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

Percentage Accurate: 85.1% → 99.5%

Time: 9.1s

Alternatives: 7

Speedup: 6.7×

• could not determine a ground truth

  1. x = -4.631679265846702e+43
  2. y = 1.329779266135363e+40
  3. z = -2.718056364276177e-240

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: 85.1% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -33000000000 \lor \neg \left(y \leq 2 \cdot 10^{-13}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -33000000000.0) (not (<= y 2e-13)))
   (+ x (/ (exp (- z)) y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -33000000000.0) || !(y <= 2e-13)) {
		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 <= (-33000000000.0d0)) .or. (.not. (y <= 2d-13))) 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 <= -33000000000.0) || !(y <= 2e-13)) {
		tmp = x + (Math.exp(-z) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -33000000000.0) or not (y <= 2e-13):
		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 <= -33000000000.0) || !(y <= 2e-13))
		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 <= -33000000000.0) || ~((y <= 2e-13)))
		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, -33000000000.0], N[Not[LessEqual[y, 2e-13]], $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 < -3.3e10 or 2.0000000000000001e-13 < y

   1. Initial program 85.4%

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

   3. Taylor expanded in y around inf

      \[\leadsto x + \frac{e^{\color{blue}{-1 \cdot z}}}{y} \]
   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 -3.3e10 < y < 2.0000000000000001e-13

   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 99.6%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -33000000000 \lor \neg \left(y \leq 2 \cdot 10^{-13}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 38.8% 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 86.4%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 39.6

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

5. Applied rewrites 39.6%

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

6. Final simplification 39.6%

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

Alternative 3: 87.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -33000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{0.3333333333333333}{y} + 0.5}{y} + 0.16666666666666666, -z, \frac{0.5}{y} + 0.5\right) \cdot z - 1, z, 1\right)}{y} + x\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+168}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot z - 1, z, 1\right), y, \left(z \cdot z\right) \cdot 0.5\right)}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -33000000000.0)
   (+
    (/
     (fma
      (-
       (*
        (fma
         (+ (/ (+ (/ 0.3333333333333333 y) 0.5) y) 0.16666666666666666)
         (- z)
         (+ (/ 0.5 y) 0.5))
        z)
       1.0)
      z
      1.0)
     y)
    x)
   (if (<= y 1.6e+168)
     (+ 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 <= -33000000000.0) {
		tmp = (fma(((fma(((((0.3333333333333333 / y) + 0.5) / y) + 0.16666666666666666), -z, ((0.5 / y) + 0.5)) * z) - 1.0), z, 1.0) / y) + x;
	} else if (y <= 1.6e+168) {
		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 <= -33000000000.0)
		tmp = Float64(Float64(fma(Float64(Float64(fma(Float64(Float64(Float64(Float64(0.3333333333333333 / y) + 0.5) / y) + 0.16666666666666666), Float64(-z), Float64(Float64(0.5 / y) + 0.5)) * z) - 1.0), z, 1.0) / y) + x);
	elseif (y <= 1.6e+168)
		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, -33000000000.0], N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(0.3333333333333333 / y), $MachinePrecision] + 0.5), $MachinePrecision] / y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * (-z) + N[(N[(0.5 / y), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] - 1.0), $MachinePrecision] * z + 1.0), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 1.6e+168], 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 < -3.3e10

   1. Initial program 90.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 90.8

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

      4. lift-exp.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-log.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lower-pow.f64 90.8

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

   4. Applied rewrites 90.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 65.7%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   10. Applied rewrites 87.6%

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

   if -3.3e10 < y < 1.6000000000000001e168

   1. Initial program 89.2%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 96.5%

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

   if 1.6000000000000001e168 < y

   1. Initial program 69.9%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 61.1%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 61.1%

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

   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 86.1%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -33000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{0.3333333333333333}{y} + 0.5}{y} + 0.16666666666666666, -z, \frac{0.5}{y} + 0.5\right) \cdot z - 1, z, 1\right)}{y} + x\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+168}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot z - 1, z, 1\right), y, \left(z \cdot z\right) \cdot 0.5\right)}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5 \cdot z - 1, z, 1\right)\\ \mathbf{if}\;y \leq -33000000000:\\ \;\;\;\;x + \frac{t\_0}{y}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+168}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{\mathsf{fma}\left(t\_0, y, \left(z \cdot z\right) \cdot 0.5\right)}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (- (* 0.5 z) 1.0) z 1.0)))
   (if (<= y -33000000000.0)
     (+ x (/ t_0 y))
     (if (<= y 1.6e+168)
       (+ x (/ 1.0 y))
       (+ x (/ (/ (fma t_0 y (* (* z z) 0.5)) y) y))))))

C

double code(double x, double y, double z) {
	double t_0 = fma(((0.5 * z) - 1.0), z, 1.0);
	double tmp;
	if (y <= -33000000000.0) {
		tmp = x + (t_0 / y);
	} else if (y <= 1.6e+168) {
		tmp = x + (1.0 / y);
	} else {
		tmp = x + ((fma(t_0, y, ((z * z) * 0.5)) / y) / y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(Float64(0.5 * z) - 1.0), z, 1.0)
	tmp = 0.0
	if (y <= -33000000000.0)
		tmp = Float64(x + Float64(t_0 / y));
	elseif (y <= 1.6e+168)
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = Float64(x + Float64(Float64(fma(t_0, y, Float64(Float64(z * z) * 0.5)) / y) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(0.5 * z), $MachinePrecision] - 1.0), $MachinePrecision] * z + 1.0), $MachinePrecision]}, If[LessEqual[y, -33000000000.0], N[(x + N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.6e+168], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(t$95$0 * 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 < -3.3e10

   1. Initial program 90.8%

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

   3. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{\left(z \cdot \left(z \cdot \left(\frac{1}{2} \cdot \frac{1}{y} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}\right) - \frac{1}{y}\right) + \frac{1}{y}\right)} \]
   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 80.1%

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

   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 84.6%

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

   if -3.3e10 < y < 1.6000000000000001e168

   1. Initial program 89.2%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 96.5%

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

   if 1.6000000000000001e168 < y

   1. Initial program 69.9%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 61.1%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 61.1%

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

   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 86.1%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -33000000000:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(0.5 \cdot z - 1, z, 1\right)}{y}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+168}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot z - 1, z, 1\right), y, \left(z \cdot z\right) \cdot 0.5\right)}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.75 \cdot 10^{+189}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(0.5 \cdot z - 1, z, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.75e+189)
   (+ x (/ (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 (z <= -2.75e+189) {
		tmp = x + (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 (z <= -2.75e+189)
		tmp = Float64(x + Float64(fma(Float64(Float64(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[z, -2.75e+189], N[(x + N[(N[(N[(N[(0.5 * 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 z < -2.75e189

   1. Initial program 72.4%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 58.9%

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

   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 81.8%

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

   if -2.75e189 < z

   1. Initial program 87.7%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 89.9%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.75 \cdot 10^{+189}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(0.5 \cdot z - 1, z, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.8% accurate, 15.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.4%

   \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
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 84.3%

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

6. Final simplification 84.3%

   \[\leadsto x + \frac{1}{y} \]
7. 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 86.4%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 39.6

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

5. Applied rewrites 39.6%

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

7. Applied rewrites 25.0%

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

9. Applied rewrites 2.0%

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

Developer Target 1: 91.6% 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 2024339 
(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)))