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

Percentage Accurate: 84.7% → 99.8%

Time: 10.4s

Alternatives: 5

Speedup: 9.1×

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: 84.7% 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: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+28} \lor \neg \left(y \leq 2.6 \cdot 10^{-8}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{y + z}\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2e+28) (not (<= y 2.6e-8)))
   (+ 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 <= -2e+28) || !(y <= 2.6e-8)) {
		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 <= (-2d+28)) .or. (.not. (y <= 2.6d-8))) 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 <= -2e+28) || !(y <= 2.6e-8)) {
		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 <= -2e+28) or not (y <= 2.6e-8):
		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 <= -2e+28) || !(y <= 2.6e-8))
		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 <= -2e+28) || ~((y <= 2.6e-8)))
		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, -2e+28], N[Not[LessEqual[y, 2.6e-8]], $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 < -1.99999999999999992e28 or 2.6000000000000001e-8 < y

   1. Initial program 83.9%

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

      1. *-commutative 83.9%

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

      2. exp-to-pow 83.9%

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

      3. +-commutative 83.9%

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

   3. Simplified 83.9%

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

   5. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   if -1.99999999999999992e28 < y < 2.6000000000000001e-8

   1. Initial program 84.6%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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. +-commutative 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+28} \lor \neg \left(y \leq 2.6 \cdot 10^{-8}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{y + z}\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+19} \lor \neg \left(y \leq 2.6 \cdot 10^{-8}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -7.2e+19) (not (<= y 2.6e-8)))
   (+ x (/ (exp (- z)) y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7.2e+19) || !(y <= 2.6e-8)) {
		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 <= (-7.2d+19)) .or. (.not. (y <= 2.6d-8))) 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 <= -7.2e+19) || !(y <= 2.6e-8)) {
		tmp = x + (Math.exp(-z) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -7.2e+19) or not (y <= 2.6e-8):
		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 <= -7.2e+19) || !(y <= 2.6e-8))
		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 <= -7.2e+19) || ~((y <= 2.6e-8)))
		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, -7.2e+19], N[Not[LessEqual[y, 2.6e-8]], $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 < -7.2e19 or 2.6000000000000001e-8 < y

   1. Initial program 84.3%

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

      1. *-commutative 84.3%

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

      2. exp-to-pow 84.3%

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

      3. +-commutative 84.3%

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

   3. Simplified 84.3%

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

   5. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   if -7.2e19 < y < 2.6000000000000001e-8

   1. Initial program 84.2%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 99.6%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+19} \lor \neg \left(y \leq 2.6 \cdot 10^{-8}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.7% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+19}:\\ \;\;\;\;x + \frac{\frac{\left(z + 1\right) \cdot \frac{z + 1}{y}}{y}}{\frac{1}{y} - \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.2e+19)
   (+ x (/ (/ (* (+ z 1.0) (/ (+ z 1.0) y)) y) (- (/ 1.0 y) (/ z y))))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.2e+19) {
		tmp = x + ((((z + 1.0) * ((z + 1.0) / y)) / y) / ((1.0 / y) - (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 <= (-7.2d+19)) then
        tmp = x + ((((z + 1.0d0) * ((z + 1.0d0) / y)) / y) / ((1.0d0 / y) - (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 <= -7.2e+19) {
		tmp = x + ((((z + 1.0) * ((z + 1.0) / y)) / y) / ((1.0 / y) - (z / y)));
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -7.2e+19:
		tmp = x + ((((z + 1.0) * ((z + 1.0) / y)) / y) / ((1.0 / y) - (z / y)))
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7.2e+19)
		tmp = Float64(x + Float64(Float64(Float64(Float64(z + 1.0) * Float64(Float64(z + 1.0) / y)) / y) / Float64(Float64(1.0 / y) - 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 <= -7.2e+19)
		tmp = x + ((((z + 1.0) * ((z + 1.0) / y)) / y) / ((1.0 / y) - (z / y)));
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7.2e+19], N[(x + N[(N[(N[(N[(z + 1.0), $MachinePrecision] * N[(N[(z + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / N[(N[(1.0 / y), $MachinePrecision] - N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.2e19

   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 82.5%

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

      2. +-commutative 82.5%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in y around inf 64.1%

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

      1. mul-1-neg 64.1%

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

      2. unsub-neg 64.1%

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

   7. Simplified 64.1%

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

      1. frac-2neg 64.1%

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

      2. div-inv 64.1%

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

      3. sub-neg 64.1%

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

      4. distribute-neg-in 64.1%

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

      5. metadata-eval 64.1%

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

      6. add-sqr-sqrt 28.4%

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

      7. sqrt-unprod 79.9%

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

      8. sqr-neg 79.9%

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

      9. sqrt-unprod 35.7%

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

      10. add-sqr-sqrt 63.5%

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

      11. add-sqr-sqrt 27.8%

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

      12. sqrt-unprod 63.3%

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

      13. sqr-neg 63.3%

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

      14. sqrt-unprod 35.7%

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

      15. add-sqr-sqrt 64.1%

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

   9. Applied egg-rr 64.1%

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

       1. *-commutative 64.1%

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

   11. Simplified 64.1%

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

       1. distribute-lft-in 64.1%

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

       2. flip-+ 67.5%

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

   13. Applied egg-rr 61.6%

       \[\leadsto x + \color{blue}{\frac{{y}^{-2} - \frac{z}{y} \cdot \frac{z}{y}}{\frac{1}{y} - \frac{z}{y}}} \]
   14. Step-by-step derivation

       1. metadata-eval 61.6%

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

       2. pow-prod-up 61.7%

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

       3. inv-pow 61.7%

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

       4. inv-pow 61.7%

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

       5. difference-of-squares 61.7%

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

       6. div-inv 61.7%

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

       7. distribute-rgt1-in 61.7%

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

       8. add-sqr-sqrt 34.0%

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

       9. sqrt-unprod 78.2%

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

       10. sqr-neg 78.2%

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

       11. sqrt-unprod 33.5%

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

       12. add-sqr-sqrt 67.5%

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

       13. distribute-rgt1-in 67.5%

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

       14. cancel-sign-sub-inv 67.5%

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

       15. div-inv 67.5%

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

       16. sub-div 67.5%

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

       17. associate-*r/ 70.2%

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

   15. Applied egg-rr 70.2%

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

   if -7.2e19 < y

   1. Initial program 84.9%

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

      1. exp-prod 94.6%

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

      2. +-commutative 94.6%

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

   3. Simplified 94.6%

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

   5. Taylor expanded in y around 0 92.9%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+19}:\\ \;\;\;\;x + \frac{\frac{\left(z + 1\right) \cdot \frac{z + 1}{y}}{y}}{\frac{1}{y} - \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.2% 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 84.2%

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

   1. exp-prod 91.2%

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

   2. +-commutative 91.2%

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

3. Simplified 91.2%

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

5. Taylor expanded in y around 0 84.6%

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

6. Final simplification 84.6%

   \[\leadsto x + \frac{1}{y} \]
7. Add Preprocessing

Alternative 5: 50.3% 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 84.2%

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

   1. *-commutative 84.2%

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

   2. exp-to-pow 84.2%

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

   3. +-commutative 84.2%

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

3. Simplified 84.2%

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

5. Taylor expanded in y around inf 85.4%

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

   1. mul-1-neg 85.4%

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

7. Simplified 85.4%

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

8. Taylor expanded in x around inf 47.5%

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

9. Final simplification 47.5%

   \[\leadsto x \]
10. Add Preprocessing

Developer target: 91.0% 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 2024010 
(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)))