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

Percentage Accurate: 84.6% → 99.5%

Time: 11.4s

Alternatives: 8

Speedup: 9.7×

• could not determine a ground truth

  1. x = 1.4658850974392028e+296
  2. y = -2.3730256635799107e+39
  3. z = -1.6188731986877096e-183

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: 84.6% 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: 99.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{e^{-z}}{y}\\ \mathbf{if}\;y \leq -980:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-15}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ (exp (- z)) y))))
   (if (<= y -980.0) t_0 (if (<= y 3.6e-15) (+ x (/ 1.0 y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (exp(-z) / y);
	double tmp;
	if (y <= -980.0) {
		tmp = t_0;
	} else if (y <= 3.6e-15) {
		tmp = x + (1.0 / y);
	} 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 = x + (exp(-z) / y)
    if (y <= (-980.0d0)) then
        tmp = t_0
    else if (y <= 3.6d-15) then
        tmp = x + (1.0d0 / y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (Math.exp(-z) / y);
	double tmp;
	if (y <= -980.0) {
		tmp = t_0;
	} else if (y <= 3.6e-15) {
		tmp = x + (1.0 / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (math.exp(-z) / y)
	tmp = 0
	if y <= -980.0:
		tmp = t_0
	elif y <= 3.6e-15:
		tmp = x + (1.0 / y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(exp(Float64(-z)) / y))
	tmp = 0.0
	if (y <= -980.0)
		tmp = t_0;
	elseif (y <= 3.6e-15)
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (exp(-z) / y);
	tmp = 0.0;
	if (y <= -980.0)
		tmp = t_0;
	elseif (y <= 3.6e-15)
		tmp = x + (1.0 / y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(N[Exp[(-z)], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -980.0], t$95$0, If[LessEqual[y, 3.6e-15], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -980 or 3.6000000000000001e-15 < y

   1. Initial program 79.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-exp.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if -980 < y < 3.6000000000000001e-15

   1. Initial program 86.1%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -980:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-15}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -350:\\ \;\;\;\;\frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -350.0) (/ (exp (- z)) y) (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -350.0) {
		tmp = exp(-z) / y;
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-350.0d0)) then
        tmp = exp(-z) / y
    else
        tmp = x + (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -350.0) {
		tmp = Math.exp(-z) / y;
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -350.0:
		tmp = math.exp(-z) / y
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -350.0)
		tmp = Float64(exp(Float64(-z)) / y);
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -350.0)
		tmp = exp(-z) / y;
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -350.0], N[(N[Exp[(-z)], $MachinePrecision] / y), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -350

   1. Initial program 38.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-exp.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 73.1

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{e^{\mathsf{neg}\left(z\right)}}{y}} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. neg-mul-1 N/A

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

      5. lower-neg.f64 73.1

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

   8. Applied rewrites 73.1%

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

   if -350 < z

   1. Initial program 91.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 95.5

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

   5. Applied rewrites 95.5%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -350:\\ \;\;\;\;\frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.3% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.75 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{x}{y}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.75e-10)
   (+ x (/ (fma z (fma z (fma z -0.16666666666666666 0.5) -1.0) 1.0) y))
   (+ x (/ (/ x y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.75e-10) {
		tmp = x + (fma(z, fma(z, fma(z, -0.16666666666666666, 0.5), -1.0), 1.0) / y);
	} else {
		tmp = x + ((x / y) / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.75e-10)
		tmp = Float64(x + Float64(fma(z, fma(z, fma(z, -0.16666666666666666, 0.5), -1.0), 1.0) / y));
	else
		tmp = Float64(x + Float64(Float64(x / y) / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.75e-10], N[(x + N[(N[(z * N[(z * N[(z * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(x / y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.7499999999999999e-10

   1. Initial program 86.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-exp.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 93.9

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 87.5

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

   8. Applied rewrites 87.5%

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

   if 1.7499999999999999e-10 < z

   1. Initial program 65.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 82.2

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

   5. Applied rewrites 82.2%

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

   6. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 67.1

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

   8. Applied rewrites 67.1%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 88.0

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

   10. Applied rewrites 88.0%

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

Alternative 4: 86.1% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+83}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.2e+83)
   (/ (fma z (fma z (fma z -0.16666666666666666 0.5) -1.0) 1.0) y)
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.2e+83) {
		tmp = fma(z, fma(z, fma(z, -0.16666666666666666, 0.5), -1.0), 1.0) / y;
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.2e+83)
		tmp = Float64(fma(z, fma(z, fma(z, -0.16666666666666666, 0.5), -1.0), 1.0) / y);
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.2e+83], N[(N[(z * N[(z * N[(z * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.1999999999999999e83

   1. Initial program 41.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 32.3

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

   5. Applied rewrites 32.3%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 59.2%

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

   9. Taylor expanded in y around inf

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 59.2

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

   11. Applied rewrites 59.2%

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

   if -3.1999999999999999e83 < z

   1. Initial program 88.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 91.8

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

   5. Applied rewrites 91.8%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+83}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.8% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -980:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.5, 1\right), -1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -980.0) (- x (/ (fma z (fma z -0.5 1.0) -1.0) y)) (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -980.0) {
		tmp = x - (fma(z, fma(z, -0.5, 1.0), -1.0) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -980.0)
		tmp = Float64(x - Float64(fma(z, fma(z, -0.5, 1.0), -1.0) / y));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -980.0], N[(x - N[(N[(z * N[(z * -0.5 + 1.0), $MachinePrecision] + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -980

   1. Initial program 80.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 70.5%

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

   6. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 75.4

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

   8. Applied rewrites 75.4%

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

   if -980 < y

   1. Initial program 83.1%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 89.6

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

   5. Applied rewrites 89.6%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -980:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.5, 1\right), -1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.6e+39) (/ (fma y x 1.0) y) (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.6e+39) {
		tmp = fma(y, x, 1.0) / y;
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.6e+39)
		tmp = Float64(fma(y, x, 1.0) / y);
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.6e+39], N[(N[(y * x + 1.0), $MachinePrecision] / y), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.59999999999999984e39

   1. Initial program 41.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 29.9

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

   5. Applied rewrites 29.9%

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

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 40.6

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

   8. Applied rewrites 40.6%

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

   if -3.59999999999999984e39 < z

   1. Initial program 90.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 94.2

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

   5. Applied rewrites 94.2%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.1% accurate, 15.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.3%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. lower-/.f64 83.4

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

5. Applied rewrites 83.4%

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

6. Final simplification 83.4%

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

Alternative 8: 40.2% 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 82.3%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 36.3

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

5. Applied rewrites 36.3%

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

Developer Target 1: 90.9% 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 2024212 
(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)))