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

Percentage Accurate: 99.9% → 99.9%

Time: 6.5s

Alternatives: 7

Speedup: 1.0×

• 25.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot x - 3}{6} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (* x x) 3.0) 6.0))

C

double code(double x) {
	return ((x * x) - 3.0) / 6.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x * x) - 3.0d0) / 6.0d0
end function

Java

public static double code(double x) {
	return ((x * x) - 3.0) / 6.0;
}

Python

def code(x):
	return ((x * x) - 3.0) / 6.0

Julia

function code(x)
	return Float64(Float64(Float64(x * x) - 3.0) / 6.0)
end

MATLAB

function tmp = code(x)
	tmp = ((x * x) - 3.0) / 6.0;
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] - 3.0), $MachinePrecision] / 6.0), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot x - 3}{6} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (* x x) 3.0) 6.0))

C

double code(double x) {
	return ((x * x) - 3.0) / 6.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x * x) - 3.0d0) / 6.0d0
end function

Java

public static double code(double x) {
	return ((x * x) - 3.0) / 6.0;
}

Python

def code(x):
	return ((x * x) - 3.0) / 6.0

Julia

function code(x)
	return Float64(Float64(Float64(x * x) - 3.0) / 6.0)
end

MATLAB

function tmp = code(x)
	tmp = ((x * x) - 3.0) / 6.0;
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] - 3.0), $MachinePrecision] / 6.0), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -0.5 - \frac{x}{\frac{-6}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- -0.5 (/ x (/ -6.0 x))))

C

double code(double x) {
	return -0.5 - (x / (-6.0 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-0.5d0) - (x / ((-6.0d0) / x))
end function

Java

public static double code(double x) {
	return -0.5 - (x / (-6.0 / x));
}

Python

def code(x):
	return -0.5 - (x / (-6.0 / x))

Julia

function code(x)
	return Float64(-0.5 - Float64(x / Float64(-6.0 / x)))
end

MATLAB

function tmp = code(x)
	tmp = -0.5 - (x / (-6.0 / x));
end

Wolfram

code[x_] := N[(-0.5 - N[(x / N[(-6.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\frac{x \cdot x - 3}{6} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip3-- N/A

      \[\leadsto \frac{\frac{{\left(x \cdot x\right)}^{3} - {3}^{3}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(3 \cdot 3 + \left(x \cdot x\right) \cdot 3\right)}}{6} \]

   2. clear-num N/A

      \[\leadsto \frac{\frac{1}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(3 \cdot 3 + \left(x \cdot x\right) \cdot 3\right)}{{\left(x \cdot x\right)}^{3} - {3}^{3}}}}{6} \]

   3. associate-/l/ N/A

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

   4. associate-/r* N/A

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

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{6}\right), \color{blue}{\left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(3 \cdot 3 + \left(x \cdot x\right) \cdot 3\right)}{{\left(x \cdot x\right)}^{3} - {3}^{3}}\right)}\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{6}, \left(\frac{\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(3 \cdot 3 + \left(x \cdot x\right) \cdot 3\right)}}{{\left(x \cdot x\right)}^{3} - {3}^{3}}\right)\right) \]

   7. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{6}, \left(\frac{1}{\color{blue}{\frac{{\left(x \cdot x\right)}^{3} - {3}^{3}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(3 \cdot 3 + \left(x \cdot x\right) \cdot 3\right)}}}\right)\right) \]

   8. flip3-- N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{6}, \left(\frac{1}{x \cdot x - \color{blue}{3}}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot x - 3\right)}\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(1, \left(x \cdot x + \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

   13. metadata-eval 99.8%

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -3\right)\right)\right) \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{0.16666666666666666}{\frac{1}{x \cdot x + -3}}} \]
5. Step-by-step derivation

   1. associate-/r/ N/A

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

   2. metadata-eval N/A

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

   3. distribute-rgt-in N/A

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

   4. metadata-eval N/A

      \[\leadsto \left(x \cdot x\right) \cdot \frac{1}{6} + \frac{-1}{2} \]

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

   7. metadata-eval N/A

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

   8. div-inv N/A

      \[\leadsto \frac{-1}{2} + \frac{x \cdot x}{\color{blue}{6}} \]

   9. associate-*l/ N/A

      \[\leadsto \frac{-1}{2} + \frac{x}{6} \cdot \color{blue}{x} \]

   10. associate-/r/ N/A

       \[\leadsto \frac{-1}{2} + \frac{x}{\color{blue}{\frac{6}{x}}} \]

   11. frac-2neg N/A

       \[\leadsto \frac{-1}{2} + \frac{\mathsf{neg}\left(x\right)}{\color{blue}{\mathsf{neg}\left(\frac{6}{x}\right)}} \]

   12. distribute-frac-neg N/A

       \[\leadsto \frac{-1}{2} + \left(\mathsf{neg}\left(\frac{x}{\mathsf{neg}\left(\frac{6}{x}\right)}\right)\right) \]

   13. unsub-neg N/A

       \[\leadsto \frac{-1}{2} - \color{blue}{\frac{x}{\mathsf{neg}\left(\frac{6}{x}\right)}} \]

   14. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\left(\frac{-1}{2}\right), \color{blue}{\left(\frac{x}{\mathsf{neg}\left(\frac{6}{x}\right)}\right)}\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(\frac{-1}{2}, \left(\frac{\color{blue}{x}}{\mathsf{neg}\left(\frac{6}{x}\right)}\right)\right) \]

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{6}{x}\right)\right)}\right)\right) \]

   17. distribute-neg-frac N/A

       \[\leadsto \mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(x, \left(\frac{\mathsf{neg}\left(6\right)}{\color{blue}{x}}\right)\right)\right) \]

   18. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(6\right)\right), \color{blue}{x}\right)\right)\right) \]

   19. metadata-eval 99.9%

       \[\leadsto \mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(-6, x\right)\right)\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{-0.5 - \frac{x}{\frac{-6}{x}}} \]
7. Add Preprocessing

Alternative 2: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-13}:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{6}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (* x x) 2e-13) -0.5 (/ x (/ 6.0 x))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 2e-13) {
		tmp = -0.5;
	} else {
		tmp = x / (6.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x * x) <= 2d-13) then
        tmp = -0.5d0
    else
        tmp = x / (6.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 2e-13) {
		tmp = -0.5;
	} else {
		tmp = x / (6.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 2e-13:
		tmp = -0.5
	else:
		tmp = x / (6.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 2e-13)
		tmp = -0.5;
	else
		tmp = Float64(x / Float64(6.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 2e-13)
		tmp = -0.5;
	else
		tmp = x / (6.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e-13], -0.5, N[(x / N[(6.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.0000000000000001e-13

   1. Initial program 100.0%

      \[\frac{x \cdot x - 3}{6} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{2}} \]
   4. Step-by-step derivation

   5. Simplified 99.8%

      \[\leadsto \color{blue}{-0.5} \]

   if 2.0000000000000001e-13 < (*.f64 x x)

   1. Initial program 99.8%

      \[\frac{x \cdot x - 3}{6} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left({x}^{2}\right)}, 6\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot x\right), 6\right) \]

      2. *-lowering-*.f64 98.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), 6\right) \]

   5. Simplified 98.7%

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

      1. associate-/l* N/A

         \[\leadsto x \cdot \color{blue}{\frac{x}{6}} \]

      2. clear-num N/A

         \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{6}{x}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{6}{x}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{6}{x}\right)}\right) \]

      5. /-lowering-/.f64 98.7%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(6, \color{blue}{x}\right)\right) \]

   7. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\frac{x}{\frac{6}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-13}:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{6}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (* x x) 2e-13) -0.5 (* x (/ x 6.0))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 2e-13) {
		tmp = -0.5;
	} else {
		tmp = x * (x / 6.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x * x) <= 2d-13) then
        tmp = -0.5d0
    else
        tmp = x * (x / 6.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 2e-13) {
		tmp = -0.5;
	} else {
		tmp = x * (x / 6.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 2e-13:
		tmp = -0.5
	else:
		tmp = x * (x / 6.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 2e-13)
		tmp = -0.5;
	else
		tmp = Float64(x * Float64(x / 6.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 2e-13)
		tmp = -0.5;
	else
		tmp = x * (x / 6.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e-13], -0.5, N[(x * N[(x / 6.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.0000000000000001e-13

   1. Initial program 100.0%

      \[\frac{x \cdot x - 3}{6} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{2}} \]
   4. Step-by-step derivation

   5. Simplified 99.8%

      \[\leadsto \color{blue}{-0.5} \]

   if 2.0000000000000001e-13 < (*.f64 x x)

   1. Initial program 99.8%

      \[\frac{x \cdot x - 3}{6} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left({x}^{2}\right)}, 6\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot x\right), 6\right) \]

      2. *-lowering-*.f64 98.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), 6\right) \]

   5. Simplified 98.7%

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

      1. associate-*l/ N/A

         \[\leadsto \frac{x}{6} \cdot \color{blue}{x} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{6}\right), \color{blue}{x}\right) \]

      3. /-lowering-/.f64 98.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, 6\right), x\right) \]

   7. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\frac{x}{6} \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-13}:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{6}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-13}:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 2e-13) -0.5 (* x (* x 0.16666666666666666))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 2e-13) {
		tmp = -0.5;
	} else {
		tmp = x * (x * 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x * x) <= 2d-13) then
        tmp = -0.5d0
    else
        tmp = x * (x * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 2e-13) {
		tmp = -0.5;
	} else {
		tmp = x * (x * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 2e-13:
		tmp = -0.5
	else:
		tmp = x * (x * 0.16666666666666666)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 2e-13)
		tmp = -0.5;
	else
		tmp = Float64(x * Float64(x * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 2e-13)
		tmp = -0.5;
	else
		tmp = x * (x * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e-13], -0.5, N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.0000000000000001e-13

   1. Initial program 100.0%

      \[\frac{x \cdot x - 3}{6} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{2}} \]
   4. Step-by-step derivation

   5. Simplified 99.8%

      \[\leadsto \color{blue}{-0.5} \]

   if 2.0000000000000001e-13 < (*.f64 x x)

   1. Initial program 99.8%

      \[\frac{x \cdot x - 3}{6} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left({x}^{2}\right)}, 6\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot x\right), 6\right) \]

      2. *-lowering-*.f64 98.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), 6\right) \]

   5. Simplified 98.7%

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

      1. associate-*l/ N/A

         \[\leadsto \frac{x}{6} \cdot \color{blue}{x} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{6}\right), \color{blue}{x}\right) \]

      3. /-lowering-/.f64 98.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, 6\right), x\right) \]

   7. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\frac{x}{6} \cdot x} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \frac{1}{6}\right), x\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \frac{1}{6}\right), x\right) \]

      3. *-lowering-*.f64 98.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{6}\right), x\right) \]

   9. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\left(x \cdot 0.16666666666666666\right)} \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-13}:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 3:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 3.0) -0.5 (* (* x x) 0.16666666666666666)))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 3.0) {
		tmp = -0.5;
	} else {
		tmp = (x * x) * 0.16666666666666666;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x * x) <= 3.0d0) then
        tmp = -0.5d0
    else
        tmp = (x * x) * 0.16666666666666666d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 3.0) {
		tmp = -0.5;
	} else {
		tmp = (x * x) * 0.16666666666666666;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 3.0:
		tmp = -0.5
	else:
		tmp = (x * x) * 0.16666666666666666
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 3.0)
		tmp = -0.5;
	else
		tmp = Float64(Float64(x * x) * 0.16666666666666666);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 3.0)
		tmp = -0.5;
	else
		tmp = (x * x) * 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 3.0], -0.5, N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 3

   1. Initial program 100.0%

      \[\frac{x \cdot x - 3}{6} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{2}} \]
   4. Step-by-step derivation

   5. Simplified 99.8%

      \[\leadsto \color{blue}{-0.5} \]

   if 3 < (*.f64 x x)

   1. Initial program 99.8%

      \[\frac{x \cdot x - 3}{6} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 98.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   5. Simplified 98.6%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 3:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -0.5 - x \cdot \frac{x}{-6} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- -0.5 (* x (/ x -6.0))))

C

double code(double x) {
	return -0.5 - (x * (x / -6.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-0.5d0) - (x * (x / (-6.0d0)))
end function

Java

public static double code(double x) {
	return -0.5 - (x * (x / -6.0));
}

Python

def code(x):
	return -0.5 - (x * (x / -6.0))

Julia

function code(x)
	return Float64(-0.5 - Float64(x * Float64(x / -6.0)))
end

MATLAB

function tmp = code(x)
	tmp = -0.5 - (x * (x / -6.0));
end

Wolfram

code[x_] := N[(-0.5 - N[(x * N[(x / -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\frac{x \cdot x - 3}{6} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip3-- N/A

      \[\leadsto \frac{\frac{{\left(x \cdot x\right)}^{3} - {3}^{3}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(3 \cdot 3 + \left(x \cdot x\right) \cdot 3\right)}}{6} \]

   2. clear-num N/A

      \[\leadsto \frac{\frac{1}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(3 \cdot 3 + \left(x \cdot x\right) \cdot 3\right)}{{\left(x \cdot x\right)}^{3} - {3}^{3}}}}{6} \]

   3. associate-/l/ N/A

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

   4. associate-/r* N/A

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

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{6}\right), \color{blue}{\left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(3 \cdot 3 + \left(x \cdot x\right) \cdot 3\right)}{{\left(x \cdot x\right)}^{3} - {3}^{3}}\right)}\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{6}, \left(\frac{\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(3 \cdot 3 + \left(x \cdot x\right) \cdot 3\right)}}{{\left(x \cdot x\right)}^{3} - {3}^{3}}\right)\right) \]

   7. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{6}, \left(\frac{1}{\color{blue}{\frac{{\left(x \cdot x\right)}^{3} - {3}^{3}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(3 \cdot 3 + \left(x \cdot x\right) \cdot 3\right)}}}\right)\right) \]

   8. flip3-- N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{6}, \left(\frac{1}{x \cdot x - \color{blue}{3}}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot x - 3\right)}\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(1, \left(x \cdot x + \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

   13. metadata-eval 99.8%

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -3\right)\right)\right) \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{0.16666666666666666}{\frac{1}{x \cdot x + -3}}} \]
5. Step-by-step derivation

   1. associate-/r/ N/A

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

   2. metadata-eval N/A

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

   3. distribute-rgt-in N/A

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

   4. metadata-eval N/A

      \[\leadsto \left(x \cdot x\right) \cdot \frac{1}{6} + \frac{-1}{2} \]

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

   7. metadata-eval N/A

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

   8. div-inv N/A

      \[\leadsto \frac{-1}{2} + \frac{x \cdot x}{\color{blue}{6}} \]

   9. associate-*l/ N/A

      \[\leadsto \frac{-1}{2} + \frac{x}{6} \cdot \color{blue}{x} \]

   10. associate-/r/ N/A

       \[\leadsto \frac{-1}{2} + \frac{x}{\color{blue}{\frac{6}{x}}} \]

   11. frac-2neg N/A

       \[\leadsto \frac{-1}{2} + \frac{\mathsf{neg}\left(x\right)}{\color{blue}{\mathsf{neg}\left(\frac{6}{x}\right)}} \]

   12. distribute-frac-neg N/A

       \[\leadsto \frac{-1}{2} + \left(\mathsf{neg}\left(\frac{x}{\mathsf{neg}\left(\frac{6}{x}\right)}\right)\right) \]

   13. unsub-neg N/A

       \[\leadsto \frac{-1}{2} - \color{blue}{\frac{x}{\mathsf{neg}\left(\frac{6}{x}\right)}} \]

   14. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\left(\frac{-1}{2}\right), \color{blue}{\left(\frac{x}{\mathsf{neg}\left(\frac{6}{x}\right)}\right)}\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(\frac{-1}{2}, \left(\frac{\color{blue}{x}}{\mathsf{neg}\left(\frac{6}{x}\right)}\right)\right) \]

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{6}{x}\right)\right)}\right)\right) \]

   17. distribute-neg-frac N/A

       \[\leadsto \mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(x, \left(\frac{\mathsf{neg}\left(6\right)}{\color{blue}{x}}\right)\right)\right) \]

   18. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(6\right)\right), \color{blue}{x}\right)\right)\right) \]

   19. metadata-eval 99.9%

       \[\leadsto \mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(-6, x\right)\right)\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{-0.5 - \frac{x}{\frac{-6}{x}}} \]
7. Step-by-step derivation

   1. associate-/r/ N/A

      \[\leadsto \mathsf{\_.f64}\left(\frac{-1}{2}, \left(\frac{x}{-6} \cdot \color{blue}{x}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(\frac{x}{-6}\right), \color{blue}{x}\right)\right) \]

   3. /-lowering-/.f64 99.9%

      \[\leadsto \mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, -6\right), x\right)\right) \]

8. Applied egg-rr 99.9%

   \[\leadsto -0.5 - \color{blue}{\frac{x}{-6} \cdot x} \]

9. Final simplification 99.9%

   \[\leadsto -0.5 - x \cdot \frac{x}{-6} \]
10. Add Preprocessing

Alternative 7: 49.8% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ -0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -0.5)

C

double code(double x) {
	return -0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -0.5d0
end function

Java

public static double code(double x) {
	return -0.5;
}

Python

def code(x):
	return -0.5

Julia

function code(x)
	return -0.5
end

MATLAB

function tmp = code(x)
	tmp = -0.5;
end

Wolfram

code[x_] := -0.5

Derivation

1. Initial program 99.9%

   \[\frac{x \cdot x - 3}{6} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-1}{2}} \]
4. Step-by-step derivation

5. Simplified 50.0%

   \[\leadsto \color{blue}{-0.5} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024161 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, H"
  :precision binary64
  (/ (- (* x x) 3.0) 6.0))