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

Percentage Accurate: 84.8% → 99.7%

Time: 9.9s

Alternatives: 7

Speedup: 15.6×

• could not determine a ground truth

  1. x = 2.5283155188632537e-270
  2. y = -1.0964796674731153e+39
  3. z = -2.3871106470890048e-88

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: 99.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{\frac{1}{y}}{e^{z}}\\ \mathbf{if}\;y \leq -10:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.002:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ (/ 1.0 y) (exp z)))))
   (if (<= y -10.0) t_0 (if (<= y 0.002) (+ x (/ 1.0 y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + ((1.0 / y) / exp(z));
	double tmp;
	if (y <= -10.0) {
		tmp = t_0;
	} else if (y <= 0.002) {
		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 + ((1.0d0 / y) / exp(z))
    if (y <= (-10.0d0)) then
        tmp = t_0
    else if (y <= 0.002d0) 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 + ((1.0 / y) / Math.exp(z));
	double tmp;
	if (y <= -10.0) {
		tmp = t_0;
	} else if (y <= 0.002) {
		tmp = x + (1.0 / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + ((1.0 / y) / math.exp(z))
	tmp = 0
	if y <= -10.0:
		tmp = t_0
	elif y <= 0.002:
		tmp = x + (1.0 / y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(Float64(1.0 / y) / exp(z)))
	tmp = 0.0
	if (y <= -10.0)
		tmp = t_0;
	elseif (y <= 0.002)
		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 + ((1.0 / y) / exp(z));
	tmp = 0.0;
	if (y <= -10.0)
		tmp = t_0;
	elseif (y <= 0.002)
		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[(1.0 / y), $MachinePrecision] / N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -10.0], t$95$0, If[LessEqual[y, 0.002], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -10 or 2e-3 < y

   1. Initial program 91.4%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 99.9

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

   5. Simplified 99.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. exp-neg N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 N/A

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

      11. lower-exp.f64 99.9

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

   7. Applied egg-rr 99.9%

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

   if -10 < y < 2e-3

   1. Initial program 84.7%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 99.0

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

   5. Simplified 99.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -10:\\ \;\;\;\;x + \frac{\frac{1}{y}}{e^{z}}\\ \mathbf{elif}\;y \leq 0.002:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{1}{y}}{e^{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{e^{-z}}{y}\\ \mathbf{if}\;y \leq -10:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.002:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ (exp (- z)) y))))
   (if (<= y -10.0) t_0 (if (<= y 0.002) (+ 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 <= -10.0) {
		tmp = t_0;
	} else if (y <= 0.002) {
		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 <= (-10.0d0)) then
        tmp = t_0
    else if (y <= 0.002d0) 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 <= -10.0) {
		tmp = t_0;
	} else if (y <= 0.002) {
		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 <= -10.0:
		tmp = t_0
	elif y <= 0.002:
		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 <= -10.0)
		tmp = t_0;
	elseif (y <= 0.002)
		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 <= -10.0)
		tmp = t_0;
	elseif (y <= 0.002)
		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, -10.0], t$95$0, If[LessEqual[y, 0.002], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -10 or 2e-3 < y

   1. Initial program 91.4%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 99.9

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

   5. Simplified 99.9%

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

   if -10 < y < 2e-3

   1. Initial program 84.7%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 99.0

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

   5. Simplified 99.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -10:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{elif}\;y \leq 0.002:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.2% accurate, 6.0× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -10.0)
   (+ x (/ (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 (y <= -10.0) {
		tmp = x + (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 (y <= -10.0)
		tmp = Float64(x + 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[y, -10.0], 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[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -10

   1. Initial program 92.3%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 99.7

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

   5. Simplified 99.7%

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

   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 88.5

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

   8. Simplified 88.5%

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

   if -10 < 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 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 93.3

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

   5. Simplified 93.3%

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

4. Final simplification 92.1%

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

Alternative 4: 87.3% accurate, 7.1× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -10.0) (+ 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 <= -10.0) {
		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 <= -10.0)
		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, -10.0], 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 < -10

   1. Initial program 92.3%

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

   3. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. sub-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 83.6

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

   5. Simplified 83.6%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 88.3

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

   8. Simplified 88.3%

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

   if -10 < 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 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 93.3

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

   5. Simplified 93.3%

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

4. Final simplification 92.1%

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

Alternative 5: 64.6% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{y}\\ \mathbf{if}\;y \leq -3.7 \cdot 10^{+36}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 y))))
   (if (<= y -3.7e+36) t_0 (if (<= y 3.3e+14) (/ 1.0 y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / y);
	double tmp;
	if (y <= -3.7e+36) {
		tmp = t_0;
	} else if (y <= 3.3e+14) {
		tmp = 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 + ((-1.0d0) / y)
    if (y <= (-3.7d+36)) then
        tmp = t_0
    else if (y <= 3.3d+14) then
        tmp = 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 + (-1.0 / y);
	double tmp;
	if (y <= -3.7e+36) {
		tmp = t_0;
	} else if (y <= 3.3e+14) {
		tmp = 1.0 / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (-1.0 / y)
	tmp = 0
	if y <= -3.7e+36:
		tmp = t_0
	elif y <= 3.3e+14:
		tmp = 1.0 / y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / y))
	tmp = 0.0
	if (y <= -3.7e+36)
		tmp = t_0;
	elseif (y <= 3.3e+14)
		tmp = Float64(1.0 / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (-1.0 / y);
	tmp = 0.0;
	if (y <= -3.7e+36)
		tmp = t_0;
	elseif (y <= 3.3e+14)
		tmp = 1.0 / y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.7e+36], t$95$0, If[LessEqual[y, 3.3e+14], N[(1.0 / y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.70000000000000029e36 or 3.3e14 < y

   1. Initial program 90.8%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 82.9

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

   5. Simplified 82.9%

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

      1. inv-pow N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow-prod-down N/A

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

      7. lift-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval 76.7

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

   7. Applied egg-rr 76.7%

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

   8. Taylor expanded in y around -inf

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

      1. lower-/.f64 72.5

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

   10. Simplified 72.5%

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

   if -3.70000000000000029e36 < y < 3.3e14

   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 0

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

      1. lower-/.f64 69.5

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

   5. Simplified 69.5%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+36}:\\ \;\;\;\;x + \frac{-1}{y}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.8% 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 88.3%

   \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
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 89.1

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

5. Simplified 89.1%

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

6. Final simplification 89.1%

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

Alternative 7: 39.0% 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 88.3%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 42.2

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

5. Simplified 42.2%

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

Developer Target 1: 91.7% 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 2024207 
(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)))