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

Percentage Accurate: 84.0% → 98.6%

Time: 9.3s

Alternatives: 7

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: 84.0% 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: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+149} \lor \neg \left(y \leq 5 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{e^{-z}}{y} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{y + z}\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2e+149) (not (<= y 5e-12)))
   (+ (/ (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 <= -2e+149) || !(y <= 5e-12)) {
		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 <= (-2d+149)) .or. (.not. (y <= 5d-12))) 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 <= -2e+149) || !(y <= 5e-12)) {
		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 <= -2e+149) or not (y <= 5e-12):
		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 <= -2e+149) || !(y <= 5e-12))
		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 <= -2e+149) || ~((y <= 5e-12)))
		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, -2e+149], N[Not[LessEqual[y, 5e-12]], $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 < -2.0000000000000001e149 or 4.9999999999999997e-12 < y

   1. Initial program 88.6%

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

      1. *-commutative 88.6%

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

      2. exp-prod 88.6%

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

      3. rem-exp-log 88.6%

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

      4. +-commutative 88.6%

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

   3. Simplified 88.6%

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

   4. Taylor expanded in y around inf 100.0%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot z}}{y} + x} \]
   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 -2.0000000000000001e149 < y < 4.9999999999999997e-12

   1. Initial program 88.3%

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

      1. exp-prod 99.9%

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

      2. sqr-pow 99.9%

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

      3. sqr-pow 99.9%

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

      4. +-commutative 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+149} \lor \neg \left(y \leq 5 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{e^{-z}}{y} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{y + z}\right)}}{y}\\ \end{array} \]

Alternative 2: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7200 \lor \neg \left(y \leq 1.02 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{e^{-z}}{y} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -7200.0) (not (<= y 1.02e-11)))
   (+ (/ (exp (- z)) y) x)
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7200.0) || !(y <= 1.02e-11)) {
		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 <= (-7200.0d0)) .or. (.not. (y <= 1.02d-11))) 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 <= -7200.0) || !(y <= 1.02e-11)) {
		tmp = (Math.exp(-z) / y) + x;
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -7200.0) or not (y <= 1.02e-11):
		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 <= -7200.0) || !(y <= 1.02e-11))
		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 <= -7200.0) || ~((y <= 1.02e-11)))
		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, -7200.0], N[Not[LessEqual[y, 1.02e-11]], $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 < -7200 or 1.01999999999999994e-11 < y

   1. Initial program 90.2%

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

      1. *-commutative 90.2%

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

      2. exp-prod 90.2%

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

      3. rem-exp-log 90.2%

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

      4. +-commutative 90.2%

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

   3. Simplified 90.2%

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

   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 -7200 < y < 1.01999999999999994e-11

   1. Initial program 86.1%

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

      1. *-commutative 86.1%

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

      2. exp-prod 86.1%

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

      3. rem-exp-log 86.1%

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

      4. +-commutative 86.1%

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

   3. Simplified 86.1%

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

   4. Taylor expanded in z around 0 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7200 \lor \neg \left(y \leq 1.02 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{e^{-z}}{y} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 3: 87.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.5e+18) (/ (exp (- z)) y) (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.5e+18) {
		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 <= (-1.5d+18)) 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 <= -1.5e+18) {
		tmp = Math.exp(-z) / y;
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.5e+18:
		tmp = math.exp(-z) / y
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.5e+18)
		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 <= -1.5e+18)
		tmp = exp(-z) / y;
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.5e+18], 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 < -1.5e18

   1. Initial program 56.3%

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

      1. *-commutative 56.3%

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

      2. exp-prod 56.3%

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

      3. rem-exp-log 56.3%

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

      4. +-commutative 56.3%

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

   3. Simplified 56.3%

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

   4. Taylor expanded in y around inf 68.0%

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

      1. mul-1-neg 68.0%

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

   6. Simplified 68.0%

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

   7. Taylor expanded in y around 0 68.0%

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

   if -1.5e18 < z

   1. Initial program 95.1%

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

      1. *-commutative 95.1%

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

      2. exp-prod 95.1%

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

      3. rem-exp-log 95.1%

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

      4. +-commutative 95.1%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in z around 0 98.2%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 4: 67.5% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-58}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-44}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+115}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+133}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.6e-58)
   x
   (if (<= y 1.7e-44)
     (/ 1.0 y)
     (if (<= y 2.4e+115) x (if (<= y 1.7e+133) (/ 1.0 y) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.6e-58) {
		tmp = x;
	} else if (y <= 1.7e-44) {
		tmp = 1.0 / y;
	} else if (y <= 2.4e+115) {
		tmp = x;
	} else if (y <= 1.7e+133) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.6d-58)) then
        tmp = x
    else if (y <= 1.7d-44) then
        tmp = 1.0d0 / y
    else if (y <= 2.4d+115) then
        tmp = x
    else if (y <= 1.7d+133) then
        tmp = 1.0d0 / y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.6e-58) {
		tmp = x;
	} else if (y <= 1.7e-44) {
		tmp = 1.0 / y;
	} else if (y <= 2.4e+115) {
		tmp = x;
	} else if (y <= 1.7e+133) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.6e-58:
		tmp = x
	elif y <= 1.7e-44:
		tmp = 1.0 / y
	elif y <= 2.4e+115:
		tmp = x
	elif y <= 1.7e+133:
		tmp = 1.0 / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.6e-58)
		tmp = x;
	elseif (y <= 1.7e-44)
		tmp = Float64(1.0 / y);
	elseif (y <= 2.4e+115)
		tmp = x;
	elseif (y <= 1.7e+133)
		tmp = Float64(1.0 / y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.6e-58)
		tmp = x;
	elseif (y <= 1.7e-44)
		tmp = 1.0 / y;
	elseif (y <= 2.4e+115)
		tmp = x;
	elseif (y <= 1.7e+133)
		tmp = 1.0 / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.6e-58], x, If[LessEqual[y, 1.7e-44], N[(1.0 / y), $MachinePrecision], If[LessEqual[y, 2.4e+115], x, If[LessEqual[y, 1.7e+133], N[(1.0 / y), $MachinePrecision], x]]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6e-58 or 1.70000000000000008e-44 < y < 2.4e115 or 1.69999999999999994e133 < y

   1. Initial program 91.1%

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

      1. *-commutative 91.1%

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

      2. exp-prod 91.1%

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

      3. rem-exp-log 91.1%

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

      4. +-commutative 91.1%

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

   3. Simplified 91.1%

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

   4. Taylor expanded in x around inf 65.8%

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

   if -1.6e-58 < y < 1.70000000000000008e-44 or 2.4e115 < y < 1.69999999999999994e133

   1. Initial program 84.0%

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

      1. *-commutative 84.0%

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

      2. exp-prod 84.0%

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

      3. rem-exp-log 84.0%

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

      4. +-commutative 84.0%

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

   3. Simplified 84.0%

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

   4. Taylor expanded in x around 0 71.5%

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

      1. exp-to-pow 71.5%

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

      2. +-commutative 71.5%

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

   6. Simplified 71.5%

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

   7. Taylor expanded in y around 0 85.4%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-58}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-44}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+115}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+133}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 85.2% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.35e+154) (* (* z z) (/ 0.5 y)) (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.35e+154) {
		tmp = (z * z) * (0.5 / 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 <= (-1.35d+154)) then
        tmp = (z * z) * (0.5d0 / 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 <= -1.35e+154) {
		tmp = (z * z) * (0.5 / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.35e+154:
		tmp = (z * z) * (0.5 / y)
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.35e+154)
		tmp = Float64(Float64(z * z) * Float64(0.5 / 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 <= -1.35e+154)
		tmp = (z * z) * (0.5 / y);
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.35e+154], N[(N[(z * z), $MachinePrecision] * N[(0.5 / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.35000000000000003e154

   1. Initial program 61.8%

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

      1. *-commutative 61.8%

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

      2. exp-prod 61.8%

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

      3. rem-exp-log 61.8%

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

      4. +-commutative 61.8%

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

   3. Simplified 61.8%

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

   4. Taylor expanded in x around 0 58.7%

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

      1. exp-to-pow 58.7%

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

      2. +-commutative 58.7%

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

   6. Simplified 58.7%

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

   7. Taylor expanded in z around 0 70.3%

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

      1. associate-+r+ 70.3%

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

      2. +-commutative 70.3%

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

      3. neg-mul-1 70.3%

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

      4. sub-neg 70.3%

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

      5. *-commutative 70.3%

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

      6. unpow2 70.3%

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

      7. +-commutative 70.3%

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

      8. associate-*r/ 70.3%

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

      9. metadata-eval 70.3%

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

   9. Simplified 70.3%

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

   10. Taylor expanded in z around inf 70.3%

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

       1. unpow2 70.3%

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

       2. associate-/l* 70.3%

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

       3. associate-*r/ 70.3%

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

       4. metadata-eval 70.3%

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

       5. +-commutative 70.3%

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

   12. Simplified 70.3%

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

   13. Taylor expanded in y around inf 71.9%

       \[\leadsto \color{blue}{0.5 \cdot \frac{{z}^{2}}{y}} \]
   14. Step-by-step derivation

       1. associate-*r/ 71.9%

          \[\leadsto \color{blue}{\frac{0.5 \cdot {z}^{2}}{y}} \]

       2. *-commutative 71.9%

          \[\leadsto \frac{\color{blue}{{z}^{2} \cdot 0.5}}{y} \]

       3. associate-*r/ 71.9%

          \[\leadsto \color{blue}{{z}^{2} \cdot \frac{0.5}{y}} \]

       4. unpow2 71.9%

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

   15. Simplified 71.9%

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

   if -1.35000000000000003e154 < z

   1. Initial program 92.0%

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

      1. *-commutative 92.0%

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

      2. exp-prod 92.0%

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

      3. rem-exp-log 92.0%

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

      4. +-commutative 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in z around 0 94.8%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 6: 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 88.4%

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

   1. *-commutative 88.4%

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

   2. exp-prod 88.4%

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

   3. rem-exp-log 88.4%

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

   4. +-commutative 88.4%

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

3. Simplified 88.4%

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

4. Taylor expanded in z around 0 87.4%

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

5. Final simplification 87.4%

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

Alternative 7: 49.4% 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 88.4%

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

   1. *-commutative 88.4%

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

   2. exp-prod 88.4%

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

   3. rem-exp-log 88.4%

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

   4. +-commutative 88.4%

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

3. Simplified 88.4%

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

4. Taylor expanded in x around inf 47.0%

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

5. Final simplification 47.0%

   \[\leadsto x \]

Developer target: 90.8% 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 2023224 
(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)))