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

Percentage Accurate: 84.7% → 98.8%

Time: 13.1s

Alternatives: 9

Speedup: 42.2×

Specification

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (/ (exp (* y (log (/ y (+ z y))))) y)))

C

double code(double x, double y, double z) {
	return x + (exp((y * log((y / (z + y))))) / y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (exp((y * log((y / (z + y))))) / y)
end function

Java

public static double code(double x, double y, double z) {
	return x + (Math.exp((y * Math.log((y / (z + y))))) / y);
}

Python

def code(x, y, z):
	return x + (math.exp((y * math.log((y / (z + y))))) / y)

Julia

function code(x, y, z)
	return Float64(x + Float64(exp(Float64(y * log(Float64(y / Float64(z + y))))) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (exp((y * log((y / (z + y))))) / y);
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[Exp[N[(y * N[Log[N[(y / N[(z + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Initial Program: 84.7% 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: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+124} \lor \neg \left(y \leq 5.8 \cdot 10^{-20}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{y + z}\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5e+124) (not (<= y 5.8e-20)))
   (+ 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 <= -5e+124) || !(y <= 5.8e-20)) {
		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 <= (-5d+124)) .or. (.not. (y <= 5.8d-20))) 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 <= -5e+124) || !(y <= 5.8e-20)) {
		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 <= -5e+124) or not (y <= 5.8e-20):
		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 <= -5e+124) || !(y <= 5.8e-20))
		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 <= -5e+124) || ~((y <= 5.8e-20)))
		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, -5e+124], N[Not[LessEqual[y, 5.8e-20]], $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 < -4.9999999999999996e124 or 5.8e-20 < y

   1. Initial program 78.2%

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

      1. *-commutative 78.2%

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

      2. exp-to-pow 78.2%

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

      3. +-commutative 78.2%

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

   3. Simplified 78.2%

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
   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 -4.9999999999999996e124 < y < 5.8e-20

   1. Initial program 87.5%

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

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

   3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+124} \lor \neg \left(y \leq 5.8 \cdot 10^{-20}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{y + z}\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -330 \lor \neg \left(y \leq 10^{-20}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -330.0) (not (<= y 1e-20)))
   (+ x (/ (exp (- z)) y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -330.0) || !(y <= 1e-20)) {
		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 <= (-330.0d0)) .or. (.not. (y <= 1d-20))) 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 <= -330.0) || !(y <= 1e-20)) {
		tmp = x + (Math.exp(-z) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -330.0) or not (y <= 1e-20):
		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 <= -330.0) || !(y <= 1e-20))
		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 <= -330.0) || ~((y <= 1e-20)))
		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, -330.0], N[Not[LessEqual[y, 1e-20]], $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 < -330 or 9.99999999999999945e-21 < y

   1. Initial program 82.3%

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

      1. *-commutative 82.3%

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

      2. exp-to-pow 82.3%

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

      3. +-commutative 82.3%

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

   3. Simplified 82.3%

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
   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 -330 < y < 9.99999999999999945e-21

   1. Initial program 83.2%

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

      1. exp-prod 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -330 \lor \neg \left(y \leq 10^{-20}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -52:\\ \;\;\;\;\frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -52.0) (/ (exp (- z)) y) (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -52.0) {
		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 <= (-52.0d0)) 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 <= -52.0) {
		tmp = Math.exp(-z) / y;
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -52.0:
		tmp = math.exp(-z) / y
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -52.0)
		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 <= -52.0)
		tmp = exp(-z) / y;
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -52.0], 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 < -52

   1. Initial program 47.7%

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

      1. *-commutative 47.7%

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

      2. exp-to-pow 47.7%

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

      3. +-commutative 47.7%

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

   3. Simplified 47.7%

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

   5. Taylor expanded in y around inf 65.0%

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

      1. mul-1-neg 65.0%

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

   7. Simplified 65.0%

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

   8. Taylor expanded in x around 0 65.0%

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

   if -52 < z

   1. Initial program 92.6%

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

      1. exp-prod 96.1%

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

      2. +-commutative 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in y around inf 95.7%

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

      1. +-commutative 95.7%

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

   7. Simplified 95.7%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -52:\\ \;\;\;\;\frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.1% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -330:\\ \;\;\;\;x + \left(\frac{1}{y} + z \cdot \left(z \cdot \left(-0.16666666666666666 \cdot \frac{z}{y} + \frac{1}{y} \cdot 0.5\right) + \frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 10^{+217}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 + z \cdot \left(\frac{0.5 \cdot \left(y \cdot z\right)}{y} + -1\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -330.0)
   (+
    x
    (+
     (/ 1.0 y)
     (*
      z
      (+
       (* z (+ (* -0.16666666666666666 (/ z y)) (* (/ 1.0 y) 0.5)))
       (/ -1.0 y)))))
   (if (<= y 1e+217)
     (+ x (/ 1.0 y))
     (+ x (/ (+ 1.0 (* z (+ (/ (* 0.5 (* y z)) y) -1.0))) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -330.0) {
		tmp = x + ((1.0 / y) + (z * ((z * ((-0.16666666666666666 * (z / y)) + ((1.0 / y) * 0.5))) + (-1.0 / y))));
	} else if (y <= 1e+217) {
		tmp = x + (1.0 / y);
	} else {
		tmp = x + ((1.0 + (z * (((0.5 * (y * z)) / y) + -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 <= (-330.0d0)) then
        tmp = x + ((1.0d0 / y) + (z * ((z * (((-0.16666666666666666d0) * (z / y)) + ((1.0d0 / y) * 0.5d0))) + ((-1.0d0) / y))))
    else if (y <= 1d+217) then
        tmp = x + (1.0d0 / y)
    else
        tmp = x + ((1.0d0 + (z * (((0.5d0 * (y * z)) / y) + (-1.0d0)))) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -330.0) {
		tmp = x + ((1.0 / y) + (z * ((z * ((-0.16666666666666666 * (z / y)) + ((1.0 / y) * 0.5))) + (-1.0 / y))));
	} else if (y <= 1e+217) {
		tmp = x + (1.0 / y);
	} else {
		tmp = x + ((1.0 + (z * (((0.5 * (y * z)) / y) + -1.0))) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -330.0:
		tmp = x + ((1.0 / y) + (z * ((z * ((-0.16666666666666666 * (z / y)) + ((1.0 / y) * 0.5))) + (-1.0 / y))))
	elif y <= 1e+217:
		tmp = x + (1.0 / y)
	else:
		tmp = x + ((1.0 + (z * (((0.5 * (y * z)) / y) + -1.0))) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -330.0)
		tmp = Float64(x + Float64(Float64(1.0 / y) + Float64(z * Float64(Float64(z * Float64(Float64(-0.16666666666666666 * Float64(z / y)) + Float64(Float64(1.0 / y) * 0.5))) + Float64(-1.0 / y)))));
	elseif (y <= 1e+217)
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = Float64(x + Float64(Float64(1.0 + Float64(z * Float64(Float64(Float64(0.5 * Float64(y * z)) / y) + -1.0))) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -330.0)
		tmp = x + ((1.0 / y) + (z * ((z * ((-0.16666666666666666 * (z / y)) + ((1.0 / y) * 0.5))) + (-1.0 / y))));
	elseif (y <= 1e+217)
		tmp = x + (1.0 / y);
	else
		tmp = x + ((1.0 + (z * (((0.5 * (y * z)) / y) + -1.0))) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -330.0], N[(x + N[(N[(1.0 / y), $MachinePrecision] + N[(z * N[(N[(z * N[(N[(-0.16666666666666666 * N[(z / y), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / y), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+217], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(1.0 + N[(z * N[(N[(N[(0.5 * N[(y * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -330

   1. Initial program 88.5%

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

      1. *-commutative 88.5%

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

      2. exp-to-pow 88.5%

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

      3. +-commutative 88.5%

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

   3. Simplified 88.5%

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

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

   8. Taylor expanded in z around 0 83.9%

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

   if -330 < y < 9.9999999999999996e216

   1. Initial program 83.1%

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

      1. exp-prod 93.4%

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

      2. +-commutative 93.4%

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

   3. Simplified 93.4%

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

   5. Taylor expanded in y around inf 92.1%

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

      1. +-commutative 92.1%

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

   7. Simplified 92.1%

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

   if 9.9999999999999996e216 < y

   1. Initial program 67.7%

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

      1. exp-prod 67.7%

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

      2. +-commutative 67.7%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in z around 0 71.7%

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

   6. Taylor expanded in y around 0 83.1%

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

      1. distribute-lft-out 83.1%

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

   8. Simplified 83.1%

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

   9. Taylor expanded in y around inf 83.1%

      \[\leadsto x + \frac{1 + z \cdot \left(\frac{0.5 \cdot \color{blue}{\left(y \cdot z\right)}}{y} - 1\right)}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -330:\\ \;\;\;\;x + \left(\frac{1}{y} + z \cdot \left(z \cdot \left(-0.16666666666666666 \cdot \frac{z}{y} + \frac{1}{y} \cdot 0.5\right) + \frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 10^{+217}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 + z \cdot \left(\frac{0.5 \cdot \left(y \cdot z\right)}{y} + -1\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.4% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -330:\\ \;\;\;\;x + \frac{1 + z \cdot \left(z \cdot 0.5 + -1\right)}{y}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+218}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 + z \cdot \left(\frac{0.5 \cdot \left(y \cdot z\right)}{y} + -1\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -330.0)
   (+ x (/ (+ 1.0 (* z (+ (* z 0.5) -1.0))) y))
   (if (<= y 8.5e+218)
     (+ x (/ 1.0 y))
     (+ x (/ (+ 1.0 (* z (+ (/ (* 0.5 (* y z)) y) -1.0))) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -330.0) {
		tmp = x + ((1.0 + (z * ((z * 0.5) + -1.0))) / y);
	} else if (y <= 8.5e+218) {
		tmp = x + (1.0 / y);
	} else {
		tmp = x + ((1.0 + (z * (((0.5 * (y * z)) / y) + -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 <= (-330.0d0)) then
        tmp = x + ((1.0d0 + (z * ((z * 0.5d0) + (-1.0d0)))) / y)
    else if (y <= 8.5d+218) then
        tmp = x + (1.0d0 / y)
    else
        tmp = x + ((1.0d0 + (z * (((0.5d0 * (y * z)) / y) + (-1.0d0)))) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -330.0) {
		tmp = x + ((1.0 + (z * ((z * 0.5) + -1.0))) / y);
	} else if (y <= 8.5e+218) {
		tmp = x + (1.0 / y);
	} else {
		tmp = x + ((1.0 + (z * (((0.5 * (y * z)) / y) + -1.0))) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -330.0:
		tmp = x + ((1.0 + (z * ((z * 0.5) + -1.0))) / y)
	elif y <= 8.5e+218:
		tmp = x + (1.0 / y)
	else:
		tmp = x + ((1.0 + (z * (((0.5 * (y * z)) / y) + -1.0))) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -330.0)
		tmp = Float64(x + Float64(Float64(1.0 + Float64(z * Float64(Float64(z * 0.5) + -1.0))) / y));
	elseif (y <= 8.5e+218)
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = Float64(x + Float64(Float64(1.0 + Float64(z * Float64(Float64(Float64(0.5 * Float64(y * z)) / y) + -1.0))) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -330.0)
		tmp = x + ((1.0 + (z * ((z * 0.5) + -1.0))) / y);
	elseif (y <= 8.5e+218)
		tmp = x + (1.0 / y);
	else
		tmp = x + ((1.0 + (z * (((0.5 * (y * z)) / y) + -1.0))) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -330.0], N[(x + N[(N[(1.0 + N[(z * N[(N[(z * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.5e+218], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(1.0 + N[(z * N[(N[(N[(0.5 * N[(y * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -330

   1. Initial program 88.5%

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

      1. exp-prod 88.5%

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

      2. +-commutative 88.5%

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

   3. Simplified 88.5%

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

   5. Taylor expanded in z around 0 82.5%

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

   6. Taylor expanded in y around inf 82.5%

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

      1. *-commutative 82.5%

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

   8. Simplified 82.5%

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

   if -330 < y < 8.50000000000000041e218

   1. Initial program 83.1%

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

      1. exp-prod 93.4%

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

      2. +-commutative 93.4%

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

   3. Simplified 93.4%

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

   5. Taylor expanded in y around inf 92.1%

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

      1. +-commutative 92.1%

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

   7. Simplified 92.1%

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

   if 8.50000000000000041e218 < y

   1. Initial program 67.7%

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

      1. exp-prod 67.7%

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

      2. +-commutative 67.7%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in z around 0 71.7%

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

   6. Taylor expanded in y around 0 83.1%

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

      1. distribute-lft-out 83.1%

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

   8. Simplified 83.1%

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

   9. Taylor expanded in y around inf 83.1%

      \[\leadsto x + \frac{1 + z \cdot \left(\frac{0.5 \cdot \color{blue}{\left(y \cdot z\right)}}{y} - 1\right)}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -330:\\ \;\;\;\;x + \frac{1 + z \cdot \left(z \cdot 0.5 + -1\right)}{y}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+218}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 + z \cdot \left(\frac{0.5 \cdot \left(y \cdot z\right)}{y} + -1\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.4% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -330:\\ \;\;\;\;x + \frac{1 + z \cdot \left(z \cdot 0.5 + -1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -330.0)
   (+ x (/ (+ 1.0 (* z (+ (* z 0.5) -1.0))) y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -330.0) {
		tmp = x + ((1.0 + (z * ((z * 0.5) + -1.0))) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-330.0d0)) then
        tmp = x + ((1.0d0 + (z * ((z * 0.5d0) + (-1.0d0)))) / y)
    else
        tmp = x + (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -330.0) {
		tmp = x + ((1.0 + (z * ((z * 0.5) + -1.0))) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -330.0:
		tmp = x + ((1.0 + (z * ((z * 0.5) + -1.0))) / y)
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -330.0)
		tmp = Float64(x + Float64(Float64(1.0 + Float64(z * Float64(Float64(z * 0.5) + -1.0))) / y));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -330.0)
		tmp = x + ((1.0 + (z * ((z * 0.5) + -1.0))) / y);
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -330.0], N[(x + N[(N[(1.0 + N[(z * N[(N[(z * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -330

   1. Initial program 88.5%

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

      1. exp-prod 88.5%

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

      2. +-commutative 88.5%

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

   3. Simplified 88.5%

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

   5. Taylor expanded in z around 0 82.5%

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

   6. Taylor expanded in y around inf 82.5%

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

      1. *-commutative 82.5%

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

   8. Simplified 82.5%

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

   if -330 < y

   1. Initial program 80.3%

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

      1. exp-prod 88.6%

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

      2. +-commutative 88.6%

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

   3. Simplified 88.6%

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

   5. Taylor expanded in y around inf 87.6%

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

      1. +-commutative 87.6%

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

   7. Simplified 87.6%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -330:\\ \;\;\;\;x + \frac{1 + z \cdot \left(z \cdot 0.5 + -1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.9% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-16}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+26}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -6e-16) x (if (<= y 8e+26) (/ 1.0 y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -6e-16) {
		tmp = x;
	} else if (y <= 8e+26) {
		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 <= (-6d-16)) then
        tmp = x
    else if (y <= 8d+26) 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 <= -6e-16) {
		tmp = x;
	} else if (y <= 8e+26) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -6e-16:
		tmp = x
	elif y <= 8e+26:
		tmp = 1.0 / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -6e-16)
		tmp = x;
	elseif (y <= 8e+26)
		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 <= -6e-16)
		tmp = x;
	elseif (y <= 8e+26)
		tmp = 1.0 / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -6e-16], x, If[LessEqual[y, 8e+26], N[(1.0 / y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -5.99999999999999987e-16 or 8.00000000000000038e26 < y

   1. Initial program 81.2%

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

      1. exp-prod 81.2%

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

      2. +-commutative 81.2%

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

   3. Simplified 81.2%

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

   5. Taylor expanded in x around inf 64.2%

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

   if -5.99999999999999987e-16 < y < 8.00000000000000038e26

   1. Initial program 84.8%

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

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 76.2%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-16}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+26}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 82.6%

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

   1. exp-prod 88.6%

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

   2. +-commutative 88.6%

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

3. Simplified 88.6%

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

5. Taylor expanded in y around inf 82.9%

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

   1. +-commutative 82.9%

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

7. Simplified 82.9%

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

8. Final simplification 82.9%

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

Alternative 9: 49.7% accurate, 211.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 x)

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x
end function

Java

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

Python

def code(x, y, z):
	return x

Julia

function code(x, y, z)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 82.6%

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

   1. exp-prod 88.6%

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

   2. +-commutative 88.6%

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

3. Simplified 88.6%

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

5. Taylor expanded in x around inf 48.1%

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

6. Final simplification 48.1%

   \[\leadsto x \]
7. Add Preprocessing

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 2024059 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G"
  :precision binary64

  :alt
  (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)))