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

Percentage Accurate: 84.8% → 99.6%

Time: 10.1s

Alternatives: 6

Speedup: 1.8×

• could not determine a ground truth

  1. x = -8.347332939592268e-59
  2. y = -1.588896478752496e+38
  3. z = -9.060649122713745e-210

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: 99.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \lor \neg \left(y \leq 10^{-11}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.5) (not (<= y 1e-11)))
   (+ x (/ (exp (- z)) y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.5) || !(y <= 1e-11)) {
		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 <= (-1.5d0)) .or. (.not. (y <= 1d-11))) 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 <= -1.5) || !(y <= 1e-11)) {
		tmp = x + (Math.exp(-z) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.5) or not (y <= 1e-11):
		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 <= -1.5) || !(y <= 1e-11))
		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 <= -1.5) || ~((y <= 1e-11)))
		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, -1.5], N[Not[LessEqual[y, 1e-11]], $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 < -1.5 or 9.99999999999999939e-12 < y

   1. Initial program 89.8%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.5 < y < 9.99999999999999939e-12

   1. Initial program 84.8%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 98.5%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \lor \neg \left(y \leq 10^{-11}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{-11}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + {\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{0.3333333333333333}{y \cdot y} + 0.16666666666666666\right) - \frac{0.5}{y}, z, 0.5\right) - \frac{0.5}{y}, z, 1\right), z, 1\right) \cdot y\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1e-11)
   (+ x (/ 1.0 y))
   (+
    x
    (pow
     (*
      (fma
       (fma
        (-
         (fma
          (- (+ (/ 0.3333333333333333 (* y y)) 0.16666666666666666) (/ 0.5 y))
          z
          0.5)
         (/ 0.5 y))
        z
        1.0)
       z
       1.0)
      y)
     -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e-11) {
		tmp = x + (1.0 / y);
	} else {
		tmp = x + pow((fma(fma((fma((((0.3333333333333333 / (y * y)) + 0.16666666666666666) - (0.5 / y)), z, 0.5) - (0.5 / y)), z, 1.0), z, 1.0) * y), -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1e-11)
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = Float64(x + (Float64(fma(fma(Float64(fma(Float64(Float64(Float64(0.3333333333333333 / Float64(y * y)) + 0.16666666666666666) - Float64(0.5 / y)), z, 0.5) - Float64(0.5 / y)), z, 1.0), z, 1.0) * y) ^ -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1e-11], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(x + N[Power[N[(N[(N[(N[(N[(N[(N[(N[(0.3333333333333333 / N[(y * y), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] - N[(0.5 / y), $MachinePrecision]), $MachinePrecision] * z + 0.5), $MachinePrecision] - N[(0.5 / y), $MachinePrecision]), $MachinePrecision] * z + 1.0), $MachinePrecision] * z + 1.0), $MachinePrecision] * y), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.99999999999999939e-12

   1. Initial program 85.5%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 91.0%

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

   if 9.99999999999999939e-12 < y

   1. Initial program 92.4%

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

      1. lift-/.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh---cosh-rev N/A

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

      6. associate-/l/ N/A

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

   4. Applied rewrites 92.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-neg.f64 92.4

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

   6. Applied rewrites 92.4%

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

   7. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   9. Applied rewrites 92.3%

      \[\leadsto x + \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{0.3333333333333333}{y \cdot y} + 0.16666666666666666\right) - \frac{0.5}{y}, z, 0.5\right) - \frac{0.5}{y}, z, 1\right), z, 1\right)} \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-11}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + {\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{0.3333333333333333}{y \cdot y} + 0.16666666666666666\right) - \frac{0.5}{y}, z, 0.5\right) - \frac{0.5}{y}, z, 1\right), z, 1\right) \cdot y\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{-11}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + {\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5 - \frac{0.5}{y}, z, 1\right), z, 1\right) \cdot y\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1e-11)
   (+ x (/ 1.0 y))
   (+ x (pow (* (fma (fma (- 0.5 (/ 0.5 y)) z 1.0) z 1.0) y) -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e-11) {
		tmp = x + (1.0 / y);
	} else {
		tmp = x + pow((fma(fma((0.5 - (0.5 / y)), z, 1.0), z, 1.0) * y), -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1e-11)
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = Float64(x + (Float64(fma(fma(Float64(0.5 - Float64(0.5 / y)), z, 1.0), z, 1.0) * y) ^ -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1e-11], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(x + N[Power[N[(N[(N[(N[(0.5 - N[(0.5 / y), $MachinePrecision]), $MachinePrecision] * z + 1.0), $MachinePrecision] * z + 1.0), $MachinePrecision] * y), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.99999999999999939e-12

   1. Initial program 85.5%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 91.0%

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

   if 9.99999999999999939e-12 < y

   1. Initial program 92.4%

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

      1. lift-/.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh---cosh-rev N/A

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

      6. associate-/l/ N/A

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

   4. Applied rewrites 92.4%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 91.8

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

   7. Applied rewrites 91.8%

      \[\leadsto x + \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 - \frac{0.5}{y}, z, 1\right), z, 1\right)} \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-11}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + {\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5 - \frac{0.5}{y}, z, 1\right), z, 1\right) \cdot y\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 39.4% 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 87.6%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 43.7

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

5. Applied rewrites 43.7%

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

6. Final simplification 43.7%

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

Alternative 5: 84.6% 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 87.6%

   \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
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 88.8%

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

Alternative 6: 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 87.6%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 43.7

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

5. Applied rewrites 43.7%

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

7. Applied rewrites 15.9%

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

8. Taylor expanded in y around -inf

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

10. Applied rewrites 2.2%

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

Developer Target 1: 91.5% 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 2024338 
(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)))