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

Percentage Accurate: 84.5% → 99.0%

Time: 9.8s

Alternatives: 8

Speedup: 7.1×

• could not determine a ground truth

  1. x = 9.383934602021285e+33
  2. y = 2.5899333466242246e+38
  3. z = -7.043155380243673e-112

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: 84.5% 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: 99.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{-z}}{y} + x\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{+27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.00185:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ (exp (- z)) y) x)))
   (if (<= y -1.8e+27) t_0 (if (<= y 0.00185) (+ (/ 1.0 y) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (exp(-z) / y) + x;
	double tmp;
	if (y <= -1.8e+27) {
		tmp = t_0;
	} else if (y <= 0.00185) {
		tmp = (1.0 / y) + x;
	} 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 = (exp(-z) / y) + x
    if (y <= (-1.8d+27)) then
        tmp = t_0
    else if (y <= 0.00185d0) then
        tmp = (1.0d0 / y) + x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (Math.exp(-z) / y) + x;
	double tmp;
	if (y <= -1.8e+27) {
		tmp = t_0;
	} else if (y <= 0.00185) {
		tmp = (1.0 / y) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (math.exp(-z) / y) + x
	tmp = 0
	if y <= -1.8e+27:
		tmp = t_0
	elif y <= 0.00185:
		tmp = (1.0 / y) + x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(exp(Float64(-z)) / y) + x)
	tmp = 0.0
	if (y <= -1.8e+27)
		tmp = t_0;
	elseif (y <= 0.00185)
		tmp = Float64(Float64(1.0 / y) + x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (exp(-z) / y) + x;
	tmp = 0.0;
	if (y <= -1.8e+27)
		tmp = t_0;
	elseif (y <= 0.00185)
		tmp = (1.0 / y) + x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Exp[(-z)], $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -1.8e+27], t$95$0, If[LessEqual[y, 0.00185], N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.79999999999999991e27 or 0.0018500000000000001 < y

   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

      \[\leadsto x + \frac{e^{\color{blue}{-1 \cdot z}}}{y} \]
   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 -1.79999999999999991e27 < y < 0.0018500000000000001

   1. Initial program 79.8%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 98.4%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{e^{-z}}{y} + x\\ \mathbf{elif}\;y \leq 0.00185:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-z}}{y} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -70000000:\\ \;\;\;\;\frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -70000000.0) (/ (exp (- z)) y) (+ (/ 1.0 y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -70000000.0) {
		tmp = exp(-z) / y;
	} else {
		tmp = (1.0 / y) + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-70000000.0d0)) then
        tmp = exp(-z) / y
    else
        tmp = (1.0d0 / y) + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -70000000.0) {
		tmp = Math.exp(-z) / y;
	} else {
		tmp = (1.0 / y) + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -70000000.0:
		tmp = math.exp(-z) / y
	else:
		tmp = (1.0 / y) + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -70000000.0)
		tmp = Float64(exp(Float64(-z)) / y);
	else
		tmp = Float64(Float64(1.0 / y) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -70000000.0)
		tmp = exp(-z) / y;
	else
		tmp = (1.0 / y) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -70000000.0], N[(N[Exp[(-z)], $MachinePrecision] / y), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7e7

   1. Initial program 45.4%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 38.0

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

   5. Applied rewrites 38.0%

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

   6. Taylor expanded in y around inf

      \[\leadsto \frac{e^{-1 \cdot z}}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.1%

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

   if -7e7 < z

   1. Initial program 92.2%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 95.5%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -70000000:\\ \;\;\;\;\frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{y} + x\\ \mathbf{if}\;y \leq -1.9 \cdot 10^{+193}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{\left(\left(z \cdot y\right) \cdot y\right) \cdot -0.16666666666666666}{y}}{y}, z, -1\right), z, 1\right)}{y} + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 y) x)))
   (if (<= y -1.9e+193)
     t_0
     (if (<= y -1.8e+27)
       (+
        (/
         (fma
          (fma (/ (/ (* (* (* z y) y) -0.16666666666666666) y) y) z -1.0)
          z
          1.0)
         y)
        x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (1.0 / y) + x;
	double tmp;
	if (y <= -1.9e+193) {
		tmp = t_0;
	} else if (y <= -1.8e+27) {
		tmp = (fma(fma((((((z * y) * y) * -0.16666666666666666) / y) / y), z, -1.0), z, 1.0) / y) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(1.0 / y) + x)
	tmp = 0.0
	if (y <= -1.9e+193)
		tmp = t_0;
	elseif (y <= -1.8e+27)
		tmp = Float64(Float64(fma(fma(Float64(Float64(Float64(Float64(Float64(z * y) * y) * -0.16666666666666666) / y) / y), z, -1.0), z, 1.0) / y) + x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -1.9e+193], t$95$0, If[LessEqual[y, -1.8e+27], N[(N[(N[(N[(N[(N[(N[(N[(N[(z * y), $MachinePrecision] * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision] * z + -1.0), $MachinePrecision] * z + 1.0), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.89999999999999986e193 or -1.79999999999999991e27 < y

   1. Initial program 82.2%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 88.9%

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

   if -1.89999999999999986e193 < y < -1.79999999999999991e27

   1. Initial program 84.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 86.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 77.3%

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

   9. Taylor expanded in z around inf

      \[\leadsto x + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{z \cdot \left(y \cdot \left(\frac{-1}{6} \cdot y - \frac{1}{2}\right) - \frac{1}{3}\right)}{y}}{y}, z, -1\right), z, 1\right)}{y} \]
   10. Step-by-step derivation

   11. Applied rewrites 77.1%

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

   12. Taylor expanded in y around inf

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

   14. Applied rewrites 95.5%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+193}:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{\left(\left(z \cdot y\right) \cdot y\right) \cdot -0.16666666666666666}{y}}{y}, z, -1\right), z, 1\right)}{y} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.8% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+27}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-0.16666666666666666 \cdot z\right) \cdot z, z, 1\right)}{y} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.95e+27)
   (+ (/ (fma (* (* -0.16666666666666666 z) z) z 1.0) y) x)
   (+ (/ 1.0 y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.95e+27) {
		tmp = (fma(((-0.16666666666666666 * z) * z), z, 1.0) / y) + x;
	} else {
		tmp = (1.0 / y) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.95e+27)
		tmp = Float64(Float64(fma(Float64(Float64(-0.16666666666666666 * z) * z), z, 1.0) / y) + x);
	else
		tmp = Float64(Float64(1.0 / y) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.95e+27], N[(N[(N[(N[(N[(-0.16666666666666666 * z), $MachinePrecision] * z), $MachinePrecision] * z + 1.0), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.9499999999999999e27

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 84.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 84.7%

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

   if -1.9499999999999999e27 < y

   1. Initial program 81.8%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 89.4%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+27}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-0.16666666666666666 \cdot z\right) \cdot z, z, 1\right)}{y} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.4% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, z, -1\right), z, 1\right)}{y} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.8e+27) (+ (/ (fma (fma 0.5 z -1.0) z 1.0) y) x) (+ (/ 1.0 y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.8e+27) {
		tmp = (fma(fma(0.5, z, -1.0), z, 1.0) / y) + x;
	} else {
		tmp = (1.0 / y) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.8e+27)
		tmp = Float64(Float64(fma(fma(0.5, z, -1.0), z, 1.0) / y) + x);
	else
		tmp = Float64(Float64(1.0 / y) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.8e+27], N[(N[(N[(N[(0.5 * z + -1.0), $MachinePrecision] * z + 1.0), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.79999999999999991e27

   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

      \[\leadsto x + \frac{\color{blue}{1 + z \cdot \left(z \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{y}\right) - 1\right)}}{y} \]
   4. 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 80.6

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 80.6%

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

   if -1.79999999999999991e27 < y

   1. Initial program 81.8%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 89.4%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, z, -1\right), z, 1\right)}{y} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.4% accurate, 15.6× speedup

Math

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

FPCore

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

C

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

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) + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.6%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 84.0%

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

6. Final simplification 84.0%

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

Alternative 7: 39.6% accurate, 19.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.6%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 43.1

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

5. Applied rewrites 43.1%

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

Alternative 8: 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 82.6%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 43.1

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

5. Applied rewrites 43.1%

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

7. Applied rewrites 14.5%

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

8. Taylor expanded in y around -inf

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

10. Applied rewrites 2.3%

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

Developer Target 1: 91.2% 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 2024255 
(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)))