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

Percentage Accurate: 84.8% → 99.3%

Time: 7.9s

Alternatives: 5

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: 84.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-z}\\ \mathbf{if}\;y \leq -2 \cdot 10^{+70}:\\ \;\;\;\;x + t_0 \cdot \frac{1}{y}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-34}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{y + z}\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t_0}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (exp (- z))))
   (if (<= y -2e+70)
     (+ x (* t_0 (/ 1.0 y)))
     (if (<= y 9e-34)
       (+ x (/ (pow (exp y) (log (/ y (+ y z)))) y))
       (+ x (/ t_0 y))))))

C

double code(double x, double y, double z) {
	double t_0 = exp(-z);
	double tmp;
	if (y <= -2e+70) {
		tmp = x + (t_0 * (1.0 / y));
	} else if (y <= 9e-34) {
		tmp = x + (pow(exp(y), log((y / (y + z)))) / y);
	} else {
		tmp = x + (t_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) :: t_0
    real(8) :: tmp
    t_0 = exp(-z)
    if (y <= (-2d+70)) then
        tmp = x + (t_0 * (1.0d0 / y))
    else if (y <= 9d-34) then
        tmp = x + ((exp(y) ** log((y / (y + z)))) / y)
    else
        tmp = x + (t_0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.exp(-z);
	double tmp;
	if (y <= -2e+70) {
		tmp = x + (t_0 * (1.0 / y));
	} else if (y <= 9e-34) {
		tmp = x + (Math.pow(Math.exp(y), Math.log((y / (y + z)))) / y);
	} else {
		tmp = x + (t_0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.exp(-z)
	tmp = 0
	if y <= -2e+70:
		tmp = x + (t_0 * (1.0 / y))
	elif y <= 9e-34:
		tmp = x + (math.pow(math.exp(y), math.log((y / (y + z)))) / y)
	else:
		tmp = x + (t_0 / y)
	return tmp

Julia

function code(x, y, z)
	t_0 = exp(Float64(-z))
	tmp = 0.0
	if (y <= -2e+70)
		tmp = Float64(x + Float64(t_0 * Float64(1.0 / y)));
	elseif (y <= 9e-34)
		tmp = Float64(x + Float64((exp(y) ^ log(Float64(y / Float64(y + z)))) / y));
	else
		tmp = Float64(x + Float64(t_0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = exp(-z);
	tmp = 0.0;
	if (y <= -2e+70)
		tmp = x + (t_0 * (1.0 / y));
	elseif (y <= 9e-34)
		tmp = x + ((exp(y) ^ log((y / (y + z)))) / y);
	else
		tmp = x + (t_0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Exp[(-z)], $MachinePrecision]}, If[LessEqual[y, -2e+70], N[(x + N[(t$95$0 * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9e-34], N[(x + N[(N[Power[N[Exp[y], $MachinePrecision], N[Log[N[(y / N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.00000000000000015e70

   1. Initial program 86.8%

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

      1. exp-prod 86.8%

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

      2. +-commutative 86.8%

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

   3. Simplified 86.8%

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

      1. clear-num 86.8%

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

      2. add-exp-log 0.0%

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

      3. add-exp-log 0.0%

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

      4. div-exp 0.0%

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

      5. pow-exp 0.0%

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

      6. add-log-exp 0.0%

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

      7. log-pow 0.0%

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

      8. div-exp 0.0%

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

      9. add-exp-log 86.8%

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

      10. add-exp-log 86.8%

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

      11. associate-/r/ 86.8%

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

      12. *-commutative 86.8%

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

   5. Applied egg-rr 86.8%

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

   6. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   8. Simplified 100.0%

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

   if -2.00000000000000015e70 < y < 9.00000000000000085e-34

   1. Initial program 84.6%

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

      1. exp-prod 99.9%

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

      2. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   if 9.00000000000000085e-34 < y

   1. Initial program 83.4%

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

      1. *-commutative 83.4%

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

      2. exp-prod 83.4%

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

      3. rem-exp-log 83.4%

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

      4. +-commutative 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. mul-1-neg 99.6%

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

   6. Simplified 99.6%

      \[\leadsto x + \frac{\color{blue}{e^{-z}}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+70}:\\ \;\;\;\;x + e^{-z} \cdot \frac{1}{y}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-34}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{y + z}\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array} \]

Alternative 2: 98.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+29} \lor \neg \left(y \leq 8 \cdot 10^{-34}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -9.5e+29) (not (<= y 8e-34)))
   (+ x (/ (exp (- z)) y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -9.5e+29) || !(y <= 8e-34)) {
		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 <= (-9.5d+29)) .or. (.not. (y <= 8d-34))) 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 <= -9.5e+29) || !(y <= 8e-34)) {
		tmp = x + (Math.exp(-z) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -9.5e+29) or not (y <= 8e-34):
		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 <= -9.5e+29) || !(y <= 8e-34))
		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 <= -9.5e+29) || ~((y <= 8e-34)))
		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, -9.5e+29], N[Not[LessEqual[y, 8e-34]], $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 < -9.5000000000000003e29 or 7.99999999999999942e-34 < y

   1. Initial program 85.3%

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

      1. *-commutative 85.3%

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

      2. exp-prod 85.3%

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

      3. rem-exp-log 85.3%

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

      4. +-commutative 85.3%

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

   3. Simplified 85.3%

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

   4. Taylor expanded in y around inf 99.8%

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

      1. mul-1-neg 99.8%

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

   6. Simplified 99.8%

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

   if -9.5000000000000003e29 < y < 7.99999999999999942e-34

   1. Initial program 84.2%

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

      1. exp-prod 99.9%

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

      2. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+29} \lor \neg \left(y \leq 8 \cdot 10^{-34}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 3: 98.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-z}\\ \mathbf{if}\;y \leq -9.5 \cdot 10^{+29}:\\ \;\;\;\;x + t_0 \cdot \frac{1}{y}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-34}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t_0}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (exp (- z))))
   (if (<= y -9.5e+29)
     (+ x (* t_0 (/ 1.0 y)))
     (if (<= y 5e-34) (+ x (/ 1.0 y)) (+ x (/ t_0 y))))))

C

double code(double x, double y, double z) {
	double t_0 = exp(-z);
	double tmp;
	if (y <= -9.5e+29) {
		tmp = x + (t_0 * (1.0 / y));
	} else if (y <= 5e-34) {
		tmp = x + (1.0 / y);
	} else {
		tmp = x + (t_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) :: t_0
    real(8) :: tmp
    t_0 = exp(-z)
    if (y <= (-9.5d+29)) then
        tmp = x + (t_0 * (1.0d0 / y))
    else if (y <= 5d-34) then
        tmp = x + (1.0d0 / y)
    else
        tmp = x + (t_0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.exp(-z);
	double tmp;
	if (y <= -9.5e+29) {
		tmp = x + (t_0 * (1.0 / y));
	} else if (y <= 5e-34) {
		tmp = x + (1.0 / y);
	} else {
		tmp = x + (t_0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.exp(-z)
	tmp = 0
	if y <= -9.5e+29:
		tmp = x + (t_0 * (1.0 / y))
	elif y <= 5e-34:
		tmp = x + (1.0 / y)
	else:
		tmp = x + (t_0 / y)
	return tmp

Julia

function code(x, y, z)
	t_0 = exp(Float64(-z))
	tmp = 0.0
	if (y <= -9.5e+29)
		tmp = Float64(x + Float64(t_0 * Float64(1.0 / y)));
	elseif (y <= 5e-34)
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = Float64(x + Float64(t_0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = exp(-z);
	tmp = 0.0;
	if (y <= -9.5e+29)
		tmp = x + (t_0 * (1.0 / y));
	elseif (y <= 5e-34)
		tmp = x + (1.0 / y);
	else
		tmp = x + (t_0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Exp[(-z)], $MachinePrecision]}, If[LessEqual[y, -9.5e+29], N[(x + N[(t$95$0 * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e-34], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.5000000000000003e29

   1. Initial program 87.3%

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

      1. exp-prod 87.3%

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

      2. +-commutative 87.3%

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

   3. Simplified 87.3%

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

      1. clear-num 87.3%

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

      2. add-exp-log 0.0%

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

      3. add-exp-log 0.0%

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

      4. div-exp 0.0%

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

      5. pow-exp 0.0%

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

      6. add-log-exp 0.0%

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

      7. log-pow 0.0%

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

      8. div-exp 0.0%

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

      9. add-exp-log 87.3%

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

      10. add-exp-log 87.3%

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

      11. associate-/r/ 87.3%

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

      12. *-commutative 87.3%

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

   5. Applied egg-rr 87.3%

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

   6. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   8. Simplified 100.0%

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

   if -9.5000000000000003e29 < y < 5.0000000000000003e-34

   1. Initial program 84.2%

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

      1. exp-prod 99.9%

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

      2. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 99.4%

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

   if 5.0000000000000003e-34 < y

   1. Initial program 83.4%

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

      1. *-commutative 83.4%

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

      2. exp-prod 83.4%

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

      3. rem-exp-log 83.4%

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

      4. +-commutative 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. mul-1-neg 99.6%

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

   6. Simplified 99.6%

      \[\leadsto x + \frac{\color{blue}{e^{-z}}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+29}:\\ \;\;\;\;x + e^{-z} \cdot \frac{1}{y}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-34}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array} \]

Alternative 4: 84.9% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

   1. exp-prod 90.9%

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

   2. +-commutative 90.9%

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

3. Simplified 90.9%

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

4. Taylor expanded in y around 0 83.0%

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

5. Final simplification 83.0%

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

Alternative 5: 50.1% accurate, 211.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 84.8%

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

   1. *-commutative 84.8%

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

   2. exp-prod 84.8%

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

   3. rem-exp-log 84.8%

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

   4. +-commutative 84.8%

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

3. Simplified 84.8%

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

4. Taylor expanded in y around inf 88.5%

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

   1. mul-1-neg 88.5%

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

6. Simplified 88.5%

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

7. Taylor expanded in x around inf 50.7%

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

8. Final simplification 50.7%

   \[\leadsto x \]

Developer target: 91.6% 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 2023336 
(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)))