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

Percentage Accurate: 85.2% → 98.4%

Time: 8.1s

Alternatives: 6

Speedup: 16.2×

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: 85.2% 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: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+109} \lor \neg \left(y \leq 1.3 \cdot 10^{-60}\right):\\ \;\;\;\;\frac{e^{-z}}{y} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{y + z}\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4.7e+109) (not (<= y 1.3e-60)))
   (+ (/ (exp (- z)) y) x)
   (+ x (/ (pow (exp y) (log (/ y (+ y z)))) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.7e+109) || !(y <= 1.3e-60)) {
		tmp = (exp(-z) / y) + x;
	} 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 <= (-4.7d+109)) .or. (.not. (y <= 1.3d-60))) then
        tmp = (exp(-z) / y) + x
    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 <= -4.7e+109) || !(y <= 1.3e-60)) {
		tmp = (Math.exp(-z) / y) + x;
	} 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 <= -4.7e+109) or not (y <= 1.3e-60):
		tmp = (math.exp(-z) / y) + x
	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 <= -4.7e+109) || !(y <= 1.3e-60))
		tmp = Float64(Float64(exp(Float64(-z)) / y) + x);
	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 <= -4.7e+109) || ~((y <= 1.3e-60)))
		tmp = (exp(-z) / y) + x;
	else
		tmp = x + ((exp(y) ^ log((y / (y + z)))) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4.7e+109], N[Not[LessEqual[y, 1.3e-60]], $MachinePrecision]], N[(N[(N[Exp[(-z)], $MachinePrecision] / y), $MachinePrecision] + x), $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 < -4.69999999999999998e109 or 1.2999999999999999e-60 < y

   1. Initial program 88.0%

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

      1. *-commutative 88.0%

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

      2. exp-prod 88.0%

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

      3. rem-exp-log 88.0%

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

      4. +-commutative 88.0%

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

   3. Simplified 88.0%

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

   4. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   if -4.69999999999999998e109 < y < 1.2999999999999999e-60

   1. Initial program 85.8%

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

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+109} \lor \neg \left(y \leq 1.3 \cdot 10^{-60}\right):\\ \;\;\;\;\frac{e^{-z}}{y} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{y + z}\right)}}{y}\\ \end{array} \]

Alternative 2: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+33} \lor \neg \left(y \leq 10^{-60}\right):\\ \;\;\;\;\frac{e^{-z}}{y} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.4e+33) (not (<= y 1e-60)))
   (+ (/ (exp (- z)) y) x)
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.4e+33) || !(y <= 1e-60)) {
		tmp = (exp(-z) / y) + x;
	} 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 <= (-2.4d+33)) .or. (.not. (y <= 1d-60))) then
        tmp = (exp(-z) / y) + x
    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 <= -2.4e+33) || !(y <= 1e-60)) {
		tmp = (Math.exp(-z) / y) + x;
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.4e+33) or not (y <= 1e-60):
		tmp = (math.exp(-z) / y) + x
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.4e+33) || !(y <= 1e-60))
		tmp = Float64(Float64(exp(Float64(-z)) / y) + x);
	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 <= -2.4e+33) || ~((y <= 1e-60)))
		tmp = (exp(-z) / y) + x;
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.4e+33], N[Not[LessEqual[y, 1e-60]], $MachinePrecision]], N[(N[(N[Exp[(-z)], $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.4e33 or 9.9999999999999997e-61 < y

   1. Initial program 89.1%

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

      1. *-commutative 89.1%

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

      2. exp-prod 89.1%

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

      3. rem-exp-log 89.1%

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

      4. +-commutative 89.1%

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

   3. Simplified 89.1%

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

   4. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   if -2.4e33 < y < 9.9999999999999997e-61

   1. Initial program 84.1%

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

      1. *-commutative 84.1%

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

      2. exp-prod 84.1%

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

      3. rem-exp-log 84.1%

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

      4. +-commutative 84.1%

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

   3. Simplified 84.1%

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

   4. Taylor expanded in z around 0 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+33} \lor \neg \left(y \leq 10^{-60}\right):\\ \;\;\;\;\frac{e^{-z}}{y} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 3: 86.8% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+36}:\\ \;\;\;\;\frac{1}{y} + \left(x + y \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.1e+36) (+ (/ 1.0 y) (+ x (* y (* x x)))) (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.1e+36) {
		tmp = (1.0 / y) + (x + (y * (x * x)));
	} 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 <= (-2.1d+36)) then
        tmp = (1.0d0 / y) + (x + (y * (x * x)))
    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 <= -2.1e+36) {
		tmp = (1.0 / y) + (x + (y * (x * x)));
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.1e+36:
		tmp = (1.0 / y) + (x + (y * (x * x)))
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.1e+36)
		tmp = Float64(Float64(1.0 / y) + Float64(x + Float64(y * Float64(x * x))));
	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 <= -2.1e+36)
		tmp = (1.0 / y) + (x + (y * (x * x)));
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.1e+36], N[(N[(1.0 / y), $MachinePrecision] + N[(x + N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.10000000000000004e36

   1. Initial program 46.4%

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

      1. *-commutative 46.4%

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

      2. exp-prod 46.4%

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

      3. rem-exp-log 46.4%

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

      4. +-commutative 46.4%

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

   3. Simplified 46.4%

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

   4. Taylor expanded in z around 0 48.0%

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

      1. flip-+ 25.7%

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

      2. div-sub 25.7%

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

      3. inv-pow 25.7%

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

      4. inv-pow 25.7%

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

      5. pow-prod-up 25.7%

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

      6. metadata-eval 25.7%

         \[\leadsto \frac{{y}^{\color{blue}{-2}}}{\frac{1}{y} - x} - \frac{x \cdot x}{\frac{1}{y} - x} \]

   6. Applied egg-rr 25.7%

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

      1. div-sub 25.7%

         \[\leadsto \color{blue}{\frac{{y}^{-2} - x \cdot x}{\frac{1}{y} - x}} \]

   8. Simplified 25.7%

      \[\leadsto \color{blue}{\frac{{y}^{-2} - x \cdot x}{\frac{1}{y} - x}} \]

   9. Taylor expanded in y around 0 16.3%

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

       1. unpow2 16.3%

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

   11. Simplified 16.3%

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

   12. Taylor expanded in y around 0 68.8%

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

       1. unpow2 68.8%

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

   14. Simplified 68.8%

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

   if -2.10000000000000004e36 < z

   1. Initial program 95.2%

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

      1. *-commutative 95.2%

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

      2. exp-prod 95.2%

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

      3. rem-exp-log 95.2%

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

      4. +-commutative 95.2%

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

   3. Simplified 95.2%

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

   4. Taylor expanded in z around 0 96.6%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+36}:\\ \;\;\;\;\frac{1}{y} + \left(x + y \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 4: 67.2% accurate, 29.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -27.5:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-80}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -27.5) x (if (<= y 4.8e-80) (/ 1.0 y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -27.5) {
		tmp = x;
	} else if (y <= 4.8e-80) {
		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 <= (-27.5d0)) then
        tmp = x
    else if (y <= 4.8d-80) 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 <= -27.5) {
		tmp = x;
	} else if (y <= 4.8e-80) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -27.5:
		tmp = x
	elif y <= 4.8e-80:
		tmp = 1.0 / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -27.5)
		tmp = x;
	elseif (y <= 4.8e-80)
		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 <= -27.5)
		tmp = x;
	elseif (y <= 4.8e-80)
		tmp = 1.0 / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -27.5], x, If[LessEqual[y, 4.8e-80], N[(1.0 / y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -27.5 or 4.7999999999999998e-80 < y

   1. Initial program 89.8%

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

      1. *-commutative 89.8%

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

      2. exp-prod 89.8%

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

      3. rem-exp-log 89.8%

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

      4. +-commutative 89.8%

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

   3. Simplified 89.8%

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

   4. Taylor expanded in x around inf 68.2%

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

   if -27.5 < y < 4.7999999999999998e-80

   1. Initial program 82.5%

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

      1. *-commutative 82.5%

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

      2. exp-prod 82.5%

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

      3. rem-exp-log 82.5%

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

      4. +-commutative 82.5%

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

   3. Simplified 82.5%

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

   4. Taylor expanded in z around 0 100.0%

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

      1. flip-+ 40.8%

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

      2. div-sub 40.8%

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

      3. inv-pow 40.8%

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

      4. inv-pow 40.8%

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

      5. pow-prod-up 40.5%

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

      6. metadata-eval 40.5%

         \[\leadsto \frac{{y}^{\color{blue}{-2}}}{\frac{1}{y} - x} - \frac{x \cdot x}{\frac{1}{y} - x} \]

   6. Applied egg-rr 40.5%

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

      1. div-sub 40.5%

         \[\leadsto \color{blue}{\frac{{y}^{-2} - x \cdot x}{\frac{1}{y} - x}} \]

   8. Simplified 40.5%

      \[\leadsto \color{blue}{\frac{{y}^{-2} - x \cdot x}{\frac{1}{y} - x}} \]

   9. Taylor expanded in y around 0 85.4%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -27.5:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-80}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 85.0% 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 87.0%

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

   1. *-commutative 87.0%

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

   2. exp-prod 87.0%

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

   3. rem-exp-log 87.0%

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

   4. +-commutative 87.0%

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

3. Simplified 87.0%

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

4. Taylor expanded in z around 0 88.5%

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

5. Final simplification 88.5%

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

Alternative 6: 49.6% 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 87.0%

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

   1. *-commutative 87.0%

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

   2. exp-prod 87.0%

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

   3. rem-exp-log 87.0%

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

   4. +-commutative 87.0%

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

3. Simplified 87.0%

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

4. Taylor expanded in x around inf 48.1%

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

5. Final simplification 48.1%

   \[\leadsto x \]

Developer target: 91.8% 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 2023185 
(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)))