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

Percentage Accurate: 85.6% → 99.6%

Time: 8.7s

Alternatives: 7

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: 85.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (exp((y * log((y / (z + y))))) / y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -10000000.0) (not (<= y 2.25e-26)))
   (+ 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 <= -10000000.0) || !(y <= 2.25e-26)) {
		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 <= (-10000000.0d0)) .or. (.not. (y <= 2.25d-26))) 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 <= -10000000.0) || !(y <= 2.25e-26)) {
		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 <= -10000000.0) or not (y <= 2.25e-26):
		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 <= -10000000.0) || !(y <= 2.25e-26))
		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 <= -10000000.0) || ~((y <= 2.25e-26)))
		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, -10000000.0], N[Not[LessEqual[y, 2.25e-26]], $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 < -1e7 or 2.2499999999999999e-26 < y

   1. Initial program 89.2%

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

      1. *-commutative 89.2%

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

      2. exp-to-pow 89.2%

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

      3. +-commutative 89.2%

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

   3. Simplified 89.2%

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

   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 -1e7 < y < 2.2499999999999999e-26

   1. Initial program 82.4%

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

      1. exp-prod 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-520.0d0)) .or. (.not. (y <= 2.25d-26))) then
        tmp = x + (exp(-z) / y)
    else
        tmp = x + (1.0d0 / y)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 89.4%

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

      1. *-commutative 89.4%

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

      2. exp-to-pow 89.4%

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

      3. +-commutative 89.4%

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

   3. Simplified 89.4%

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

   4. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   if -520 < y < 2.2499999999999999e-26

   1. Initial program 82.0%

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

      1. exp-prod 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 87.5% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-520.0d0)) then
        tmp = (x - (z / y)) + ((1.0d0 + (0.5d0 * (z * z))) / y)
    else
        tmp = x + (1.0d0 / y)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -520

   1. Initial program 88.4%

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

      1. 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 z around 0 85.3%

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

      1. associate-+r+ 85.3%

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

      2. mul-1-neg 85.3%

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

      3. unsub-neg 85.3%

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

      4. fma-def 85.3%

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

      5. unpow2 85.3%

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

      6. associate-*r/ 85.3%

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

      7. metadata-eval 85.3%

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

      8. associate-*r/ 85.3%

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

      9. metadata-eval 85.3%

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

      10. unpow2 85.3%

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

   6. Simplified 85.3%

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

   7. Taylor expanded in y around inf 85.3%

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

      1. unpow2 85.3%

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

   9. Simplified 85.3%

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

   if -520 < y

   1. Initial program 86.1%

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

      1. exp-prod 95.0%

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

      2. +-commutative 95.0%

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

   3. Simplified 95.0%

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

   4. Taylor expanded in y around inf 92.4%

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

      1. +-commutative 92.4%

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

   6. Simplified 92.4%

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

4. Final simplification 90.4%

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

Alternative 4: 87.5% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -20000:\\ \;\;\;\;x - \frac{\left(z \cdot z\right) \cdot -0.5 + \left(z + -1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -20000.0)
   (- x (/ (+ (* (* z z) -0.5) (+ z -1.0)) y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -20000.0) {
		tmp = x - ((((z * z) * -0.5) + (z + -1.0)) / 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 <= (-20000.0d0)) then
        tmp = x - ((((z * z) * (-0.5d0)) + (z + (-1.0d0))) / 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 <= -20000.0) {
		tmp = x - ((((z * z) * -0.5) + (z + -1.0)) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -20000.0:
		tmp = x - ((((z * z) * -0.5) + (z + -1.0)) / y)
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -20000.0)
		tmp = Float64(x - Float64(Float64(Float64(Float64(z * z) * -0.5) + Float64(z + -1.0)) / 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 <= -20000.0)
		tmp = x - ((((z * z) * -0.5) + (z + -1.0)) / y);
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -20000.0], N[(x - N[(N[(N[(N[(z * z), $MachinePrecision] * -0.5), $MachinePrecision] + N[(z + -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2e4

   1. Initial program 88.4%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 z around 0 85.3%

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

      1. associate-+r+ 85.3%

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

      2. mul-1-neg 85.3%

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

      3. unsub-neg 85.3%

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

      4. fma-def 85.3%

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

      5. unpow2 85.3%

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

      6. associate-*r/ 85.3%

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

      7. metadata-eval 85.3%

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

      8. associate-*r/ 85.3%

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

      9. metadata-eval 85.3%

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

      10. unpow2 85.3%

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

   6. Simplified 85.3%

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

   7. Taylor expanded in y around -inf 85.2%

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

      1. mul-1-neg 85.2%

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

      2. neg-mul-1 85.2%

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

      3. *-commutative 85.2%

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

      4. unpow2 85.2%

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

      5. sub-neg 85.2%

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

   9. Simplified 85.2%

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

   if -2e4 < y

   1. Initial program 86.1%

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

      1. exp-prod 95.0%

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

      2. +-commutative 95.0%

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

   3. Simplified 95.0%

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

   4. Taylor expanded in y around inf 92.4%

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

      1. +-commutative 92.4%

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

   6. Simplified 92.4%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -20000:\\ \;\;\;\;x - \frac{\left(z \cdot z\right) \cdot -0.5 + \left(z + -1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 5: 67.8% accurate, 29.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-83}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-30}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.8e-83) x (if (<= y 7e-30) (/ 1.0 y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.8e-83) {
		tmp = x;
	} else if (y <= 7e-30) {
		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 <= (-4.8d-83)) then
        tmp = x
    else if (y <= 7d-30) 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 <= -4.8e-83) {
		tmp = x;
	} else if (y <= 7e-30) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.8e-83:
		tmp = x
	elif y <= 7e-30:
		tmp = 1.0 / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.8e-83)
		tmp = x;
	elseif (y <= 7e-30)
		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 <= -4.8e-83)
		tmp = x;
	elseif (y <= 7e-30)
		tmp = 1.0 / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.8e-83], x, If[LessEqual[y, 7e-30], N[(1.0 / y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -4.8000000000000002e-83 or 7.0000000000000006e-30 < y

   1. Initial program 90.4%

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

      1. exp-prod 89.9%

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

      2. +-commutative 89.9%

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

   3. Simplified 89.9%

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

   4. Taylor expanded in x around inf 69.2%

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

   if -4.8000000000000002e-83 < y < 7.0000000000000006e-30

   1. Initial program 77.6%

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

      1. exp-prod 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 80.1%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-83}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-30}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 85.3% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (1.0d0 / y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

   1. exp-prod 92.8%

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

   2. +-commutative 92.8%

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

3. Simplified 92.8%

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

4. Taylor expanded in y around inf 87.9%

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

   1. +-commutative 87.9%

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

6. Simplified 87.9%

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

7. Final simplification 87.9%

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

Alternative 7: 49.8% accurate, 211.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 86.8%

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

   1. exp-prod 92.8%

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

   2. +-commutative 92.8%

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

3. Simplified 92.8%

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

4. Taylor expanded in x around inf 55.4%

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

5. Final simplification 55.4%

   \[\leadsto x \]

Developer target: 92.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y / (z + y)) < 7.11541576d-315) then
        tmp = x + (exp(((-1.0d0) / z)) / y)
    else
        tmp = x + (exp(log(((y / (y + z)) ** y))) / y)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

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

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

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