.1e-50], N[(1.0 / y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -8.6000000000000005e-18 or 4.09999999999999985e-50 < y

   1. Initial program 87.6%

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

      1. exp-prod 87.6%

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

      2. +-commutative 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in y around 0 76.7%

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

   5. Taylor expanded in x around inf 67.7%

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

   if -8.6000000000000005e-18 < y < 4.09999999999999985e-50

   1. Initial program 85.6%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 0 100.0%

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

   5. Taylor expanded in x around 0 78.3%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{-18}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{-50}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 84.5% 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 0 86.3%

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

5. Final simplification 86.3%

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

Alternative 7: 49.2% 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 y around 0 86.3%

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

5. Taylor expanded in x around inf 49.4%

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

6. Final simplification 49.4%

   \[\leadsto x \]

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 2023271 
(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)))