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

Percentage Accurate: 99.8% → 99.8%

Time: 3.8s

Alternatives: 5

Speedup: 1.0×

• 25.2% 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: 99.8% 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]

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, 0.8× 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. sqr-neg 100.0%

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

      2. *-lft-identity 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-*r/ 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/r/ 100.0%

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

      7. *-commutative 100.0%

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

      8. sqr-neg 100.0%

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

      9. fma-neg 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 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. sqr-neg 99.7%

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

      2. *-lft-identity 99.7%

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

      3. metadata-eval 99.7%

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

      4. associate-*r/ 99.7%

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

      5. associate-/l* 99.7%

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

      6. associate-/r/ 99.7%

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

      7. *-commutative 99.7%

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

      8. sqr-neg 99.7%

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

      9. fma-neg 99.7%

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

      10. metadata-eval 99.7%

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

      11. metadata-eval 99.7%

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

      12. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around inf 98.0%

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

      1. unpow2 98.0%

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

   6. Applied egg-rr 98.0%

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

4. Final simplification 98.5%

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

Alternative 3: 98.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.001:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 0.001) -0.5 (* x (* x 0.16666666666666666))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.001) {
		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) <= 0.001d0) 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) <= 0.001) {
		tmp = -0.5;
	} else {
		tmp = x * (x * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.001:
		tmp = -0.5
	else:
		tmp = x * (x * 0.16666666666666666)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.001)
		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) <= 0.001)
		tmp = -0.5;
	else
		tmp = x * (x * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.001], -0.5, N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1e-3

   1. Initial program 100.0%

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

      1. sqr-neg 100.0%

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

      2. *-lft-identity 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-*r/ 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/r/ 100.0%

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

      7. *-commutative 100.0%

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

      8. sqr-neg 100.0%

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

      9. fma-neg 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.0%

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

   if 1e-3 < (*.f64 x x)

   1. Initial program 99.7%

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

      1. sqr-neg 99.7%

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

      2. *-lft-identity 99.7%

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

      3. metadata-eval 99.7%

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

      4. associate-*r/ 99.7%

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

      5. associate-/l* 99.7%

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

      6. associate-/r/ 99.7%

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

      7. *-commutative 99.7%

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

      8. sqr-neg 99.7%

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

      9. fma-neg 99.7%

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

      10. metadata-eval 99.7%

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

      11. metadata-eval 99.7%

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

      12. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around inf 98.0%

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

      1. *-commutative 98.0%

         \[\leadsto \color{blue}{{x}^{2} \cdot 0.16666666666666666} \]

      2. add-sqr-sqrt 97.8%

         \[\leadsto \color{blue}{\sqrt{{x}^{2} \cdot 0.16666666666666666} \cdot \sqrt{{x}^{2} \cdot 0.16666666666666666}} \]

      3. pow2 97.8%

         \[\leadsto \color{blue}{{\left(\sqrt{{x}^{2} \cdot 0.16666666666666666}\right)}^{2}} \]

      4. sqrt-prod 97.9%

         \[\leadsto {\color{blue}{\left(\sqrt{{x}^{2}} \cdot \sqrt{0.16666666666666666}\right)}}^{2} \]

      5. unpow2 97.9%

         \[\leadsto {\left(\sqrt{\color{blue}{x \cdot x}} \cdot \sqrt{0.16666666666666666}\right)}^{2} \]

      6. sqrt-prod 49.4%

         \[\leadsto {\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \sqrt{0.16666666666666666}\right)}^{2} \]

      7. add-sqr-sqrt 97.9%

         \[\leadsto {\left(\color{blue}{x} \cdot \sqrt{0.16666666666666666}\right)}^{2} \]

   6. Applied egg-rr 97.9%

      \[\leadsto \color{blue}{{\left(x \cdot \sqrt{0.16666666666666666}\right)}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 97.9%

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

      2. swap-sqr 98.0%

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

      3. rem-square-sqrt 98.0%

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

      4. associate-*l* 98.1%

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

      5. *-commutative 98.1%

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

   8. Applied egg-rr 98.1%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.001:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \]

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

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(Float64(x * x) * 0.16666666666666666) - 0.5)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. sqr-neg 99.9%

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

   2. *-lft-identity 99.9%

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

   3. metadata-eval 99.9%

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

   4. associate-*r/ 99.9%

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

   5. associate-/l* 99.8%

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

   6. associate-/r/ 99.8%

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

   7. *-commutative 99.8%

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

   8. sqr-neg 99.8%

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

   9. fma-neg 99.8%

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

   10. metadata-eval 99.8%

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

   11. metadata-eval 99.8%

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

   12. metadata-eval 99.8%

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

3. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, -3\right) \cdot 0.16666666666666666} \]
4. Step-by-step derivation

   1. metadata-eval 99.8%

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

   2. fma-neg 99.8%

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

   3. metadata-eval 99.8%

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

   4. div-inv 99.9%

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

   5. div-sub 99.9%

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

   6. div-inv 99.8%

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

   7. pow2 99.8%

      \[\leadsto \color{blue}{{x}^{2}} \cdot \frac{1}{6} - \frac{3}{6} \]

   8. metadata-eval 99.8%

      \[\leadsto {x}^{2} \cdot \color{blue}{0.16666666666666666} - \frac{3}{6} \]

   9. metadata-eval 99.8%

      \[\leadsto {x}^{2} \cdot 0.16666666666666666 - \color{blue}{0.5} \]

5. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{{x}^{2} \cdot 0.16666666666666666 - 0.5} \]
6. Step-by-step derivation

   1. unpow2 51.1%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 5: 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. Step-by-step derivation

   1. sqr-neg 99.9%

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

   2. *-lft-identity 99.9%

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

   3. metadata-eval 99.9%

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

   4. associate-*r/ 99.9%

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

   5. associate-/l* 99.8%

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

   6. associate-/r/ 99.8%

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

   7. *-commutative 99.8%

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

   8. sqr-neg 99.8%

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

   9. fma-neg 99.8%

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

   10. metadata-eval 99.8%

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

   11. metadata-eval 99.8%

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

   12. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around 0 49.2%

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

5. Final simplification 49.2%

   \[\leadsto -0.5 \]

Reproduce

herbie shell --seed 2023320 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, H"
  :precision binary64
  (/ (- (* x x) 3.0) 6.0))