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

Percentage Accurate: 99.9% → 99.9%

Time: 4.7s

Alternatives: 6

Speedup: 1.0×

• 25.4% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sqr-neg 99.8%

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

   2. *-lft-identity 99.8%

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

   3. metadata-eval 99.8%

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

   4. associate-*r/ 99.8%

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

   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 99.8%

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

   5. clear-num 99.8%

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

   6. fma-neg 99.8%

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

   7. 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. Step-by-step derivation

   1. clear-num 99.8%

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

   2. associate-/r/ 99.8%

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

7. Applied egg-rr 99.8%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{fma}\left(x, x, -3\right)} \cdot 6}} \]
8. Step-by-step derivation

   1. associate-*l/ 99.8%

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

   2. metadata-eval 99.8%

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

   3. associate-/r/ 99.8%

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

   4. metadata-eval 99.8%

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

   5. fma-udef 99.8%

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

   6. unpow2 99.8%

      \[\leadsto 0.16666666666666666 \cdot \left(\color{blue}{{x}^{2}} + -3\right) \]

   7. distribute-lft-in 99.8%

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

   8. unpow2 99.8%

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

   9. associate-*r* 99.8%

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

   10. metadata-eval 99.8%

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

   11. metadata-eval 99.8%

       \[\leadsto \left(0.16666666666666666 \cdot x\right) \cdot x + \color{blue}{\frac{-1}{2}} \]

   12. fma-def 99.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666 \cdot x, x, \frac{-1}{2}\right)} \]

   13. metadata-eval 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.16666666666666666 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.75) -0.5 (* x (* 0.16666666666666666 x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.75) {
		tmp = -0.5;
	} else {
		tmp = x * (0.16666666666666666 * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.75d0) then
        tmp = -0.5d0
    else
        tmp = x * (0.16666666666666666d0 * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.75) {
		tmp = -0.5;
	} else {
		tmp = x * (0.16666666666666666 * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.75:
		tmp = -0.5
	else:
		tmp = x * (0.16666666666666666 * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.75)
		tmp = -0.5;
	else
		tmp = Float64(x * Float64(0.16666666666666666 * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.75)
		tmp = -0.5;
	else
		tmp = x * (0.16666666666666666 * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.75], -0.5, N[(x * N[(0.16666666666666666 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.75

   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* 99.9%

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

      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 58.7%

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

   if 1.75 < 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/ 99.6%

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

      7. *-commutative 99.6%

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

      8. sqr-neg 99.6%

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

      9. fma-neg 99.6%

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

      10. metadata-eval 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. metadata-eval 99.6%

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

      2. fma-neg 99.6%

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

      3. metadata-eval 99.6%

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

      4. div-inv 99.7%

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

      5. clear-num 99.7%

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

      6. fma-neg 99.7%

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

      7. 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 98.9%

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

      1. associate-/r/ 98.9%

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

      2. metadata-eval 98.9%

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

      3. unpow2 98.9%

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

      4. associate-*r* 99.1%

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

   8. Applied egg-rr 99.1%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.16666666666666666 \cdot x\right)\\ \end{array} \]

Alternative 3: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{6}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.75) -0.5 (* x (/ x 6.0))))

C

double code(double x) {
	double tmp;
	if (x <= 1.75) {
		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 <= 1.75d0) 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 <= 1.75) {
		tmp = -0.5;
	} else {
		tmp = x * (x / 6.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.75:
		tmp = -0.5
	else:
		tmp = x * (x / 6.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.75)
		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 <= 1.75)
		tmp = -0.5;
	else
		tmp = x * (x / 6.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.75], -0.5, N[(x * N[(x / 6.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.75

   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* 99.9%

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

      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 58.7%

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

   if 1.75 < 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/ 99.6%

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

      7. *-commutative 99.6%

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

      8. sqr-neg 99.6%

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

      9. fma-neg 99.6%

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

      10. metadata-eval 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 98.9%

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

      1. rem-exp-log 98.6%

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

      2. add-sqr-sqrt 98.6%

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

      3. sqrt-unprod 69.7%

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

      4. *-commutative 69.7%

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

      5. *-commutative 69.7%

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

      6. swap-sqr 69.8%

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

      7. pow-prod-up 69.8%

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

      8. metadata-eval 69.8%

         \[\leadsto \sqrt{{x}^{\color{blue}{4}} \cdot \left(e^{\log 0.16666666666666666} \cdot e^{\log 0.16666666666666666}\right)} \]

      9. rem-exp-log 69.9%

         \[\leadsto \sqrt{{x}^{4} \cdot \left(\color{blue}{0.16666666666666666} \cdot e^{\log 0.16666666666666666}\right)} \]

      10. rem-exp-log 69.9%

          \[\leadsto \sqrt{{x}^{4} \cdot \left(0.16666666666666666 \cdot \color{blue}{0.16666666666666666}\right)} \]

      11. metadata-eval 69.9%

          \[\leadsto \sqrt{{x}^{4} \cdot \color{blue}{0.027777777777777776}} \]

   6. Applied egg-rr 69.9%

      \[\leadsto \color{blue}{\sqrt{{x}^{4} \cdot 0.027777777777777776}} \]
   7. Step-by-step derivation

      1. sqrt-prod 69.9%

         \[\leadsto \color{blue}{\sqrt{{x}^{4}} \cdot \sqrt{0.027777777777777776}} \]

      2. metadata-eval 69.9%

         \[\leadsto \sqrt{{x}^{4}} \cdot \color{blue}{0.16666666666666666} \]

      3. sqrt-pow1 98.9%

         \[\leadsto \color{blue}{{x}^{\left(\frac{4}{2}\right)}} \cdot 0.16666666666666666 \]

      4. metadata-eval 98.9%

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

      5. metadata-eval 98.9%

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

      6. div-inv 99.0%

         \[\leadsto \color{blue}{\frac{{x}^{2}}{6}} \]

      7. unpow2 99.0%

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

      8. associate-/l* 99.0%

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

   8. Applied egg-rr 99.0%

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

      1. associate-/r/ 99.1%

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

   10. Simplified 99.1%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{6}\\ \end{array} \]

Alternative 4: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{6}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.75) -0.5 (/ x (/ 6.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.75) {
		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 <= 1.75d0) 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 <= 1.75) {
		tmp = -0.5;
	} else {
		tmp = x / (6.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.75:
		tmp = -0.5
	else:
		tmp = x / (6.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.75)
		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 <= 1.75)
		tmp = -0.5;
	else
		tmp = x / (6.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.75], -0.5, N[(x / N[(6.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.75

   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* 99.9%

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

      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 58.7%

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

   if 1.75 < 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/ 99.6%

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

      7. *-commutative 99.6%

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

      8. sqr-neg 99.6%

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

      9. fma-neg 99.6%

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

      10. metadata-eval 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 98.9%

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

      1. rem-exp-log 98.6%

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

      2. add-sqr-sqrt 98.6%

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

      3. sqrt-unprod 69.7%

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

      4. *-commutative 69.7%

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

      5. *-commutative 69.7%

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

      6. swap-sqr 69.8%

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

      7. pow-prod-up 69.8%

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

      8. metadata-eval 69.8%

         \[\leadsto \sqrt{{x}^{\color{blue}{4}} \cdot \left(e^{\log 0.16666666666666666} \cdot e^{\log 0.16666666666666666}\right)} \]

      9. rem-exp-log 69.9%

         \[\leadsto \sqrt{{x}^{4} \cdot \left(\color{blue}{0.16666666666666666} \cdot e^{\log 0.16666666666666666}\right)} \]

      10. rem-exp-log 69.9%

          \[\leadsto \sqrt{{x}^{4} \cdot \left(0.16666666666666666 \cdot \color{blue}{0.16666666666666666}\right)} \]

      11. metadata-eval 69.9%

          \[\leadsto \sqrt{{x}^{4} \cdot \color{blue}{0.027777777777777776}} \]

   6. Applied egg-rr 69.9%

      \[\leadsto \color{blue}{\sqrt{{x}^{4} \cdot 0.027777777777777776}} \]
   7. Step-by-step derivation

      1. sqrt-prod 69.9%

         \[\leadsto \color{blue}{\sqrt{{x}^{4}} \cdot \sqrt{0.027777777777777776}} \]

      2. metadata-eval 69.9%

         \[\leadsto \sqrt{{x}^{4}} \cdot \color{blue}{0.16666666666666666} \]

      3. sqrt-pow1 98.9%

         \[\leadsto \color{blue}{{x}^{\left(\frac{4}{2}\right)}} \cdot 0.16666666666666666 \]

      4. metadata-eval 98.9%

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

      5. metadata-eval 98.9%

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

      6. div-inv 99.0%

         \[\leadsto \color{blue}{\frac{{x}^{2}}{6}} \]

      7. unpow2 99.0%

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

      8. associate-/l* 99.0%

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

   8. Applied egg-rr 99.0%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{6}{x}}\\ \end{array} \]

Alternative 5: 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 99.8%

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

2. Final simplification 99.8%

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

Alternative 6: 49.3% 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. sqr-neg 99.8%

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

   2. *-lft-identity 99.8%

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

   3. metadata-eval 99.8%

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

   4. associate-*r/ 99.8%

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

   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 43.5%

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

5. Final simplification 43.5%

   \[\leadsto -0.5 \]

Reproduce

herbie shell --seed 2023336 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, H"
  :precision binary64
  (/ (- (* x x) 3.0) 6.0))