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

Percentage Accurate: 99.9% → 99.9%

Time: 6.9s

Alternatives: 5

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(1 - x\right) + y \cdot \sqrt{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (- 1.0 x) (* y (sqrt x))))

C

double code(double x, double y) {
	return (1.0 - x) + (y * sqrt(x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - x) + (y * sqrt(x))
end function

Java

public static double code(double x, double y) {
	return (1.0 - x) + (y * Math.sqrt(x));
}

Python

def code(x, y):
	return (1.0 - x) + (y * math.sqrt(x))

Julia

function code(x, y)
	return Float64(Float64(1.0 - x) + Float64(y * sqrt(x)))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - x) + (y * sqrt(x));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - x), $MachinePrecision] + N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - x\right) + y \cdot \sqrt{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (- 1.0 x) (* y (sqrt x))))

C

double code(double x, double y) {
	return (1.0 - x) + (y * sqrt(x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - x) + (y * sqrt(x))
end function

Java

public static double code(double x, double y) {
	return (1.0 - x) + (y * Math.sqrt(x));
}

Python

def code(x, y):
	return (1.0 - x) + (y * math.sqrt(x))

Julia

function code(x, y)
	return Float64(Float64(1.0 - x) + Float64(y * sqrt(x)))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - x) + (y * sqrt(x));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - x), $MachinePrecision] + N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - x\right) + \frac{y}{{x}^{-0.5}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (- 1.0 x) (/ y (pow x -0.5))))

C

double code(double x, double y) {
	return (1.0 - x) + (y / pow(x, -0.5));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - x) + (y / (x ** (-0.5d0)))
end function

Java

public static double code(double x, double y) {
	return (1.0 - x) + (y / Math.pow(x, -0.5));
}

Python

def code(x, y):
	return (1.0 - x) + (y / math.pow(x, -0.5))

Julia

function code(x, y)
	return Float64(Float64(1.0 - x) + Float64(y / (x ^ -0.5)))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - x) + (y / (x ^ -0.5));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - x), $MachinePrecision] + N[(y / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(1 - x\right) + y \cdot \sqrt{x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add0 99.9%

      \[\leadsto \left(1 - x\right) + \color{blue}{\left(y \cdot \sqrt{x} + 0\right)} \]

   2. fma-define 99.9%

      \[\leadsto \left(1 - x\right) + \color{blue}{\mathsf{fma}\left(y, \sqrt{x}, 0\right)} \]

   3. metadata-eval 99.9%

      \[\leadsto \left(1 - x\right) + \mathsf{fma}\left(y, \sqrt{x}, \color{blue}{-0}\right) \]

   4. fma-neg 99.9%

      \[\leadsto \left(1 - x\right) + \color{blue}{\left(y \cdot \sqrt{x} - 0\right)} \]

   5. *-commutative 99.9%

      \[\leadsto \left(1 - x\right) + \left(\color{blue}{\sqrt{x} \cdot y} - 0\right) \]

   6. add-sqr-sqrt 99.7%

      \[\leadsto \left(1 - x\right) + \left(\color{blue}{\left(\sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{x}}\right)} \cdot y - 0\right) \]

   7. associate-*l* 99.7%

      \[\leadsto \left(1 - x\right) + \left(\color{blue}{\sqrt{\sqrt{x}} \cdot \left(\sqrt{\sqrt{x}} \cdot y\right)} - 0\right) \]

   8. fma-neg 99.7%

      \[\leadsto \left(1 - x\right) + \color{blue}{\mathsf{fma}\left(\sqrt{\sqrt{x}}, \sqrt{\sqrt{x}} \cdot y, -0\right)} \]

   9. pow1/2 99.7%

      \[\leadsto \left(1 - x\right) + \mathsf{fma}\left(\sqrt{\color{blue}{{x}^{0.5}}}, \sqrt{\sqrt{x}} \cdot y, -0\right) \]

   10. sqrt-pow1 99.8%

       \[\leadsto \left(1 - x\right) + \mathsf{fma}\left(\color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}, \sqrt{\sqrt{x}} \cdot y, -0\right) \]

   11. metadata-eval 99.8%

       \[\leadsto \left(1 - x\right) + \mathsf{fma}\left({x}^{\color{blue}{0.25}}, \sqrt{\sqrt{x}} \cdot y, -0\right) \]

   12. pow1/2 99.8%

       \[\leadsto \left(1 - x\right) + \mathsf{fma}\left({x}^{0.25}, \sqrt{\color{blue}{{x}^{0.5}}} \cdot y, -0\right) \]

   13. sqrt-pow1 99.7%

       \[\leadsto \left(1 - x\right) + \mathsf{fma}\left({x}^{0.25}, \color{blue}{{x}^{\left(\frac{0.5}{2}\right)}} \cdot y, -0\right) \]

   14. metadata-eval 99.7%

       \[\leadsto \left(1 - x\right) + \mathsf{fma}\left({x}^{0.25}, {x}^{\color{blue}{0.25}} \cdot y, -0\right) \]

   15. metadata-eval 99.7%

       \[\leadsto \left(1 - x\right) + \mathsf{fma}\left({x}^{0.25}, {x}^{0.25} \cdot y, \color{blue}{0}\right) \]

4. Applied egg-rr 99.7%

   \[\leadsto \left(1 - x\right) + \color{blue}{\mathsf{fma}\left({x}^{0.25}, {x}^{0.25} \cdot y, 0\right)} \]
5. Step-by-step derivation

   1. fma-undefine 99.7%

      \[\leadsto \left(1 - x\right) + \color{blue}{\left({x}^{0.25} \cdot \left({x}^{0.25} \cdot y\right) + 0\right)} \]

   2. add0 99.7%

      \[\leadsto \left(1 - x\right) + \color{blue}{{x}^{0.25} \cdot \left({x}^{0.25} \cdot y\right)} \]

   3. associate-*r* 99.7%

      \[\leadsto \left(1 - x\right) + \color{blue}{\left({x}^{0.25} \cdot {x}^{0.25}\right) \cdot y} \]

   4. pow-prod-up 99.9%

      \[\leadsto \left(1 - x\right) + \color{blue}{{x}^{\left(0.25 + 0.25\right)}} \cdot y \]

   5. metadata-eval 99.9%

      \[\leadsto \left(1 - x\right) + {x}^{\color{blue}{0.5}} \cdot y \]

   6. metadata-eval 99.9%

      \[\leadsto \left(1 - x\right) + {x}^{\color{blue}{\left(1 - 0.5\right)}} \cdot y \]

   7. pow-div 99.8%

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

   8. pow1 99.8%

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

   9. pow1/2 99.8%

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

   10. pow1 99.8%

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

   11. metadata-eval 99.8%

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

   12. pow-div 84.4%

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

   13. pow1 84.4%

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

   14. times-frac 67.2%

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

   15. associate-/l* 73.9%

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

   16. div-inv 73.8%

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

   17. associate-/l* 84.4%

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

   18. pow1 84.4%

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

   19. pow-div 95.0%

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

   20. metadata-eval 95.0%

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

   21. pow1 95.0%

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

6. Applied egg-rr 95.0%

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

   1. associate-*r/ 95.5%

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

   2. *-rgt-identity 95.5%

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

   3. associate-/r/ 99.8%

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

8. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*r/ 96.2%

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

   3. associate-/l* 99.9%

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

   4. pow1/2 99.9%

      \[\leadsto \left(1 - x\right) + \frac{y}{\frac{\color{blue}{{x}^{0.5}}}{x}} \]

   5. pow1 99.9%

      \[\leadsto \left(1 - x\right) + \frac{y}{\frac{{x}^{0.5}}{\color{blue}{{x}^{1}}}} \]

   6. pow-div 99.9%

      \[\leadsto \left(1 - x\right) + \frac{y}{\color{blue}{{x}^{\left(0.5 - 1\right)}}} \]

   7. metadata-eval 99.9%

      \[\leadsto \left(1 - x\right) + \frac{y}{{x}^{\color{blue}{-0.5}}} \]

10. Applied egg-rr 99.9%

    \[\leadsto \left(1 - x\right) + \color{blue}{\frac{y}{{x}^{-0.5}}} \]

11. Final simplification 99.9%

    \[\leadsto \left(1 - x\right) + \frac{y}{{x}^{-0.5}} \]
12. Add Preprocessing

Alternative 2: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - x\right) + y \cdot \sqrt{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (- 1.0 x) (* y (sqrt x))))

C

double code(double x, double y) {
	return (1.0 - x) + (y * sqrt(x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - x) + (y * sqrt(x))
end function

Java

public static double code(double x, double y) {
	return (1.0 - x) + (y * Math.sqrt(x));
}

Python

def code(x, y):
	return (1.0 - x) + (y * math.sqrt(x))

Julia

function code(x, y)
	return Float64(Float64(1.0 - x) + Float64(y * sqrt(x)))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - x) + (y * sqrt(x));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - x), $MachinePrecision] + N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(1 - x\right) + y \cdot \sqrt{x} \]
2. Add Preprocessing

3. Final simplification 99.9%

   \[\leadsto \left(1 - x\right) + y \cdot \sqrt{x} \]
4. Add Preprocessing

Alternative 3: 61.3% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.8:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 7.8) 1.0 (- x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 7.8) {
		tmp = 1.0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 7.8d0) then
        tmp = 1.0d0
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 7.8) {
		tmp = 1.0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 7.8:
		tmp = 1.0
	else:
		tmp = -x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 7.8)
		tmp = 1.0;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 7.8)
		tmp = 1.0;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 7.8], 1.0, (-x)]

Derivation

1. Split input into 2 regimes

2. if x < 7.79999999999999982

   1. Initial program 99.8%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 59.2%

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

   if 7.79999999999999982 < x

   1. Initial program 99.9%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 66.2%

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

      1. mul-1-neg 66.2%

         \[\leadsto \color{blue}{-x} \]

   5. Simplified 66.2%

      \[\leadsto \color{blue}{-x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.8:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.3% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ 1 - x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 x))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0 - x

Julia

function code(x, y)
	return Float64(1.0 - x)
end

MATLAB

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

Wolfram

code[x_, y_] := N[(1.0 - x), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(1 - x\right) + y \cdot \sqrt{x} \]
2. Add Preprocessing

3. Taylor expanded in y around 0 64.1%

   \[\leadsto \color{blue}{1 - x} \]

4. Final simplification 64.1%

   \[\leadsto 1 - x \]
5. Add Preprocessing

Alternative 5: 31.9% accurate, 107.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.9%

   \[\left(1 - x\right) + y \cdot \sqrt{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 32.3%

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

4. Final simplification 32.3%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))