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

Percentage Accurate: 99.8% → 99.9%

Time: 4.4s

Alternatives: 6

Speedup: 1.0×

• 24.8% 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: 99.8% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. div-sub N/A

      \[\leadsto \frac{x \cdot x}{6} - \color{blue}{\frac{3}{6}} \]

   2. sub-neg N/A

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

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

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

   4. associate-/l* N/A

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

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

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

   6. /-lowering-/.f64 N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{x \cdot \frac{x}{6} + -0.5} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num N/A

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

   2. un-div-inv N/A

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

   3. /-lowering-/.f64 N/A

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

   4. /-lowering-/.f64 99.9%

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

6. Applied egg-rr 99.9%

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

Alternative 2: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-8}:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{6}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (* x x) 5e-8) -0.5 (/ x (/ 6.0 x))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 5e-8) {
		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) <= 5d-8) 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) <= 5e-8) {
		tmp = -0.5;
	} else {
		tmp = x / (6.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 5e-8:
		tmp = -0.5
	else:
		tmp = x / (6.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 5e-8)
		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) <= 5e-8)
		tmp = -0.5;
	else
		tmp = x / (6.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-8], -0.5, N[(x / N[(6.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.9999999999999998e-8

   1. Initial program 100.0%

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

      1. div-sub N/A

         \[\leadsto \frac{x \cdot x}{6} - \color{blue}{\frac{3}{6}} \]

      2. sub-neg N/A

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

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

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

      4. associate-/l* N/A

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

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

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

      6. /-lowering-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 99.0%

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

   if 4.9999999999999998e-8 < (*.f64 x x)

   1. Initial program 99.7%

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

      1. div-sub N/A

         \[\leadsto \frac{x \cdot x}{6} - \color{blue}{\frac{3}{6}} \]

      2. sub-neg N/A

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

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

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

      4. associate-/l* N/A

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

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

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

      6. /-lowering-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2}} \]
   6. 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 98.3%

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

   7. Simplified 98.3%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(x \cdot x\right)} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

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

      2. associate-/r/ N/A

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

      3. associate-/r* N/A

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

      4. clear-num N/A

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

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

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

      6. /-lowering-/.f64 98.4%

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

   9. Applied egg-rr 98.4%

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

Alternative 3: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-8}:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{6}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (* x x) 5e-8) -0.5 (* x (/ x 6.0))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 5e-8) {
		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) <= 5d-8) 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) <= 5e-8) {
		tmp = -0.5;
	} else {
		tmp = x * (x / 6.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 5e-8:
		tmp = -0.5
	else:
		tmp = x * (x / 6.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 5e-8)
		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) <= 5e-8)
		tmp = -0.5;
	else
		tmp = x * (x / 6.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-8], -0.5, N[(x * N[(x / 6.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.9999999999999998e-8

   1. Initial program 100.0%

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

      1. div-sub N/A

         \[\leadsto \frac{x \cdot x}{6} - \color{blue}{\frac{3}{6}} \]

      2. sub-neg N/A

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

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

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

      4. associate-/l* N/A

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

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

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

      6. /-lowering-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 99.0%

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

   if 4.9999999999999998e-8 < (*.f64 x x)

   1. Initial program 99.7%

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

      1. div-sub N/A

         \[\leadsto \frac{x \cdot x}{6} - \color{blue}{\frac{3}{6}} \]

      2. sub-neg N/A

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

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

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

      4. associate-/l* N/A

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

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

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

      6. /-lowering-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2}} \]
   6. 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 98.3%

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

   7. Simplified 98.3%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(x \cdot x\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. div-inv N/A

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

      4. associate-*l/ N/A

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

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

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

      6. /-lowering-/.f64 98.4%

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

   9. Applied egg-rr 98.4%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-8}:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{6}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% 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. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \frac{x \cdot x}{6} - \color{blue}{\frac{3}{6}} \]

      2. sub-neg N/A

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

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

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

      4. associate-/l* N/A

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

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

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

      6. /-lowering-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 99.0%

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

   if 3 < (*.f64 x x)

   1. Initial program 99.7%

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

      1. div-sub N/A

         \[\leadsto \frac{x \cdot x}{6} - \color{blue}{\frac{3}{6}} \]

      2. sub-neg N/A

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

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

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

      4. associate-/l* N/A

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

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

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

      6. /-lowering-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2}} \]
   6. 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 98.3%

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

   7. Simplified 98.3%

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

4. Final simplification 98.6%

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

Alternative 5: 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 99.8%

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

   1. div-sub N/A

      \[\leadsto \frac{x \cdot x}{6} - \color{blue}{\frac{3}{6}} \]

   2. sub-neg N/A

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

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

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

   4. associate-/l* N/A

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

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

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

   6. /-lowering-/.f64 N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 6: 49.9% 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 99.8%

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

   1. div-sub N/A

      \[\leadsto \frac{x \cdot x}{6} - \color{blue}{\frac{3}{6}} \]

   2. sub-neg N/A

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

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

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

   4. associate-/l* N/A

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

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

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

   6. /-lowering-/.f64 N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0

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

7. Simplified 54.1%

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

Reproduce

herbie shell --seed 2024138 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, H"
  :precision binary64
  (/ (- (* x x) 3.0) 6.0))