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

Percentage Accurate: 84.3% → 99.6%

Time: 10.7s

Alternatives: 8

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: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+97} \lor \neg \left(y \leq 0.54\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{y + z}\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -8.5e+97) (not (<= y 0.54)))
   (+ 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 <= -8.5e+97) || !(y <= 0.54)) {
		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 <= (-8.5d+97)) .or. (.not. (y <= 0.54d0))) 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 <= -8.5e+97) || !(y <= 0.54)) {
		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 <= -8.5e+97) or not (y <= 0.54):
		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 <= -8.5e+97) || !(y <= 0.54))
		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 <= -8.5e+97) || ~((y <= 0.54)))
		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, -8.5e+97], N[Not[LessEqual[y, 0.54]], $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 < -8.4999999999999993e97 or 0.54000000000000004 < y

   1. Initial program 78.7%

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

      1. *-commutative 78.7%

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

      2. exp-to-pow 78.7%

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

      3. +-commutative 78.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   if -8.4999999999999993e97 < y < 0.54000000000000004

   1. Initial program 87.9%

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

      1. exp-prod 99.8%

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

      2. +-commutative 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+97} \lor \neg \left(y \leq 0.54\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{y + z}\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -50 \lor \neg \left(y \leq 0.4\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -50.0) (not (<= y 0.4)))
   (+ x (/ (exp (- z)) y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -50.0) || !(y <= 0.4)) {
		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 <= (-50.0d0)) .or. (.not. (y <= 0.4d0))) 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 <= -50.0) || !(y <= 0.4)) {
		tmp = x + (Math.exp(-z) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -50.0) or not (y <= 0.4):
		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 <= -50.0) || !(y <= 0.4))
		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 <= -50.0) || ~((y <= 0.4)))
		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, -50.0], N[Not[LessEqual[y, 0.4]], $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 < -50 or 0.40000000000000002 < y

   1. Initial program 80.6%

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

      1. *-commutative 80.6%

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

      2. exp-to-pow 80.6%

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

      3. +-commutative 80.6%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   if -50 < y < 0.40000000000000002

   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 99.8%

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

      2. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 98.5%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -50 \lor \neg \left(y \leq 0.4\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.5% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1

   1. Initial program 82.1%

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

      1. exp-prod 82.1%

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

      2. +-commutative 82.1%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in z around 0 80.0%

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

   6. Taylor expanded in y around 0 80.3%

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

      1. distribute-lft-out 80.3%

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

      2. *-commutative 80.3%

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

   8. Simplified 80.3%

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

   if -1 < y

   1. Initial program 84.0%

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

      1. exp-prod 92.4%

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

      2. +-commutative 92.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in y around inf 89.5%

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

4. Final simplification 87.0%

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

Alternative 4: 88.0% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.16:\\ \;\;\;\;x + \frac{1 + z \cdot \left(z \cdot \left(z \cdot -0.16666666666666666\right) + -1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.16)
   (+ x (/ (+ 1.0 (* z (+ (* z (* z -0.16666666666666666)) -1.0))) y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.16) {
		tmp = x + ((1.0 + (z * ((z * (z * -0.16666666666666666)) + -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 <= (-1.16d0)) then
        tmp = x + ((1.0d0 + (z * ((z * (z * (-0.16666666666666666d0))) + (-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 <= -1.16) {
		tmp = x + ((1.0 + (z * ((z * (z * -0.16666666666666666)) + -1.0))) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.16:
		tmp = x + ((1.0 + (z * ((z * (z * -0.16666666666666666)) + -1.0))) / y)
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.16)
		tmp = Float64(x + Float64(Float64(1.0 + Float64(z * Float64(Float64(z * Float64(z * -0.16666666666666666)) + -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 <= -1.16)
		tmp = x + ((1.0 + (z * ((z * (z * -0.16666666666666666)) + -1.0))) / y);
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.16], N[(x + N[(N[(1.0 + N[(z * N[(N[(z * N[(z * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.15999999999999992

   1. Initial program 82.1%

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

      1. *-commutative 82.1%

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

      2. exp-to-pow 82.1%

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

      3. +-commutative 82.1%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in z around 0 80.0%

      \[\leadsto x + \frac{\color{blue}{1 + z \cdot \left(z \cdot \left(0.5 + -0.16666666666666666 \cdot z\right) - 1\right)}}{y} \]

   9. Taylor expanded in z around inf 80.0%

      \[\leadsto x + \frac{1 + z \cdot \left(z \cdot \color{blue}{\left(-0.16666666666666666 \cdot z\right)} - 1\right)}{y} \]
   10. Step-by-step derivation

       1. *-commutative 80.0%

          \[\leadsto x + \frac{1 + z \cdot \left(z \cdot \color{blue}{\left(z \cdot -0.16666666666666666\right)} - 1\right)}{y} \]

   11. Simplified 80.0%

       \[\leadsto x + \frac{1 + z \cdot \left(z \cdot \color{blue}{\left(z \cdot -0.16666666666666666\right)} - 1\right)}{y} \]

   if -1.15999999999999992 < y

   1. Initial program 84.0%

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

      1. exp-prod 92.4%

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

      2. +-commutative 92.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in y around inf 89.5%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.16:\\ \;\;\;\;x + \frac{1 + z \cdot \left(z \cdot \left(z \cdot -0.16666666666666666\right) + -1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.0% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.9:\\ \;\;\;\;x + \frac{1 + z \cdot \left(z \cdot 0.5 + -1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -0.9) (+ x (/ (+ 1.0 (* z (+ (* z 0.5) -1.0))) y)) (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.9) {
		tmp = x + ((1.0 + (z * ((z * 0.5) + -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 <= (-0.9d0)) then
        tmp = x + ((1.0d0 + (z * ((z * 0.5d0) + (-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 <= -0.9) {
		tmp = x + ((1.0 + (z * ((z * 0.5) + -1.0))) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -0.9:
		tmp = x + ((1.0 + (z * ((z * 0.5) + -1.0))) / y)
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -0.9)
		tmp = Float64(x + Float64(Float64(1.0 + Float64(z * Float64(Float64(z * 0.5) + -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 <= -0.9)
		tmp = x + ((1.0 + (z * ((z * 0.5) + -1.0))) / y);
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -0.9], N[(x + N[(N[(1.0 + N[(z * N[(N[(z * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.900000000000000022

   1. Initial program 82.1%

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

      1. exp-prod 82.1%

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

      2. +-commutative 82.1%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in z around 0 80.0%

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

   6. Taylor expanded in y around inf 80.0%

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

      1. *-commutative 80.0%

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

   8. Simplified 80.0%

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

   if -0.900000000000000022 < y

   1. Initial program 84.0%

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

      1. exp-prod 92.4%

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

      2. +-commutative 92.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in y around inf 89.5%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.9:\\ \;\;\;\;x + \frac{1 + z \cdot \left(z \cdot 0.5 + -1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.4% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-109}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{-88}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7e-109) x (if (<= y 8.4e-88) (/ 1.0 y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7e-109) {
		tmp = x;
	} else if (y <= 8.4e-88) {
		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 <= (-7d-109)) then
        tmp = x
    else if (y <= 8.4d-88) 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 <= -7e-109) {
		tmp = x;
	} else if (y <= 8.4e-88) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -7e-109:
		tmp = x
	elif y <= 8.4e-88:
		tmp = 1.0 / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7e-109)
		tmp = x;
	elseif (y <= 8.4e-88)
		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 <= -7e-109)
		tmp = x;
	elseif (y <= 8.4e-88)
		tmp = 1.0 / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7e-109], x, If[LessEqual[y, 8.4e-88], N[(1.0 / y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -7e-109 or 8.3999999999999998e-88 < y

   1. Initial program 84.1%

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

      1. exp-prod 84.0%

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

      2. +-commutative 84.0%

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

   3. Simplified 84.0%

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

   5. Taylor expanded in x around inf 66.0%

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

   if -7e-109 < y < 8.3999999999999998e-88

   1. Initial program 82.3%

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

   5. Taylor expanded in y around 0 82.8%

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

Alternative 7: 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 83.5%

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

   1. exp-prod 89.6%

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

   2. +-commutative 89.6%

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

3. Simplified 89.6%

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

5. Taylor expanded in y around inf 83.0%

   \[\leadsto \color{blue}{x + \frac{1}{y}} \]
6. Add Preprocessing

Alternative 8: 48.7% 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 83.5%

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

   1. exp-prod 89.6%

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

   2. +-commutative 89.6%

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

3. Simplified 89.6%

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

5. Taylor expanded in x around inf 49.3%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer Target 1: 91.2% 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 2024186 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (/ y (+ z y)) 17788539399477/2500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ x (/ (exp (/ -1 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y))))

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