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×

• 24.4% 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

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

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]

Derivation

1. Initial program 99.9%

   \[\frac{x \cdot x - 3}{6} \]

2. Final simplification 99.9%

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

Alternative 2: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{-7}:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-x}{-6}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 4e-7) -0.5 (* x (/ (- x) -6.0))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 4e-7) {
		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) <= 4d-7) 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) <= 4e-7) {
		tmp = -0.5;
	} else {
		tmp = x * (-x / -6.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 4e-7:
		tmp = -0.5
	else:
		tmp = x * (-x / -6.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 4e-7)
		tmp = -0.5;
	else
		tmp = Float64(x * Float64(Float64(-x) / -6.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 4e-7)
		tmp = -0.5;
	else
		tmp = x * (-x / -6.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 4e-7], -0.5, N[(x * N[((-x) / -6.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 3.9999999999999998e-7

   1. Initial program 100.0%

      \[\frac{x \cdot x - 3}{6} \]

   2. Taylor expanded in x around 0 99.6%

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

   if 3.9999999999999998e-7 < (*.f64 x x)

   1. Initial program 99.8%

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

      1. div-sub 99.8%

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

      2. associate-*r/ 99.8%

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

      3. fma-neg 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{x}{6}, -\frac{3}{6}\right)} \]

      4. *-lft-identity 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-*r/ 99.8%

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

      7. associate-/l* 99.7%

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

      8. associate-/r/ 99.7%

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

      9. *-commutative 99.7%

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

      10. metadata-eval 99.7%

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

      11. metadata-eval 99.7%

          \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{0.16666666666666666}, -\frac{3}{6}\right) \]

      12. metadata-eval 99.7%

          \[\leadsto \mathsf{fma}\left(x, x \cdot 0.16666666666666666, -\color{blue}{0.5}\right) \]

      13. metadata-eval 99.7%

          \[\leadsto \mathsf{fma}\left(x, x \cdot 0.16666666666666666, \color{blue}{-0.5}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 0.16666666666666666, -0.5\right)} \]
   4. Step-by-step derivation

      1. metadata-eval 99.7%

         \[\leadsto \mathsf{fma}\left(x, x \cdot 0.16666666666666666, \color{blue}{-0.5}\right) \]

      2. metadata-eval 99.7%

         \[\leadsto \mathsf{fma}\left(x, x \cdot 0.16666666666666666, -\color{blue}{\frac{3}{6}}\right) \]

      3. fma-neg 99.7%

         \[\leadsto \color{blue}{x \cdot \left(x \cdot 0.16666666666666666\right) - \frac{3}{6}} \]

      4. associate-*r* 99.7%

         \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot 0.16666666666666666} - \frac{3}{6} \]

      5. metadata-eval 99.7%

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

      6. div-inv 99.8%

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

      7. div-sub 99.8%

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

      8. clear-num 99.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{6}{x \cdot x - 3}}} \]

      9. clear-num 99.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{6}{x \cdot x - 3}}{1}}} \]

      10. /-rgt-identity 99.8%

          \[\leadsto \frac{1}{\color{blue}{\frac{6}{x \cdot x - 3}}} \]

      11. fma-neg 99.8%

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

      12. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around inf 98.7%

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

      1. unpow2 98.7%

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

   8. Simplified 98.7%

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

      1. associate-/r* 98.7%

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

      2. clear-num 98.7%

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

      3. frac-2neg 98.7%

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

      4. associate-/r/ 98.7%

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

      5. metadata-eval 98.7%

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

   10. Applied egg-rr 98.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{-7}:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-x}{-6}\\ \end{array} \]

Alternative 3: 98.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{-7}:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 4e-7) -0.5 (* (* x x) 0.16666666666666666)))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 4e-7) {
		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) <= 4d-7) 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) <= 4e-7) {
		tmp = -0.5;
	} else {
		tmp = (x * x) * 0.16666666666666666;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 4e-7:
		tmp = -0.5
	else:
		tmp = (x * x) * 0.16666666666666666
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 4e-7)
		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) <= 4e-7)
		tmp = -0.5;
	else
		tmp = (x * x) * 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 4e-7], -0.5, N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 3.9999999999999998e-7

   1. Initial program 100.0%

      \[\frac{x \cdot x - 3}{6} \]

   2. Taylor expanded in x around 0 99.6%

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

   if 3.9999999999999998e-7 < (*.f64 x x)

   1. Initial program 99.8%

      \[\frac{x \cdot x - 3}{6} \]

   2. Taylor expanded in x around inf 98.7%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot {x}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 98.7%

         \[\leadsto 0.16666666666666666 \cdot \color{blue}{\left(x \cdot x\right)} \]

   4. Simplified 98.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{-7}:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \]

Alternative 4: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot 0.16666666666666666\right) - 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (* x (* x 0.16666666666666666)) 0.5))

C

double code(double x) {
	return (x * (x * 0.16666666666666666)) - 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * (x * 0.16666666666666666d0)) - 0.5d0
end function

Java

public static double code(double x) {
	return (x * (x * 0.16666666666666666)) - 0.5;
}

Python

def code(x):
	return (x * (x * 0.16666666666666666)) - 0.5

Julia

function code(x)
	return Float64(Float64(x * Float64(x * 0.16666666666666666)) - 0.5)
end

MATLAB

function tmp = code(x)
	tmp = (x * (x * 0.16666666666666666)) - 0.5;
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. div-sub 99.9%

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

   2. div-inv 99.8%

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

   3. metadata-eval 99.8%

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{0.16666666666666666} - \frac{3}{6} \]

   4. associate-*r* 99.8%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot 0.16666666666666666\right)} - \frac{3}{6} \]

   5. metadata-eval 99.8%

      \[\leadsto x \cdot \left(x \cdot 0.16666666666666666\right) - \color{blue}{0.5} \]

3. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot 0.16666666666666666\right) - 0.5} \]

4. Final simplification 99.8%

   \[\leadsto x \cdot \left(x \cdot 0.16666666666666666\right) - 0.5 \]

Alternative 5: 50.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;9\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.7) -0.5 9.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.7) {
		tmp = -0.5;
	} else {
		tmp = 9.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.7d0) then
        tmp = -0.5d0
    else
        tmp = 9.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.7) {
		tmp = -0.5;
	} else {
		tmp = 9.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.7:
		tmp = -0.5
	else:
		tmp = 9.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.7)
		tmp = -0.5;
	else
		tmp = 9.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.7)
		tmp = -0.5;
	else
		tmp = 9.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.7], -0.5, 9.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.69999999999999996

   1. Initial program 99.9%

      \[\frac{x \cdot x - 3}{6} \]

   2. Taylor expanded in x around 0 64.5%

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

   if 1.69999999999999996 < x

   1. Initial program 99.8%

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

      1. div-sub 99.8%

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

      2. associate-*r/ 99.8%

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

      3. fma-neg 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{x}{6}, -\frac{3}{6}\right)} \]

      4. *-lft-identity 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-*r/ 99.8%

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

      7. associate-/l* 99.7%

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

      8. associate-/r/ 99.7%

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

      9. *-commutative 99.7%

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

      10. metadata-eval 99.7%

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

      11. metadata-eval 99.7%

          \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{0.16666666666666666}, -\frac{3}{6}\right) \]

      12. metadata-eval 99.7%

          \[\leadsto \mathsf{fma}\left(x, x \cdot 0.16666666666666666, -\color{blue}{0.5}\right) \]

      13. metadata-eval 99.7%

          \[\leadsto \mathsf{fma}\left(x, x \cdot 0.16666666666666666, \color{blue}{-0.5}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 0.16666666666666666, -0.5\right)} \]
   4. Step-by-step derivation

      1. metadata-eval 99.7%

         \[\leadsto \mathsf{fma}\left(x, x \cdot 0.16666666666666666, \color{blue}{-0.5}\right) \]

      2. metadata-eval 99.7%

         \[\leadsto \mathsf{fma}\left(x, x \cdot 0.16666666666666666, -\color{blue}{\frac{3}{6}}\right) \]

      3. fma-neg 99.7%

         \[\leadsto \color{blue}{x \cdot \left(x \cdot 0.16666666666666666\right) - \frac{3}{6}} \]

      4. associate-*r* 99.7%

         \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot 0.16666666666666666} - \frac{3}{6} \]

      5. metadata-eval 99.7%

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

      6. div-inv 99.8%

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

      7. div-sub 99.8%

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

      8. clear-num 99.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{6}{x \cdot x - 3}}} \]

      9. clear-num 99.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{6}{x \cdot x - 3}}{1}}} \]

      10. /-rgt-identity 99.7%

          \[\leadsto \frac{1}{\color{blue}{\frac{6}{x \cdot x - 3}}} \]

      11. fma-neg 99.7%

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

      12. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x around inf 97.8%

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

      1. unpow2 97.8%

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

   8. Simplified 97.8%

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

      1. associate-/r* 97.7%

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

      2. clear-num 97.9%

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

      3. frac-2neg 97.9%

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

      4. associate-/r/ 97.9%

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

      5. metadata-eval 97.9%

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

   10. Applied egg-rr 97.9%

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

   12. Applied egg-rr 5.0%

       \[\leadsto \color{blue}{9} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;9\\ \end{array} \]

Alternative 6: 50.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;216\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.7) -0.5 216.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.7) {
		tmp = -0.5;
	} else {
		tmp = 216.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.7d0) then
        tmp = -0.5d0
    else
        tmp = 216.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.7) {
		tmp = -0.5;
	} else {
		tmp = 216.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.7:
		tmp = -0.5
	else:
		tmp = 216.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.7)
		tmp = -0.5;
	else
		tmp = 216.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.7)
		tmp = -0.5;
	else
		tmp = 216.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.7], -0.5, 216.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.69999999999999996

   1. Initial program 99.9%

      \[\frac{x \cdot x - 3}{6} \]

   2. Taylor expanded in x around 0 64.5%

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

   if 1.69999999999999996 < x

   1. Initial program 99.8%

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

      1. div-sub 99.8%

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

      2. associate-*r/ 99.8%

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

      3. fma-neg 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{x}{6}, -\frac{3}{6}\right)} \]

      4. *-lft-identity 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-*r/ 99.8%

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

      7. associate-/l* 99.7%

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

      8. associate-/r/ 99.7%

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

      9. *-commutative 99.7%

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

      10. metadata-eval 99.7%

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

      11. metadata-eval 99.7%

          \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{0.16666666666666666}, -\frac{3}{6}\right) \]

      12. metadata-eval 99.7%

          \[\leadsto \mathsf{fma}\left(x, x \cdot 0.16666666666666666, -\color{blue}{0.5}\right) \]

      13. metadata-eval 99.7%

          \[\leadsto \mathsf{fma}\left(x, x \cdot 0.16666666666666666, \color{blue}{-0.5}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 0.16666666666666666, -0.5\right)} \]
   4. Step-by-step derivation

      1. metadata-eval 99.7%

         \[\leadsto \mathsf{fma}\left(x, x \cdot 0.16666666666666666, \color{blue}{-0.5}\right) \]

      2. metadata-eval 99.7%

         \[\leadsto \mathsf{fma}\left(x, x \cdot 0.16666666666666666, -\color{blue}{\frac{3}{6}}\right) \]

      3. fma-neg 99.7%

         \[\leadsto \color{blue}{x \cdot \left(x \cdot 0.16666666666666666\right) - \frac{3}{6}} \]

      4. associate-*r* 99.7%

         \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot 0.16666666666666666} - \frac{3}{6} \]

      5. metadata-eval 99.7%

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

      6. div-inv 99.8%

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

      7. div-sub 99.8%

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

      8. clear-num 99.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{6}{x \cdot x - 3}}} \]

      9. clear-num 99.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{6}{x \cdot x - 3}}{1}}} \]

      10. /-rgt-identity 99.7%

          \[\leadsto \frac{1}{\color{blue}{\frac{6}{x \cdot x - 3}}} \]

      11. fma-neg 99.7%

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

      12. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x around inf 97.8%

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

      1. unpow2 97.8%

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

   8. Simplified 97.8%

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

      1. associate-/r* 97.7%

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

      2. clear-num 97.9%

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

      3. frac-2neg 97.9%

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

      4. associate-/r/ 97.9%

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

      5. metadata-eval 97.9%

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

   10. Applied egg-rr 97.9%

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

   12. Applied egg-rr 5.0%

       \[\leadsto \color{blue}{216} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;216\\ \end{array} \]

Alternative 7: 50.0% 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. Taylor expanded in x around 0 46.4%

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

3. Final simplification 46.4%

   \[\leadsto -0.5 \]

Reproduce

herbie shell --seed 2023297 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, H"
  :precision binary64
  (/ (- (* x x) 3.0) 6.0))