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

Percentage Accurate: 84.4% → 99.5%

Time: 9.1s

Alternatives: 7

Speedup: 5.8×

• could not determine a ground truth

  1. x = 5.1707024519232567e+303
  2. y = 5.183681306004391e+38
  3. z = -5.643233416198467e-141

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: 84.4% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2800000 \lor \neg \left(y \leq 10^{-7}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2800000.0) (not (<= y 1e-7)))
   (+ x (/ (exp (- z)) y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2800000.0) || !(y <= 1e-7)) {
		tmp = x + (exp(-z) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-2800000.0d0)) .or. (.not. (y <= 1d-7))) then
        tmp = x + (exp(-z) / y)
    else
        tmp = x + (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2800000.0) || !(y <= 1e-7)) {
		tmp = x + (Math.exp(-z) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2800000.0) or not (y <= 1e-7):
		tmp = x + (math.exp(-z) / y)
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2800000.0) || !(y <= 1e-7))
		tmp = Float64(x + Float64(exp(Float64(-z)) / y));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2800000.0) || ~((y <= 1e-7)))
		tmp = x + (exp(-z) / y);
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2800000.0], N[Not[LessEqual[y, 1e-7]], $MachinePrecision]], N[(x + N[(N[Exp[(-z)], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.8e6 or 9.9999999999999995e-8 < y

   1. Initial program 80.3%

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

   3. Taylor expanded in y around inf

      \[\leadsto x + \frac{e^{\color{blue}{-1 \cdot z}}}{y} \]
   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 -2.8e6 < y < 9.9999999999999995e-8

   1. Initial program 83.7%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 98.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2800000 \lor \neg \left(y \leq 10^{-7}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 38.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ {y}^{-1} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (pow y -1.0))

C

double code(double x, double y, double z) {
	return pow(y, -1.0);
}

Fortran

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

Java

public static double code(double x, double y, double z) {
	return Math.pow(y, -1.0);
}

Python

def code(x, y, z):
	return math.pow(y, -1.0)

Julia

function code(x, y, z)
	return y ^ -1.0
end

MATLAB

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

Wolfram

code[x_, y_, z_] := N[Power[y, -1.0], $MachinePrecision]

Derivation

1. Initial program 81.7%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 35.7

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

5. Applied rewrites 35.7%

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

6. Final simplification 35.7%

   \[\leadsto {y}^{-1} \]
7. Add Preprocessing

Alternative 3: 87.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2800000:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, z, 0.5\right) \cdot z - 1, z, 1\right)}{y}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+210}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot z - 1, z, 1\right), y, \left(z \cdot z\right) \cdot 0.5\right)}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2800000.0)
   (+ x (/ (fma (- (* (fma -0.16666666666666666 z 0.5) z) 1.0) z 1.0) y))
   (if (<= y 9e+210)
     (+ x (/ 1.0 y))
     (+ x (/ (/ (fma (fma (- (* 0.5 z) 1.0) z 1.0) y (* (* z z) 0.5)) y) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2800000.0) {
		tmp = x + (fma(((fma(-0.16666666666666666, z, 0.5) * z) - 1.0), z, 1.0) / y);
	} else if (y <= 9e+210) {
		tmp = x + (1.0 / y);
	} else {
		tmp = x + ((fma(fma(((0.5 * z) - 1.0), z, 1.0), y, ((z * z) * 0.5)) / y) / y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2800000.0)
		tmp = Float64(x + Float64(fma(Float64(Float64(fma(-0.16666666666666666, z, 0.5) * z) - 1.0), z, 1.0) / y));
	elseif (y <= 9e+210)
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = Float64(x + Float64(Float64(fma(fma(Float64(Float64(0.5 * z) - 1.0), z, 1.0), y, Float64(Float64(z * z) * 0.5)) / y) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2800000.0], N[(x + N[(N[(N[(N[(N[(-0.16666666666666666 * z + 0.5), $MachinePrecision] * z), $MachinePrecision] - 1.0), $MachinePrecision] * z + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9e+210], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(N[(N[(N[(0.5 * z), $MachinePrecision] - 1.0), $MachinePrecision] * z + 1.0), $MachinePrecision] * y + N[(N[(z * z), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.8e6

   1. Initial program 84.6%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 76.2%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 82.0%

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

   if -2.8e6 < y < 9.00000000000000007e210

   1. Initial program 86.6%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 94.5%

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

   if 9.00000000000000007e210 < y

   1. Initial program 54.3%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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. *-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 45.8%

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

   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 54.2%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 76.7%

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

Alternative 4: 87.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+26}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, z, 0.5\right) \cdot z - 1, z, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.2e+26)
   (+ x (/ (fma (- (* (fma -0.16666666666666666 z 0.5) z) 1.0) z 1.0) y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.2e+26) {
		tmp = x + (fma(((fma(-0.16666666666666666, z, 0.5) * z) - 1.0), z, 1.0) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7.2e+26)
		tmp = Float64(x + Float64(fma(Float64(Float64(fma(-0.16666666666666666, z, 0.5) * z) - 1.0), z, 1.0) / y));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7.2e+26], N[(x + N[(N[(N[(N[(N[(-0.16666666666666666 * z + 0.5), $MachinePrecision] * z), $MachinePrecision] - 1.0), $MachinePrecision] * z + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.20000000000000048e26

   1. Initial program 84.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 76.4%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 82.5%

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

   if -7.20000000000000048e26 < y

   1. Initial program 80.6%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 85.8%

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

Alternative 5: 87.0% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+26}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(\left(-0.16666666666666666 \cdot z\right) \cdot z - 1, z, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.2e+26)
   (+ x (/ (fma (- (* (* -0.16666666666666666 z) z) 1.0) z 1.0) y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.2e+26) {
		tmp = x + (fma((((-0.16666666666666666 * z) * z) - 1.0), z, 1.0) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7.2e+26)
		tmp = Float64(x + Float64(fma(Float64(Float64(Float64(-0.16666666666666666 * z) * z) - 1.0), z, 1.0) / y));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7.2e+26], N[(x + N[(N[(N[(N[(N[(-0.16666666666666666 * z), $MachinePrecision] * z), $MachinePrecision] - 1.0), $MachinePrecision] * z + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.20000000000000048e26

   1. Initial program 84.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 76.4%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 82.5%

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

   12. Taylor expanded in z around inf

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

   14. Applied rewrites 82.0%

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

   if -7.20000000000000048e26 < y

   1. Initial program 80.6%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 85.8%

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

Alternative 6: 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 81.7%

   \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
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 81.9%

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

Alternative 7: 2.3% 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 81.7%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 35.7

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

5. Applied rewrites 35.7%

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

7. Applied rewrites 17.8%

   \[\leadsto \left|\frac{-1}{y}\right| \]
8. Step-by-step derivation

9. Applied rewrites 2.3%

   \[\leadsto {\left(\frac{-1}{y}\right)}^{\color{blue}{1}} \]
10. Step-by-step derivation

11. Applied rewrites 2.3%

    \[\leadsto \frac{-1}{\color{blue}{y}} \]
12. 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 2024326 
(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)))