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

Percentage Accurate: 84.7% → 99.6%

Time: 11.0s

Alternatives: 5

Speedup: 6.0×

• could not determine a ground truth

  1. x = 109329739166445.03
  2. y = -2.2691503364179648e+38
  3. z = -5.988803261995637e-257

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: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (/ (exp (* y (log (/ y (+ z y))))) y)))

C

double code(double x, double y, double z) {
	return x + (exp((y * log((y / (z + y))))) / y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (exp((y * log((y / (z + y))))) / y)
end function

Java

public static double code(double x, double y, double z) {
	return x + (Math.exp((y * Math.log((y / (z + y))))) / y);
}

Python

def code(x, y, z):
	return x + (math.exp((y * math.log((y / (z + y))))) / y)

Julia

function code(x, y, z)
	return Float64(x + Float64(exp(Float64(y * log(Float64(y / Float64(z + y))))) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (exp((y * log((y / (z + y))))) / y);
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[Exp[N[(y * N[Log[N[(y / N[(z + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{e^{-z}}{y}\\ \mathbf{if}\;y \leq -1.6:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.2:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(z, -1 \cdot \frac{1}{\mathsf{fma}\left(z, 0.5 + \frac{0.5}{y}, 1\right)}, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ (exp (- z)) y))))
   (if (<= y -1.6)
     t_0
     (if (<= y 0.2)
       (+ x (/ (fma z (* -1.0 (/ 1.0 (fma z (+ 0.5 (/ 0.5 y)) 1.0))) 1.0) y))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (exp(-z) / y);
	double tmp;
	if (y <= -1.6) {
		tmp = t_0;
	} else if (y <= 0.2) {
		tmp = x + (fma(z, (-1.0 * (1.0 / fma(z, (0.5 + (0.5 / y)), 1.0))), 1.0) / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(exp(Float64(-z)) / y))
	tmp = 0.0
	if (y <= -1.6)
		tmp = t_0;
	elseif (y <= 0.2)
		tmp = Float64(x + Float64(fma(z, Float64(-1.0 * Float64(1.0 / fma(z, Float64(0.5 + Float64(0.5 / y)), 1.0))), 1.0) / y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(N[Exp[(-z)], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.6], t$95$0, If[LessEqual[y, 0.2], N[(x + N[(N[(z * N[(-1.0 * N[(1.0 / N[(z * N[(0.5 + N[(0.5 / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6000000000000001 or 0.20000000000000001 < y

   1. Initial program 84.9%

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

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

   if -1.6000000000000001 < y < 0.20000000000000001

   1. Initial program 84.9%

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

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{\color{blue}{1 + z \cdot \left(z \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{y}\right) - 1\right)}}{y} \]
   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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 49.3

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

   5. Applied rewrites 49.3%

      \[\leadsto x + \frac{\color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5 + \frac{0.5}{y}, -1\right), 1\right)}}{y} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.7%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 99.3%

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

Alternative 2: 88.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1150:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(z, -1 \cdot \left(y \cdot \mathsf{fma}\left(-4, \frac{\mathsf{fma}\left(y, z \cdot 0.5, y\right)}{z \cdot z}, \frac{2}{z}\right)\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1150.0)
   (+
    x
    (/
     (fma
      z
      (* -1.0 (* y (fma -4.0 (/ (fma y (* z 0.5) y) (* z z)) (/ 2.0 z))))
      1.0)
     y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1150.0) {
		tmp = x + (fma(z, (-1.0 * (y * fma(-4.0, (fma(y, (z * 0.5), y) / (z * z)), (2.0 / z)))), 1.0) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1150.0)
		tmp = Float64(x + Float64(fma(z, Float64(-1.0 * Float64(y * fma(-4.0, Float64(fma(y, Float64(z * 0.5), y) / Float64(z * z)), Float64(2.0 / z)))), 1.0) / y));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1150.0], N[(x + N[(N[(z * N[(-1.0 * N[(y * N[(-4.0 * N[(N[(y * N[(z * 0.5), $MachinePrecision] + y), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(2.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1150

   1. Initial program 49.0%

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

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{\color{blue}{1 + z \cdot \left(z \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{y}\right) - 1\right)}}{y} \]
   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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 27.8

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

   5. Applied rewrites 27.8%

      \[\leadsto x + \frac{\color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5 + \frac{0.5}{y}, -1\right), 1\right)}}{y} \]
   6. Step-by-step derivation

   7. Applied rewrites 27.7%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 40.9%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 59.8%

       \[\leadsto x + \frac{\mathsf{fma}\left(z, -1 \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(-4, \frac{\mathsf{fma}\left(y, z \cdot 0.5, y\right)}{z \cdot z}, \frac{2}{z}\right)}\right), 1\right)}{y} \]

   if -1150 < z

   1. Initial program 94.2%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 96.1%

      \[\leadsto x + \frac{\color{blue}{1}}{y} \]
3. Recombined 2 regimes into one program.
4. 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 2024233 
(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)))