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

Percentage Accurate: 84.5% → 99.0%

Time: 9.1s

Alternatives: 7

Speedup: 19.1×

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: 84.5% 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: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+65} \lor \neg \left(y \leq 2.5 \cdot 10^{-41}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{y + z}\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1e+65) (not (<= y 2.5e-41)))
   (+ 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 <= -1e+65) || !(y <= 2.5e-41)) {
		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 <= (-1d+65)) .or. (.not. (y <= 2.5d-41))) 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 <= -1e+65) || !(y <= 2.5e-41)) {
		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 <= -1e+65) or not (y <= 2.5e-41):
		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 <= -1e+65) || !(y <= 2.5e-41))
		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 <= -1e+65) || ~((y <= 2.5e-41)))
		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, -1e+65], N[Not[LessEqual[y, 2.5e-41]], $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 < -9.9999999999999999e64 or 2.4999999999999998e-41 < y

   1. Initial program 84.5%

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

   2. Taylor expanded in y around inf 99.5%

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

      1. mul-1-neg 99.5%

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

   4. Simplified 99.5%

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

   if -9.9999999999999999e64 < y < 2.4999999999999998e-41

   1. Initial program 79.4%

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

      1. exp-prod 99.9%

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

      2. sqr-pow 99.9%

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

      3. sqr-pow 99.9%

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

      4. +-commutative 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+65} \lor \neg \left(y \leq 2.5 \cdot 10^{-41}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{y + z}\right)}}{y}\\ \end{array} \]

Alternative 2: 88.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{+78} \lor \neg \left(z \leq -3.05 \cdot 10^{+56}\right) \land z \leq -255000:\\ \;\;\;\;\frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -6.4e+78) (and (not (<= z -3.05e+56)) (<= z -255000.0)))
   (/ (exp (- z)) y)
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.4e+78) || (!(z <= -3.05e+56) && (z <= -255000.0))) {
		tmp = exp(-z) / y;
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-6.4d+78)) .or. (.not. (z <= (-3.05d+56))) .and. (z <= (-255000.0d0))) then
        tmp = exp(-z) / y
    else
        tmp = x + (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.4e+78) || (!(z <= -3.05e+56) && (z <= -255000.0))) {
		tmp = Math.exp(-z) / y;
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -6.4e+78) or (not (z <= -3.05e+56) and (z <= -255000.0)):
		tmp = math.exp(-z) / y
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -6.4e+78) || (!(z <= -3.05e+56) && (z <= -255000.0)))
		tmp = Float64(exp(Float64(-z)) / y);
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -6.4e+78) || (~((z <= -3.05e+56)) && (z <= -255000.0)))
		tmp = exp(-z) / y;
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -6.4e+78], And[N[Not[LessEqual[z, -3.05e+56]], $MachinePrecision], LessEqual[z, -255000.0]]], N[(N[Exp[(-z)], $MachinePrecision] / y), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.39999999999999989e78 or -3.0500000000000001e56 < z < -255000

   1. Initial program 46.3%

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

   2. Taylor expanded in y around inf 65.2%

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

      1. mul-1-neg 65.2%

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

   4. Simplified 65.2%

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

   5. Taylor expanded in x around 0 65.2%

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

   if -6.39999999999999989e78 < z < -3.0500000000000001e56 or -255000 < z

   1. Initial program 91.6%

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

   2. Taylor expanded in y around 0 96.9%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{+78} \lor \neg \left(z \leq -3.05 \cdot 10^{+56}\right) \land z \leq -255000:\\ \;\;\;\;\frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 3: 98.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+22} \lor \neg \left(y \leq 2.5 \cdot 10^{-41}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.05e+22) (not (<= y 2.5e-41)))
   (+ x (/ (exp (- z)) y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.05e+22) || !(y <= 2.5e-41)) {
		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 <= (-1.05d+22)) .or. (.not. (y <= 2.5d-41))) 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 <= -1.05e+22) || !(y <= 2.5e-41)) {
		tmp = x + (Math.exp(-z) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.05e+22) or not (y <= 2.5e-41):
		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 <= -1.05e+22) || !(y <= 2.5e-41))
		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 <= -1.05e+22) || ~((y <= 2.5e-41)))
		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, -1.05e+22], N[Not[LessEqual[y, 2.5e-41]], $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 < -1.0499999999999999e22 or 2.4999999999999998e-41 < y

   1. Initial program 85.3%

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

   2. Taylor expanded in y around inf 99.5%

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

      1. mul-1-neg 99.5%

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

   4. Simplified 99.5%

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

   if -1.0499999999999999e22 < y < 2.4999999999999998e-41

   1. Initial program 77.7%

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

   2. Taylor expanded in y around 0 99.3%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+22} \lor \neg \left(y \leq 2.5 \cdot 10^{-41}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 4: 85.6% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+172}:\\ \;\;\;\;x + \frac{0.5}{\frac{y}{z \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3e+172) (+ x (/ 0.5 (/ y (* z z)))) (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3e+172) {
		tmp = x + (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 <= (-3d+172)) then
        tmp = x + (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 <= -3e+172) {
		tmp = x + (0.5 / (y / (z * z)));
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3e+172:
		tmp = x + (0.5 / (y / (z * z)))
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3e+172)
		tmp = Float64(x + Float64(0.5 / Float64(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 <= -3e+172)
		tmp = x + (0.5 / (y / (z * z)));
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3e+172], N[(x + N[(0.5 / N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.9999999999999999e172

   1. Initial program 48.4%

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

      1. clear-num 44.1%

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

      2. log-rec 44.1%

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

      3. +-commutative 44.1%

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

   3. Applied egg-rr 44.1%

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

   4. Taylor expanded in z around 0 59.3%

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

      1. associate-+r+ 59.3%

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

      2. +-commutative 59.3%

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

      3. neg-mul-1 59.3%

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

      4. sub-neg 59.3%

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

      5. *-commutative 59.3%

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

      6. unpow2 59.3%

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

      7. +-commutative 59.3%

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

      8. associate-*r/ 59.3%

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

      9. metadata-eval 59.3%

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

   6. Simplified 59.3%

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

   7. Taylor expanded in z around inf 59.3%

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

      1. unpow2 59.3%

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

      2. associate-/l* 59.3%

         \[\leadsto x + \color{blue}{\frac{0.5 \cdot \frac{1}{y} + 0.5}{\frac{y}{z \cdot z}}} \]

      3. associate-*r/ 59.3%

         \[\leadsto x + \frac{\color{blue}{\frac{0.5 \cdot 1}{y}} + 0.5}{\frac{y}{z \cdot z}} \]

      4. metadata-eval 59.3%

         \[\leadsto x + \frac{\frac{\color{blue}{0.5}}{y} + 0.5}{\frac{y}{z \cdot z}} \]

      5. +-commutative 59.3%

         \[\leadsto x + \frac{\color{blue}{0.5 + \frac{0.5}{y}}}{\frac{y}{z \cdot z}} \]

   9. Simplified 59.3%

      \[\leadsto x + \color{blue}{\frac{0.5 + \frac{0.5}{y}}{\frac{y}{z \cdot z}}} \]

   10. Taylor expanded in y around inf 61.3%

       \[\leadsto x + \frac{\color{blue}{0.5}}{\frac{y}{z \cdot z}} \]

   if -2.9999999999999999e172 < z

   1. Initial program 85.6%

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

   2. Taylor expanded in y around 0 88.7%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+172}:\\ \;\;\;\;x + \frac{0.5}{\frac{y}{z \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 5: 66.5% accurate, 29.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-53}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-81}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.5e-53) x (if (<= y 7.8e-81) (/ 1.0 y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.5e-53) {
		tmp = x;
	} else if (y <= 7.8e-81) {
		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 <= (-1.5d-53)) then
        tmp = x
    else if (y <= 7.8d-81) 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 <= -1.5e-53) {
		tmp = x;
	} else if (y <= 7.8e-81) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.5e-53:
		tmp = x
	elif y <= 7.8e-81:
		tmp = 1.0 / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.5e-53)
		tmp = x;
	elseif (y <= 7.8e-81)
		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 <= -1.5e-53)
		tmp = x;
	elseif (y <= 7.8e-81)
		tmp = 1.0 / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.5e-53], x, If[LessEqual[y, 7.8e-81], N[(1.0 / y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5000000000000001e-53 or 7.7999999999999997e-81 < y

   1. Initial program 86.5%

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

   2. Taylor expanded in y around 0 78.1%

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

   3. Taylor expanded in x around inf 65.1%

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

   if -1.5000000000000001e-53 < y < 7.7999999999999997e-81

   1. Initial program 72.9%

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

   2. Taylor expanded in y around 0 100.0%

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

   3. Taylor expanded in x around 0 77.9%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-53}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-81}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 83.9% 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 82.4%

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

2. Taylor expanded in y around 0 84.7%

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

3. Final simplification 84.7%

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

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 82.4%

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

2. Taylor expanded in y around 0 84.7%

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

3. Taylor expanded in x around inf 52.2%

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

4. Final simplification 52.2%

   \[\leadsto x \]

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 2023221 
(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)))