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

Percentage Accurate: 85.2% → 98.0%

Time: 8.9s

Alternatives: 7

Speedup: 7.5×

• could not determine a ground truth

  1. x = 1.8480689376499765e-50
  2. y = -6.549770624947088e+40
  3. z = -5.4498386742801324e-108

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{-z}}{y} + x\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{+56}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ (exp (- z)) y) x)))
   (if (<= y -1.8e+56) t_0 (if (<= y 5.5e-16) (+ (/ 1.0 y) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (exp(-z) / y) + x;
	double tmp;
	if (y <= -1.8e+56) {
		tmp = t_0;
	} else if (y <= 5.5e-16) {
		tmp = (1.0 / y) + x;
	} else {
		tmp = t_0;
	}
	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) / y) + x
    if (y <= (-1.8d+56)) then
        tmp = t_0
    else if (y <= 5.5d-16) then
        tmp = (1.0d0 / y) + x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (Math.exp(-z) / y) + x;
	double tmp;
	if (y <= -1.8e+56) {
		tmp = t_0;
	} else if (y <= 5.5e-16) {
		tmp = (1.0 / y) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (math.exp(-z) / y) + x
	tmp = 0
	if y <= -1.8e+56:
		tmp = t_0
	elif y <= 5.5e-16:
		tmp = (1.0 / y) + x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(exp(Float64(-z)) / y) + x)
	tmp = 0.0
	if (y <= -1.8e+56)
		tmp = t_0;
	elseif (y <= 5.5e-16)
		tmp = Float64(Float64(1.0 / y) + x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (exp(-z) / y) + x;
	tmp = 0.0;
	if (y <= -1.8e+56)
		tmp = t_0;
	elseif (y <= 5.5e-16)
		tmp = (1.0 / y) + x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Exp[(-z)], $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -1.8e+56], t$95$0, If[LessEqual[y, 5.5e-16], N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.79999999999999999e56 or 5.49999999999999964e-16 < y

   1. Initial program 85.5%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.79999999999999999e56 < y < 5.49999999999999964e-16

   1. Initial program 87.9%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+56}:\\ \;\;\;\;\frac{e^{-z}}{y} + x\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-z}}{y} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.0% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+56}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, z, 0.5\right), z, -1\right), z, 1\right)}{y} + x\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, z, 1\right) \cdot y, z, y\right)} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.8e+56)
   (+ (/ (fma (fma (fma -0.16666666666666666 z 0.5) z -1.0) z 1.0) y) x)
   (if (<= y 5.5e-16)
     (+ (/ 1.0 y) x)
     (+ (/ 1.0 (fma (* (fma 0.5 z 1.0) y) z y)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.8e+56) {
		tmp = (fma(fma(fma(-0.16666666666666666, z, 0.5), z, -1.0), z, 1.0) / y) + x;
	} else if (y <= 5.5e-16) {
		tmp = (1.0 / y) + x;
	} else {
		tmp = (1.0 / fma((fma(0.5, z, 1.0) * y), z, y)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.8e+56)
		tmp = Float64(Float64(fma(fma(fma(-0.16666666666666666, z, 0.5), z, -1.0), z, 1.0) / y) + x);
	elseif (y <= 5.5e-16)
		tmp = Float64(Float64(1.0 / y) + x);
	else
		tmp = Float64(Float64(1.0 / fma(Float64(fma(0.5, z, 1.0) * y), z, y)) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.8e+56], N[(N[(N[(N[(N[(-0.16666666666666666 * z + 0.5), $MachinePrecision] * z + -1.0), $MachinePrecision] * z + 1.0), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 5.5e-16], N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision], N[(N[(1.0 / N[(N[(N[(0.5 * z + 1.0), $MachinePrecision] * y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.79999999999999999e56

   1. Initial program 84.9%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{\color{blue}{1 + z \cdot \left(z \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(z \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{y}^{2}} + \frac{1}{2} \cdot \frac{1}{y}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{y}\right)\right) - 1\right)}}{y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \frac{\color{blue}{z \cdot \left(z \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(z \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{y}^{2}} + \frac{1}{2} \cdot \frac{1}{y}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{y}\right)\right) - 1\right) + 1}}{y} \]

      2. *-commutative N/A

         \[\leadsto x + \frac{\color{blue}{\left(z \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(z \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{y}^{2}} + \frac{1}{2} \cdot \frac{1}{y}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{y}\right)\right) - 1\right) \cdot z} + 1}{y} \]

      3. lower-fma.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\mathsf{fma}\left(z \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(z \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{y}^{2}} + \frac{1}{2} \cdot \frac{1}{y}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{y}\right)\right) - 1, z, 1\right)}}{y} \]

   5. Applied rewrites 88.4%

      \[\leadsto x + \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(0.16666666666666666 + \frac{0.5}{y}\right) + \frac{0.3333333333333333}{y \cdot y}, -z, \frac{0.5}{y} + 0.5\right), z, -1\right), z, 1\right)}}{y} \]

   6. Taylor expanded in y around inf

      \[\leadsto x + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} + \frac{-1}{6} \cdot z, z, -1\right), z, 1\right)}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 88.4%

      \[\leadsto x + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, z, 0.5\right), z, -1\right), z, 1\right)}{y} \]

   if -1.79999999999999999e56 < y < 5.49999999999999964e-16

   1. Initial program 87.9%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.9%

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

   if 5.49999999999999964e-16 < y

   1. Initial program 85.9%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 75.6

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

   5. Applied rewrites 75.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 75.6

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

   7. Applied rewrites 75.6%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto x + \frac{1}{\color{blue}{\mathsf{fma}\left(-1 \cdot \left(z \cdot \left(-1 \cdot y + y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{y}\right)\right)\right) - -1 \cdot y, z, y\right)}} \]

   10. Applied rewrites 85.4%

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

   11. Taylor expanded in y around inf

       \[\leadsto x + \frac{1}{\mathsf{fma}\left(y \cdot \left(1 + \frac{1}{2} \cdot z\right), z, y\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 85.4%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+56}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, z, 0.5\right), z, -1\right), z, 1\right)}{y} + x\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, z, 1\right) \cdot y, z, y\right)} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.5% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+56}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, z, -1\right), z, 1\right)}{y} + x\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, z, 1\right) \cdot y, z, y\right)} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.9e+56)
   (+ (/ (fma (fma 0.5 z -1.0) z 1.0) y) x)
   (if (<= y 5.5e-16)
     (+ (/ 1.0 y) x)
     (+ (/ 1.0 (fma (* (fma 0.5 z 1.0) y) z y)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.9e+56) {
		tmp = (fma(fma(0.5, z, -1.0), z, 1.0) / y) + x;
	} else if (y <= 5.5e-16) {
		tmp = (1.0 / y) + x;
	} else {
		tmp = (1.0 / fma((fma(0.5, z, 1.0) * y), z, y)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.9e+56)
		tmp = Float64(Float64(fma(fma(0.5, z, -1.0), z, 1.0) / y) + x);
	elseif (y <= 5.5e-16)
		tmp = Float64(Float64(1.0 / y) + x);
	else
		tmp = Float64(Float64(1.0 / fma(Float64(fma(0.5, z, 1.0) * y), z, y)) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.9e+56], N[(N[(N[(N[(0.5 * z + -1.0), $MachinePrecision] * z + 1.0), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 5.5e-16], N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision], N[(N[(1.0 / N[(N[(N[(0.5 * z + 1.0), $MachinePrecision] * y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.89999999999999998e56

   1. Initial program 84.9%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 60.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 77.8%

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

   if -1.89999999999999998e56 < y < 5.49999999999999964e-16

   1. Initial program 87.9%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.9%

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

   if 5.49999999999999964e-16 < y

   1. Initial program 85.9%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 75.6

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

   5. Applied rewrites 75.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 75.6

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

   7. Applied rewrites 75.6%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto x + \frac{1}{\color{blue}{\mathsf{fma}\left(-1 \cdot \left(z \cdot \left(-1 \cdot y + y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{y}\right)\right)\right) - -1 \cdot y, z, y\right)}} \]

   10. Applied rewrites 85.4%

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

   11. Taylor expanded in y around inf

       \[\leadsto x + \frac{1}{\mathsf{fma}\left(y \cdot \left(1 + \frac{1}{2} \cdot z\right), z, y\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 85.4%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+56}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, z, -1\right), z, 1\right)}{y} + x\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, z, 1\right) \cdot y, z, y\right)} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.9% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+56}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, z, -1\right), z, 1\right)}{y} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.9e+56) (+ (/ (fma (fma 0.5 z -1.0) z 1.0) y) x) (+ (/ 1.0 y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.9e+56) {
		tmp = (fma(fma(0.5, z, -1.0), z, 1.0) / y) + x;
	} else {
		tmp = (1.0 / y) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.9e+56)
		tmp = Float64(Float64(fma(fma(0.5, z, -1.0), z, 1.0) / y) + x);
	else
		tmp = Float64(Float64(1.0 / y) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.9e+56], N[(N[(N[(N[(0.5 * z + -1.0), $MachinePrecision] * z + 1.0), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.89999999999999998e56

   1. Initial program 84.9%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 60.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 77.8%

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

   if -1.89999999999999998e56 < y

   1. Initial program 87.1%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 91.4%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+56}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, z, -1\right), z, 1\right)}{y} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.1% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+160}:\\ \;\;\;\;\frac{z \cdot z}{y} \cdot 0.5 + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.6e+160) (+ (* (/ (* z z) y) 0.5) x) (+ (/ 1.0 y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.6e+160) {
		tmp = (((z * z) / y) * 0.5) + x;
	} else {
		tmp = (1.0 / y) + 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 (z <= (-4.6d+160)) then
        tmp = (((z * z) / y) * 0.5d0) + x
    else
        tmp = (1.0d0 / y) + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.6e+160) {
		tmp = (((z * z) / y) * 0.5) + x;
	} else {
		tmp = (1.0 / y) + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4.6e+160:
		tmp = (((z * z) / y) * 0.5) + x
	else:
		tmp = (1.0 / y) + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.6e+160)
		tmp = Float64(Float64(Float64(Float64(z * z) / y) * 0.5) + x);
	else
		tmp = Float64(Float64(1.0 / y) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -4.6e+160)
		tmp = (((z * z) / y) * 0.5) + x;
	else
		tmp = (1.0 / y) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.6e+160], N[(N[(N[(N[(z * z), $MachinePrecision] / y), $MachinePrecision] * 0.5), $MachinePrecision] + x), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.59999999999999975e160

   1. Initial program 61.2%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 15.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 67.5%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 67.5%

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

   if -4.59999999999999975e160 < z

   1. Initial program 90.1%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 89.6%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+160}:\\ \;\;\;\;\frac{z \cdot z}{y} \cdot 0.5 + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.9% accurate, 15.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (1.0 / y) + 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 = (1.0d0 / y) + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.5%

   \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

5. Applied rewrites 82.9%

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

6. Final simplification 82.9%

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

Alternative 7: 39.7% accurate, 19.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 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 = 1.0d0 / y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.5%

   \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. lower-/.f64 40.5

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

5. Applied rewrites 40.5%

   \[\leadsto \color{blue}{\frac{1}{y}} \]
6. Add Preprocessing

Developer Target 1: 91.9% 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 2024294 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (/ y (+ z y)) 17788539399477/2500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ x (/ (exp (/ -1 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y))))

  (+ x (/ (exp (* y (log (/ y (+ z y))))) y)))