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

Percentage Accurate: 84.6% → 99.2%

Time: 6.9s

Alternatives: 6

Speedup: 17.5×

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: 84.6% 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: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+108} \lor \neg \left(y \leq 2 \cdot 10^{-15}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{y + z}\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4e+108) (not (<= y 2e-15)))
   (+ 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 <= -4e+108) || !(y <= 2e-15)) {
		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 <= (-4d+108)) .or. (.not. (y <= 2d-15))) 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 <= -4e+108) || !(y <= 2e-15)) {
		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 <= -4e+108) or not (y <= 2e-15):
		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 <= -4e+108) || !(y <= 2e-15))
		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 <= -4e+108) || ~((y <= 2e-15)))
		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, -4e+108], N[Not[LessEqual[y, 2e-15]], $MachinePrecision]], N[(x + N[(N[Exp[(-z)], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Power[N[Exp[y], $MachinePrecision], N[Log[N[(y / N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.0000000000000001e108 or 2.0000000000000002e-15 < y

   1. Initial program 87.6%

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

      1. *-commutative 87.6%

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

      2. exp-prod 87.6%

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

      3. rem-exp-log 87.6%

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

      4. +-commutative 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   if -4.0000000000000001e108 < y < 2.0000000000000002e-15

   1. Initial program 84.4%

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

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

      3. sqr-pow 100.0%

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

      4. +-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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+108} \lor \neg \left(y \leq 2 \cdot 10^{-15}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{y + z}\right)}}{y}\\ \end{array} \]

Alternative 2: 99.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -12500000000 \lor \neg \left(y \leq 1.9 \cdot 10^{-11}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -12500000000.0) (not (<= y 1.9e-11)))
   (+ x (/ (exp (- z)) y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -12500000000.0) || !(y <= 1.9e-11)) {
		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 <= (-12500000000.0d0)) .or. (.not. (y <= 1.9d-11))) 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 <= -12500000000.0) || !(y <= 1.9e-11)) {
		tmp = x + (Math.exp(-z) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -12500000000.0) or not (y <= 1.9e-11):
		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 <= -12500000000.0) || !(y <= 1.9e-11))
		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 <= -12500000000.0) || ~((y <= 1.9e-11)))
		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, -12500000000.0], N[Not[LessEqual[y, 1.9e-11]], $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 < -1.25e10 or 1.8999999999999999e-11 < y

   1. Initial program 88.6%

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

      1. *-commutative 88.6%

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

      2. exp-prod 88.6%

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

      3. rem-exp-log 88.6%

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

      4. +-commutative 88.6%

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

   3. Simplified 88.6%

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

   4. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   if -1.25e10 < y < 1.8999999999999999e-11

   1. Initial program 83.0%

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

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

      3. sqr-pow 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -12500000000 \lor \neg \left(y \leq 1.9 \cdot 10^{-11}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 3: 85.5% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{+154}:\\ \;\;\;\;x + \frac{-1 - z \cdot z}{-y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -7.6e+154) (+ x (/ (- -1.0 (* z z)) (- y))) (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -7.6e+154) {
		tmp = x + ((-1.0 - (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 (z <= (-7.6d+154)) then
        tmp = x + (((-1.0d0) - (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 (z <= -7.6e+154) {
		tmp = x + ((-1.0 - (z * z)) / -y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -7.6e+154:
		tmp = x + ((-1.0 - (z * z)) / -y)
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -7.6e+154)
		tmp = Float64(x + Float64(Float64(-1.0 - Float64(z * z)) / Float64(-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 <= -7.6e+154)
		tmp = x + ((-1.0 - (z * z)) / -y);
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -7.6e+154], N[(x + N[(N[(-1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision] / (-y)), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.5999999999999996e154

   1. Initial program 57.4%

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

      1. exp-prod 85.7%

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

      2. sqr-pow 85.7%

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

      3. sqr-pow 85.7%

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

      4. +-commutative 85.7%

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

   3. Simplified 85.7%

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

   4. Taylor expanded in y around inf 4.4%

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

      1. +-commutative 4.4%

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

      2. mul-1-neg 4.4%

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

      3. unsub-neg 4.4%

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

   6. Simplified 4.4%

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

      1. frac-2neg 4.4%

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

      2. div-inv 4.4%

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

      3. sub-neg 4.4%

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

      4. distribute-neg-in 4.4%

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

      5. metadata-eval 4.4%

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

      6. add-sqr-sqrt 4.4%

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

      7. sqrt-unprod 71.7%

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

      8. sqr-neg 71.7%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 1.1%

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

      11. add-sqr-sqrt 1.1%

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

      12. sqrt-unprod 0.1%

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

      13. sqr-neg 0.1%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 4.4%

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

   8. Applied egg-rr 4.4%

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

      1. flip-+ 71.7%

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

      2. frac-2neg 71.7%

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

      3. metadata-eval 71.7%

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

      4. frac-times 21.7%

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

      5. metadata-eval 21.7%

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

      6. sub-neg 21.7%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 0.1%

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

      9. sqr-neg 0.1%

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

      10. sqrt-unprod 0.1%

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

      11. add-sqr-sqrt 0.1%

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

      12. distribute-lft-neg-out 0.1%

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

      13. sqr-neg 0.1%

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

      14. sub-neg 0.1%

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

      15. add-sqr-sqrt 0.1%

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

      16. sqrt-unprod 0.0%

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

      17. sqr-neg 0.0%

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

      18. sqrt-unprod 0.0%

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

      19. add-sqr-sqrt 21.7%

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

   10. Applied egg-rr 21.7%

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

       1. *-commutative 21.7%

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

       2. *-commutative 21.7%

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

       3. distribute-lft-in 21.7%

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

       4. metadata-eval 21.7%

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

       5. mul-1-neg 21.7%

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

       6. unsub-neg 21.7%

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

       7. +-commutative 21.7%

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

       8. distribute-lft-in 21.7%

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

       9. *-commutative 21.7%

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

       10. mul-1-neg 21.7%

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

       11. unsub-neg 21.7%

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

   12. Simplified 21.7%

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

   13. Taylor expanded in z around 0 71.7%

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

       1. neg-mul-1 71.7%

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

   15. Simplified 71.7%

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

   if -7.5999999999999996e154 < z

   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 94.5%

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

      2. sqr-pow 94.5%

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

      3. sqr-pow 94.5%

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

      4. +-commutative 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in y around 0 93.4%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{+154}:\\ \;\;\;\;x + \frac{-1 - z \cdot z}{-y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 4: 67.7% accurate, 29.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-22}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 7800000000:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.5e-22) x (if (<= y 7800000000.0) (/ 1.0 y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.5e-22) {
		tmp = x;
	} else if (y <= 7800000000.0) {
		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 <= (-3.5d-22)) then
        tmp = x
    else if (y <= 7800000000.0d0) 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 <= -3.5e-22) {
		tmp = x;
	} else if (y <= 7800000000.0) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.5e-22:
		tmp = x
	elif y <= 7800000000.0:
		tmp = 1.0 / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.5e-22)
		tmp = x;
	elseif (y <= 7800000000.0)
		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 <= -3.5e-22)
		tmp = x;
	elseif (y <= 7800000000.0)
		tmp = 1.0 / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.5e-22], x, If[LessEqual[y, 7800000000.0], N[(1.0 / y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -3.50000000000000005e-22 or 7.8e9 < y

   1. Initial program 88.8%

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

      1. exp-prod 88.8%

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

      2. sqr-pow 88.8%

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

      3. sqr-pow 88.8%

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

      4. +-commutative 88.8%

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

   3. Simplified 88.8%

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

   4. Taylor expanded in y around 0 79.3%

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

   5. Taylor expanded in x around inf 68.9%

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

   if -3.50000000000000005e-22 < y < 7.8e9

   1. Initial program 82.5%

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

      1. exp-prod 100.0%

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

      2. sqr-pow 100.0%

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

      3. sqr-pow 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

   5. Taylor expanded in x around 0 81.7%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-22}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 7800000000:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 84.5% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.0%

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

   1. exp-prod 93.8%

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

   2. sqr-pow 93.8%

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

   3. sqr-pow 93.8%

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

   4. +-commutative 93.8%

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

3. Simplified 93.8%

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

4. Taylor expanded in y around 0 88.6%

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

5. Final simplification 88.6%

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

Alternative 6: 48.6% 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 86.0%

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

   1. exp-prod 93.8%

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

   2. sqr-pow 93.8%

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

   3. sqr-pow 93.8%

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

   4. +-commutative 93.8%

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

3. Simplified 93.8%

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

4. Taylor expanded in y around 0 88.6%

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

5. Taylor expanded in x around inf 47.4%

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

6. Final simplification 47.4%

   \[\leadsto x \]

Developer target: 91.4% 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 2023195 
(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)))