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

Percentage Accurate: 84.8% → 99.3%

Time: 9.3s

Alternatives: 7

Speedup: 7.1×

• could not determine a ground truth

  1. x = 3.972109299407219e-174
  2. y = 2.2176794332760605e+37
  3. z = -1.860151623890355e-37

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: 99.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-z}\\ \mathbf{if}\;y \leq -1550:\\ \;\;\;\;x + \frac{t\_0}{y}\\ \mathbf{elif}\;y \leq 10^{-31}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-t\_0, \frac{-1}{y}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (exp (- z))))
   (if (<= y -1550.0)
     (+ x (/ t_0 y))
     (if (<= y 1e-31) (+ x (/ 1.0 y)) (fma (- t_0) (/ -1.0 y) x)))))

C

double code(double x, double y, double z) {
	double t_0 = exp(-z);
	double tmp;
	if (y <= -1550.0) {
		tmp = x + (t_0 / y);
	} else if (y <= 1e-31) {
		tmp = x + (1.0 / y);
	} else {
		tmp = fma(-t_0, (-1.0 / y), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = exp(Float64(-z))
	tmp = 0.0
	if (y <= -1550.0)
		tmp = Float64(x + Float64(t_0 / y));
	elseif (y <= 1e-31)
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = fma(Float64(-t_0), Float64(-1.0 / y), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Exp[(-z)], $MachinePrecision]}, If[LessEqual[y, -1550.0], N[(x + N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e-31], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[((-t$95$0) * N[(-1.0 / y), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1550

   1. Initial program 88.1%

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

   3. Taylor expanded in y around inf

      \[\leadsto x + \frac{e^{\color{blue}{-1 \cdot z}}}{y} \]
   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 -1550 < y < 1e-31

   1. Initial program 90.2%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.4%

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

   if 1e-31 < y

   1. Initial program 85.1%

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

   3. Taylor expanded in y around inf

      \[\leadsto x + \frac{e^{\color{blue}{-1 \cdot z}}}{y} \]
   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 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lift-/.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 99.3

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

   7. Applied rewrites 99.3%

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

Alternative 2: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1550 \lor \neg \left(y \leq 10^{-31}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1550.0) (not (<= y 1e-31)))
   (+ x (/ (exp (- z)) y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1550.0) || !(y <= 1e-31)) {
		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 <= (-1550.0d0)) .or. (.not. (y <= 1d-31))) 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 <= -1550.0) || !(y <= 1e-31)) {
		tmp = x + (Math.exp(-z) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1550.0) or not (y <= 1e-31):
		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 <= -1550.0) || !(y <= 1e-31))
		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 <= -1550.0) || ~((y <= 1e-31)))
		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, -1550.0], N[Not[LessEqual[y, 1e-31]], $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 < -1550 or 1e-31 < y

   1. Initial program 86.5%

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

   3. Taylor expanded in y around 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 99.6

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

   5. Applied rewrites 99.6%

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

   if -1550 < y < 1e-31

   1. Initial program 90.2%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1550 \lor \neg \left(y \leq 10^{-31}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 39.5% 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 88.2%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 41.4

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

5. Applied rewrites 41.4%

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

6. Final simplification 41.4%

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

Alternative 4: 87.9% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1550:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, z, -1\right), y, 0.5 \cdot z\right)}{y}, z, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1550.0)
   (+ x (/ (fma (/ (fma (fma 0.5 z -1.0) y (* 0.5 z)) y) z 1.0) y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1550.0) {
		tmp = x + (fma((fma(fma(0.5, z, -1.0), y, (0.5 * z)) / y), z, 1.0) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1550.0)
		tmp = Float64(x + Float64(fma(Float64(fma(fma(0.5, z, -1.0), y, Float64(0.5 * z)) / y), z, 1.0) / y));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1550.0], N[(x + N[(N[(N[(N[(N[(0.5 * z + -1.0), $MachinePrecision] * y + N[(0.5 * z), $MachinePrecision]), $MachinePrecision] / y), $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 < -1550

   1. Initial program 88.1%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-log.f64 N/A

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

      5. exp-to-pow N/A

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

      6. sqr-pow N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 88.1

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

   4. Applied rewrites 88.1%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 82.2

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

   7. Applied rewrites 82.2%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 85.2%

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

   if -1550 < y

   1. Initial program 88.2%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 92.3%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1550:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, z, -1\right), y, 0.5 \cdot z\right)}{y}, z, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.6% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1550:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, z, -1\right), z, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1550.0) (+ x (/ (fma (fma 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 <= -1550.0) {
		tmp = x + (fma(fma(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 <= -1550.0)
		tmp = Float64(x + Float64(fma(fma(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, -1550.0], N[(x + N[(N[(N[(0.5 * z + -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 < -1550

   1. Initial program 88.1%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-log.f64 N/A

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

      5. exp-to-pow N/A

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

      6. sqr-pow N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 88.1

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

   4. Applied rewrites 88.1%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 82.2

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

   7. Applied rewrites 82.2%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 82.2%

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

   if -1550 < y

   1. Initial program 88.2%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 92.3%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1550:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, z, -1\right), z, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.7% 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 88.2%

   \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
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 86.3%

   \[\leadsto x + \frac{\color{blue}{1}}{y} \]
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 88.2%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 41.4

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

5. Applied rewrites 41.4%

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

7. Applied rewrites 12.1%

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

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 2024318 
(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)))