Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A

Percentage Accurate: 99.9% → 99.7%

Time: 7.0s

Alternatives: 7

Speedup: 1.0×

• 25.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ 1 - x \cdot \left(0.253 + x \cdot 0.12\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (* x (+ 0.253 (* x 0.12)))))

C

double code(double x) {
	return 1.0 - (x * (0.253 + (x * 0.12)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (x * (0.253d0 + (x * 0.12d0)))
end function

Java

public static double code(double x) {
	return 1.0 - (x * (0.253 + (x * 0.12)));
}

Python

def code(x):
	return 1.0 - (x * (0.253 + (x * 0.12)))

Julia

function code(x)
	return Float64(1.0 - Float64(x * Float64(0.253 + Float64(x * 0.12))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - (x * (0.253 + (x * 0.12)));
end

Wolfram

code[x_] := N[(1.0 - N[(x * N[(0.253 + N[(x * 0.12), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - x \cdot \left(0.253 + x \cdot 0.12\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (* x (+ 0.253 (* x 0.12)))))

C

double code(double x) {
	return 1.0 - (x * (0.253 + (x * 0.12)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (x * (0.253d0 + (x * 0.12d0)))
end function

Java

public static double code(double x) {
	return 1.0 - (x * (0.253 + (x * 0.12)));
}

Python

def code(x):
	return 1.0 - (x * (0.253 + (x * 0.12)))

Julia

function code(x)
	return Float64(1.0 - Float64(x * Float64(0.253 + Float64(x * 0.12))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - (x * (0.253 + (x * 0.12)));
end

Wolfram

code[x_] := N[(1.0 - N[(x * N[(0.253 + N[(x * 0.12), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{x}{\mathsf{fma}\left(x, -0.12, 0.253\right)} \cdot \left({x}^{2} \cdot 0.0144 - 0.064009\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ 1.0 (* (/ x (fma x -0.12 0.253)) (- (* (pow x 2.0) 0.0144) 0.064009))))

C

double code(double x) {
	return 1.0 + ((x / fma(x, -0.12, 0.253)) * ((pow(x, 2.0) * 0.0144) - 0.064009));
}

Julia

function code(x)
	return Float64(1.0 + Float64(Float64(x / fma(x, -0.12, 0.253)) * Float64(Float64((x ^ 2.0) * 0.0144) - 0.064009)))
end

Wolfram

code[x_] := N[(1.0 + N[(N[(x / N[(x * -0.12 + 0.253), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[x, 2.0], $MachinePrecision] * 0.0144), $MachinePrecision] - 0.064009), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative 99.8%

      \[\leadsto 1 - \color{blue}{\left(0.253 + x \cdot 0.12\right) \cdot x} \]

   2. flip-+ 99.8%

      \[\leadsto 1 - \color{blue}{\frac{0.253 \cdot 0.253 - \left(x \cdot 0.12\right) \cdot \left(x \cdot 0.12\right)}{0.253 - x \cdot 0.12}} \cdot x \]

   3. associate-*l/ 93.0%

      \[\leadsto 1 - \color{blue}{\frac{\left(0.253 \cdot 0.253 - \left(x \cdot 0.12\right) \cdot \left(x \cdot 0.12\right)\right) \cdot x}{0.253 - x \cdot 0.12}} \]

   4. metadata-eval 93.0%

      \[\leadsto 1 - \frac{\left(\color{blue}{0.064009} - \left(x \cdot 0.12\right) \cdot \left(x \cdot 0.12\right)\right) \cdot x}{0.253 - x \cdot 0.12} \]

   5. swap-sqr 93.0%

      \[\leadsto 1 - \frac{\left(0.064009 - \color{blue}{\left(x \cdot x\right) \cdot \left(0.12 \cdot 0.12\right)}\right) \cdot x}{0.253 - x \cdot 0.12} \]

   6. pow2 93.0%

      \[\leadsto 1 - \frac{\left(0.064009 - \color{blue}{{x}^{2}} \cdot \left(0.12 \cdot 0.12\right)\right) \cdot x}{0.253 - x \cdot 0.12} \]

   7. metadata-eval 93.0%

      \[\leadsto 1 - \frac{\left(0.064009 - {x}^{2} \cdot \color{blue}{0.0144}\right) \cdot x}{0.253 - x \cdot 0.12} \]

   8. *-commutative 93.0%

      \[\leadsto 1 - \frac{\left(0.064009 - {x}^{2} \cdot 0.0144\right) \cdot x}{0.253 - \color{blue}{0.12 \cdot x}} \]

   9. cancel-sign-sub-inv 93.0%

      \[\leadsto 1 - \frac{\left(0.064009 - {x}^{2} \cdot 0.0144\right) \cdot x}{\color{blue}{0.253 + \left(-0.12\right) \cdot x}} \]

   10. metadata-eval 93.0%

       \[\leadsto 1 - \frac{\left(0.064009 - {x}^{2} \cdot 0.0144\right) \cdot x}{0.253 + \color{blue}{-0.12} \cdot x} \]

   11. *-commutative 93.0%

       \[\leadsto 1 - \frac{\left(0.064009 - {x}^{2} \cdot 0.0144\right) \cdot x}{0.253 + \color{blue}{x \cdot -0.12}} \]

4. Applied egg-rr 93.0%

   \[\leadsto 1 - \color{blue}{\frac{\left(0.064009 - {x}^{2} \cdot 0.0144\right) \cdot x}{0.253 + x \cdot -0.12}} \]
5. Step-by-step derivation

   1. associate-/l* 99.9%

      \[\leadsto 1 - \color{blue}{\left(0.064009 - {x}^{2} \cdot 0.0144\right) \cdot \frac{x}{0.253 + x \cdot -0.12}} \]

   2. *-commutative 99.9%

      \[\leadsto 1 - \color{blue}{\frac{x}{0.253 + x \cdot -0.12} \cdot \left(0.064009 - {x}^{2} \cdot 0.0144\right)} \]

   3. +-commutative 99.9%

      \[\leadsto 1 - \frac{x}{\color{blue}{x \cdot -0.12 + 0.253}} \cdot \left(0.064009 - {x}^{2} \cdot 0.0144\right) \]

   4. fma-define 99.9%

      \[\leadsto 1 - \frac{x}{\color{blue}{\mathsf{fma}\left(x, -0.12, 0.253\right)}} \cdot \left(0.064009 - {x}^{2} \cdot 0.0144\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto 1 - \color{blue}{\frac{x}{\mathsf{fma}\left(x, -0.12, 0.253\right)} \cdot \left(0.064009 - {x}^{2} \cdot 0.0144\right)} \]

7. Final simplification 99.9%

   \[\leadsto 1 + \frac{x}{\mathsf{fma}\left(x, -0.12, 0.253\right)} \cdot \left({x}^{2} \cdot 0.0144 - 0.064009\right) \]
8. Add Preprocessing

Alternative 2: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -4.2) (not (<= x 2.0))) (* x (* x -0.12)) 1.0))

C

double code(double x) {
	double tmp;
	if ((x <= -4.2) || !(x <= 2.0)) {
		tmp = x * (x * -0.12);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-4.2d0)) .or. (.not. (x <= 2.0d0))) then
        tmp = x * (x * (-0.12d0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -4.2) || !(x <= 2.0)) {
		tmp = x * (x * -0.12);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -4.2) or not (x <= 2.0):
		tmp = x * (x * -0.12)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -4.2) || !(x <= 2.0))
		tmp = Float64(x * Float64(x * -0.12));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -4.2) || ~((x <= 2.0)))
		tmp = x * (x * -0.12);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -4.2], N[Not[LessEqual[x, 2.0]], $MachinePrecision]], N[(x * N[(x * -0.12), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -4.20000000000000018 or 2 < x

   1. Initial program 99.7%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
   2. Step-by-step derivation

      1. sub-neg 99.7%

         \[\leadsto \color{blue}{1 + \left(-x \cdot \left(0.253 + x \cdot 0.12\right)\right)} \]

      2. +-commutative 99.7%

         \[\leadsto \color{blue}{\left(-x \cdot \left(0.253 + x \cdot 0.12\right)\right) + 1} \]

      3. distribute-rgt-neg-in 99.7%

         \[\leadsto \color{blue}{x \cdot \left(-\left(0.253 + x \cdot 0.12\right)\right)} + 1 \]

      4. fma-define 99.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, -\left(0.253 + x \cdot 0.12\right), 1\right)} \]

      5. +-commutative 99.7%

         \[\leadsto \mathsf{fma}\left(x, -\color{blue}{\left(x \cdot 0.12 + 0.253\right)}, 1\right) \]

      6. distribute-neg-in 99.7%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(-x \cdot 0.12\right) + \left(-0.253\right)}, 1\right) \]

      7. distribute-rgt-neg-in 99.7%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(-0.12\right)} + \left(-0.253\right), 1\right) \]

      8. metadata-eval 99.7%

         \[\leadsto \mathsf{fma}\left(x, x \cdot \left(-0.12\right) + \color{blue}{-0.253}, 1\right) \]

      9. metadata-eval 99.7%

         \[\leadsto \mathsf{fma}\left(x, x \cdot \left(-0.12\right) + \color{blue}{0.253 \cdot -1}, 1\right) \]

      10. fma-define 99.7%

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.12, 0.253 \cdot -1\right)}, 1\right) \]

      11. metadata-eval 99.7%

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{-0.12}, 0.253 \cdot -1\right), 1\right) \]

      12. metadata-eval 99.7%

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.12, \color{blue}{-0.253}\right), 1\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.12, -0.253\right), 1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.1%

      \[\leadsto \color{blue}{-0.12 \cdot {x}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 98.1%

         \[\leadsto \color{blue}{{x}^{2} \cdot -0.12} \]

   7. Simplified 98.1%

      \[\leadsto \color{blue}{{x}^{2} \cdot -0.12} \]
   8. Step-by-step derivation

      1. *-commutative 98.1%

         \[\leadsto \color{blue}{-0.12 \cdot {x}^{2}} \]

      2. unpow2 98.1%

         \[\leadsto -0.12 \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* 98.1%

         \[\leadsto \color{blue}{\left(-0.12 \cdot x\right) \cdot x} \]

   9. Applied egg-rr 98.1%

      \[\leadsto \color{blue}{\left(-0.12 \cdot x\right) \cdot x} \]

   if -4.20000000000000018 < x < 2

   1. Initial program 100.0%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \color{blue}{1 + \left(-x \cdot \left(0.253 + x \cdot 0.12\right)\right)} \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(-x \cdot \left(0.253 + x \cdot 0.12\right)\right) + 1} \]

      3. distribute-rgt-neg-in 100.0%

         \[\leadsto \color{blue}{x \cdot \left(-\left(0.253 + x \cdot 0.12\right)\right)} + 1 \]

      4. fma-define 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, -\left(0.253 + x \cdot 0.12\right), 1\right)} \]

      5. +-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(x, -\color{blue}{\left(x \cdot 0.12 + 0.253\right)}, 1\right) \]

      6. distribute-neg-in 100.0%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(-x \cdot 0.12\right) + \left(-0.253\right)}, 1\right) \]

      7. distribute-rgt-neg-in 100.0%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(-0.12\right)} + \left(-0.253\right), 1\right) \]

      8. metadata-eval 100.0%

         \[\leadsto \mathsf{fma}\left(x, x \cdot \left(-0.12\right) + \color{blue}{-0.253}, 1\right) \]

      9. metadata-eval 100.0%

         \[\leadsto \mathsf{fma}\left(x, x \cdot \left(-0.12\right) + \color{blue}{0.253 \cdot -1}, 1\right) \]

      10. fma-define 100.0%

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.12, 0.253 \cdot -1\right)}, 1\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{-0.12}, 0.253 \cdot -1\right), 1\right) \]

      12. metadata-eval 100.0%

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.12, \color{blue}{-0.253}\right), 1\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.12, -0.253\right), 1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 97.1%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2:\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;1 - x \cdot 0.253\\ \mathbf{else}:\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.2)
   (* x (* x -0.12))
   (if (<= x 2.0) (- 1.0 (* x 0.253)) (* -0.12 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = x * (x * -0.12);
	} else if (x <= 2.0) {
		tmp = 1.0 - (x * 0.253);
	} else {
		tmp = -0.12 * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4.2d0)) then
        tmp = x * (x * (-0.12d0))
    else if (x <= 2.0d0) then
        tmp = 1.0d0 - (x * 0.253d0)
    else
        tmp = (-0.12d0) * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = x * (x * -0.12);
	} else if (x <= 2.0) {
		tmp = 1.0 - (x * 0.253);
	} else {
		tmp = -0.12 * (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.2:
		tmp = x * (x * -0.12)
	elif x <= 2.0:
		tmp = 1.0 - (x * 0.253)
	else:
		tmp = -0.12 * (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.2)
		tmp = Float64(x * Float64(x * -0.12));
	elseif (x <= 2.0)
		tmp = Float64(1.0 - Float64(x * 0.253));
	else
		tmp = Float64(-0.12 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4.2)
		tmp = x * (x * -0.12);
	elseif (x <= 2.0)
		tmp = 1.0 - (x * 0.253);
	else
		tmp = -0.12 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.2], N[(x * N[(x * -0.12), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.0], N[(1.0 - N[(x * 0.253), $MachinePrecision]), $MachinePrecision], N[(-0.12 * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.20000000000000018

   1. Initial program 99.7%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
   2. Step-by-step derivation

      1. sub-neg 99.7%

         \[\leadsto \color{blue}{1 + \left(-x \cdot \left(0.253 + x \cdot 0.12\right)\right)} \]

      2. +-commutative 99.7%

         \[\leadsto \color{blue}{\left(-x \cdot \left(0.253 + x \cdot 0.12\right)\right) + 1} \]

      3. distribute-rgt-neg-in 99.7%

         \[\leadsto \color{blue}{x \cdot \left(-\left(0.253 + x \cdot 0.12\right)\right)} + 1 \]

      4. fma-define 99.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, -\left(0.253 + x \cdot 0.12\right), 1\right)} \]

      5. +-commutative 99.7%

         \[\leadsto \mathsf{fma}\left(x, -\color{blue}{\left(x \cdot 0.12 + 0.253\right)}, 1\right) \]

      6. distribute-neg-in 99.7%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(-x \cdot 0.12\right) + \left(-0.253\right)}, 1\right) \]

      7. distribute-rgt-neg-in 99.7%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(-0.12\right)} + \left(-0.253\right), 1\right) \]

      8. metadata-eval 99.7%

         \[\leadsto \mathsf{fma}\left(x, x \cdot \left(-0.12\right) + \color{blue}{-0.253}, 1\right) \]

      9. metadata-eval 99.7%

         \[\leadsto \mathsf{fma}\left(x, x \cdot \left(-0.12\right) + \color{blue}{0.253 \cdot -1}, 1\right) \]

      10. fma-define 99.7%

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.12, 0.253 \cdot -1\right)}, 1\right) \]

      11. metadata-eval 99.7%

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{-0.12}, 0.253 \cdot -1\right), 1\right) \]

      12. metadata-eval 99.7%

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.12, \color{blue}{-0.253}\right), 1\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.12, -0.253\right), 1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 99.1%

      \[\leadsto \color{blue}{-0.12 \cdot {x}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 99.1%

         \[\leadsto \color{blue}{{x}^{2} \cdot -0.12} \]

   7. Simplified 99.1%

      \[\leadsto \color{blue}{{x}^{2} \cdot -0.12} \]
   8. Step-by-step derivation

      1. *-commutative 99.1%

         \[\leadsto \color{blue}{-0.12 \cdot {x}^{2}} \]

      2. unpow2 99.1%

         \[\leadsto -0.12 \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* 99.2%

         \[\leadsto \color{blue}{\left(-0.12 \cdot x\right) \cdot x} \]

   9. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\left(-0.12 \cdot x\right) \cdot x} \]

   if -4.20000000000000018 < x < 2

   1. Initial program 100.0%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 98.5%

      \[\leadsto 1 - \color{blue}{0.253 \cdot x} \]
   4. Step-by-step derivation

      1. *-commutative 98.5%

         \[\leadsto 1 - \color{blue}{x \cdot 0.253} \]

   5. Simplified 98.5%

      \[\leadsto 1 - \color{blue}{x \cdot 0.253} \]

   if 2 < x

   1. Initial program 99.7%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
   2. Step-by-step derivation

      1. sub-neg 99.7%

         \[\leadsto \color{blue}{1 + \left(-x \cdot \left(0.253 + x \cdot 0.12\right)\right)} \]

      2. +-commutative 99.7%

         \[\leadsto \color{blue}{\left(-x \cdot \left(0.253 + x \cdot 0.12\right)\right) + 1} \]

      3. distribute-rgt-neg-in 99.7%

         \[\leadsto \color{blue}{x \cdot \left(-\left(0.253 + x \cdot 0.12\right)\right)} + 1 \]

      4. fma-define 99.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, -\left(0.253 + x \cdot 0.12\right), 1\right)} \]

      5. +-commutative 99.7%

         \[\leadsto \mathsf{fma}\left(x, -\color{blue}{\left(x \cdot 0.12 + 0.253\right)}, 1\right) \]

      6. distribute-neg-in 99.7%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(-x \cdot 0.12\right) + \left(-0.253\right)}, 1\right) \]

      7. distribute-rgt-neg-in 99.7%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(-0.12\right)} + \left(-0.253\right), 1\right) \]

      8. metadata-eval 99.7%

         \[\leadsto \mathsf{fma}\left(x, x \cdot \left(-0.12\right) + \color{blue}{-0.253}, 1\right) \]

      9. metadata-eval 99.7%

         \[\leadsto \mathsf{fma}\left(x, x \cdot \left(-0.12\right) + \color{blue}{0.253 \cdot -1}, 1\right) \]

      10. fma-define 99.7%

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.12, 0.253 \cdot -1\right)}, 1\right) \]

      11. metadata-eval 99.7%

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{-0.12}, 0.253 \cdot -1\right), 1\right) \]

      12. metadata-eval 99.7%

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.12, \color{blue}{-0.253}\right), 1\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.12, -0.253\right), 1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 97.2%

      \[\leadsto \color{blue}{-0.12 \cdot {x}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 97.2%

         \[\leadsto \color{blue}{{x}^{2} \cdot -0.12} \]

   7. Simplified 97.2%

      \[\leadsto \color{blue}{{x}^{2} \cdot -0.12} \]
   8. Step-by-step derivation

      1. unpow2 97.2%

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot -0.12 \]

   9. Applied egg-rr 97.2%

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot -0.12 \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2:\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;1 - x \cdot 0.253\\ \mathbf{else}:\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2:\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.2) (* x (* x -0.12)) (if (<= x 2.0) 1.0 (* -0.12 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = x * (x * -0.12);
	} else if (x <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = -0.12 * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4.2d0)) then
        tmp = x * (x * (-0.12d0))
    else if (x <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = (-0.12d0) * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = x * (x * -0.12);
	} else if (x <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = -0.12 * (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.2:
		tmp = x * (x * -0.12)
	elif x <= 2.0:
		tmp = 1.0
	else:
		tmp = -0.12 * (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.2)
		tmp = Float64(x * Float64(x * -0.12));
	elseif (x <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(-0.12 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4.2)
		tmp = x * (x * -0.12);
	elseif (x <= 2.0)
		tmp = 1.0;
	else
		tmp = -0.12 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.2], N[(x * N[(x * -0.12), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.0], 1.0, N[(-0.12 * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.20000000000000018

   1. Initial program 99.7%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
   2. Step-by-step derivation

      1. sub-neg 99.7%

         \[\leadsto \color{blue}{1 + \left(-x \cdot \left(0.253 + x \cdot 0.12\right)\right)} \]

      2. +-commutative 99.7%

         \[\leadsto \color{blue}{\left(-x \cdot \left(0.253 + x \cdot 0.12\right)\right) + 1} \]

      3. distribute-rgt-neg-in 99.7%

         \[\leadsto \color{blue}{x \cdot \left(-\left(0.253 + x \cdot 0.12\right)\right)} + 1 \]

      4. fma-define 99.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, -\left(0.253 + x \cdot 0.12\right), 1\right)} \]

      5. +-commutative 99.7%

         \[\leadsto \mathsf{fma}\left(x, -\color{blue}{\left(x \cdot 0.12 + 0.253\right)}, 1\right) \]

      6. distribute-neg-in 99.7%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(-x \cdot 0.12\right) + \left(-0.253\right)}, 1\right) \]

      7. distribute-rgt-neg-in 99.7%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(-0.12\right)} + \left(-0.253\right), 1\right) \]

      8. metadata-eval 99.7%

         \[\leadsto \mathsf{fma}\left(x, x \cdot \left(-0.12\right) + \color{blue}{-0.253}, 1\right) \]

      9. metadata-eval 99.7%

         \[\leadsto \mathsf{fma}\left(x, x \cdot \left(-0.12\right) + \color{blue}{0.253 \cdot -1}, 1\right) \]

      10. fma-define 99.7%

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.12, 0.253 \cdot -1\right)}, 1\right) \]

      11. metadata-eval 99.7%

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{-0.12}, 0.253 \cdot -1\right), 1\right) \]

      12. metadata-eval 99.7%

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.12, \color{blue}{-0.253}\right), 1\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.12, -0.253\right), 1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 99.1%

      \[\leadsto \color{blue}{-0.12 \cdot {x}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 99.1%

         \[\leadsto \color{blue}{{x}^{2} \cdot -0.12} \]

   7. Simplified 99.1%

      \[\leadsto \color{blue}{{x}^{2} \cdot -0.12} \]
   8. Step-by-step derivation

      1. *-commutative 99.1%

         \[\leadsto \color{blue}{-0.12 \cdot {x}^{2}} \]

      2. unpow2 99.1%

         \[\leadsto -0.12 \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* 99.2%

         \[\leadsto \color{blue}{\left(-0.12 \cdot x\right) \cdot x} \]

   9. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\left(-0.12 \cdot x\right) \cdot x} \]

   if -4.20000000000000018 < x < 2

   1. Initial program 100.0%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \color{blue}{1 + \left(-x \cdot \left(0.253 + x \cdot 0.12\right)\right)} \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(-x \cdot \left(0.253 + x \cdot 0.12\right)\right) + 1} \]

      3. distribute-rgt-neg-in 100.0%

         \[\leadsto \color{blue}{x \cdot \left(-\left(0.253 + x \cdot 0.12\right)\right)} + 1 \]

      4. fma-define 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, -\left(0.253 + x \cdot 0.12\right), 1\right)} \]

      5. +-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(x, -\color{blue}{\left(x \cdot 0.12 + 0.253\right)}, 1\right) \]

      6. distribute-neg-in 100.0%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(-x \cdot 0.12\right) + \left(-0.253\right)}, 1\right) \]

      7. distribute-rgt-neg-in 100.0%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(-0.12\right)} + \left(-0.253\right), 1\right) \]

      8. metadata-eval 100.0%

         \[\leadsto \mathsf{fma}\left(x, x \cdot \left(-0.12\right) + \color{blue}{-0.253}, 1\right) \]

      9. metadata-eval 100.0%

         \[\leadsto \mathsf{fma}\left(x, x \cdot \left(-0.12\right) + \color{blue}{0.253 \cdot -1}, 1\right) \]

      10. fma-define 100.0%

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.12, 0.253 \cdot -1\right)}, 1\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{-0.12}, 0.253 \cdot -1\right), 1\right) \]

      12. metadata-eval 100.0%

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.12, \color{blue}{-0.253}\right), 1\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.12, -0.253\right), 1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 97.1%

      \[\leadsto \color{blue}{1} \]

   if 2 < x

   1. Initial program 99.7%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
   2. Step-by-step derivation

      1. sub-neg 99.7%

         \[\leadsto \color{blue}{1 + \left(-x \cdot \left(0.253 + x \cdot 0.12\right)\right)} \]

      2. +-commutative 99.7%

         \[\leadsto \color{blue}{\left(-x \cdot \left(0.253 + x \cdot 0.12\right)\right) + 1} \]

      3. distribute-rgt-neg-in 99.7%

         \[\leadsto \color{blue}{x \cdot \left(-\left(0.253 + x \cdot 0.12\right)\right)} + 1 \]

      4. fma-define 99.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, -\left(0.253 + x \cdot 0.12\right), 1\right)} \]

      5. +-commutative 99.7%

         \[\leadsto \mathsf{fma}\left(x, -\color{blue}{\left(x \cdot 0.12 + 0.253\right)}, 1\right) \]

      6. distribute-neg-in 99.7%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(-x \cdot 0.12\right) + \left(-0.253\right)}, 1\right) \]

      7. distribute-rgt-neg-in 99.7%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(-0.12\right)} + \left(-0.253\right), 1\right) \]

      8. metadata-eval 99.7%

         \[\leadsto \mathsf{fma}\left(x, x \cdot \left(-0.12\right) + \color{blue}{-0.253}, 1\right) \]

      9. metadata-eval 99.7%

         \[\leadsto \mathsf{fma}\left(x, x \cdot \left(-0.12\right) + \color{blue}{0.253 \cdot -1}, 1\right) \]

      10. fma-define 99.7%

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.12, 0.253 \cdot -1\right)}, 1\right) \]

      11. metadata-eval 99.7%

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{-0.12}, 0.253 \cdot -1\right), 1\right) \]

      12. metadata-eval 99.7%

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.12, \color{blue}{-0.253}\right), 1\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.12, -0.253\right), 1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 97.2%

      \[\leadsto \color{blue}{-0.12 \cdot {x}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 97.2%

         \[\leadsto \color{blue}{{x}^{2} \cdot -0.12} \]

   7. Simplified 97.2%

      \[\leadsto \color{blue}{{x}^{2} \cdot -0.12} \]
   8. Step-by-step derivation

      1. unpow2 97.2%

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot -0.12 \]

   9. Applied egg-rr 97.2%

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot -0.12 \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2:\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ 1 - x \cdot \left(x \cdot \left(0.12 + \frac{0.253}{x}\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (* x (* x (+ 0.12 (/ 0.253 x))))))

C

double code(double x) {
	return 1.0 - (x * (x * (0.12 + (0.253 / x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (x * (x * (0.12d0 + (0.253d0 / x))))
end function

Java

public static double code(double x) {
	return 1.0 - (x * (x * (0.12 + (0.253 / x))));
}

Python

def code(x):
	return 1.0 - (x * (x * (0.12 + (0.253 / x))))

Julia

function code(x)
	return Float64(1.0 - Float64(x * Float64(x * Float64(0.12 + Float64(0.253 / x)))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - (x * (x * (0.12 + (0.253 / x))));
end

Wolfram

code[x_] := N[(1.0 - N[(x * N[(x * N[(0.12 + N[(0.253 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf 99.9%

   \[\leadsto 1 - x \cdot \color{blue}{\left(x \cdot \left(0.12 + 0.253 \cdot \frac{1}{x}\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative 99.9%

      \[\leadsto 1 - \color{blue}{\left(x \cdot \left(0.12 + 0.253 \cdot \frac{1}{x}\right)\right) \cdot x} \]

   2. +-commutative 99.9%

      \[\leadsto 1 - \left(x \cdot \color{blue}{\left(0.253 \cdot \frac{1}{x} + 0.12\right)}\right) \cdot x \]

   3. un-div-inv 99.9%

      \[\leadsto 1 - \left(x \cdot \left(\color{blue}{\frac{0.253}{x}} + 0.12\right)\right) \cdot x \]

5. Applied egg-rr 99.9%

   \[\leadsto 1 - \color{blue}{\left(x \cdot \left(\frac{0.253}{x} + 0.12\right)\right) \cdot x} \]

6. Final simplification 99.9%

   \[\leadsto 1 - x \cdot \left(x \cdot \left(0.12 + \frac{0.253}{x}\right)\right) \]
7. Add Preprocessing

Alternative 6: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - x \cdot \left(0.253 + x \cdot 0.12\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (* x (+ 0.253 (* x 0.12)))))

C

double code(double x) {
	return 1.0 - (x * (0.253 + (x * 0.12)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (x * (0.253d0 + (x * 0.12d0)))
end function

Java

public static double code(double x) {
	return 1.0 - (x * (0.253 + (x * 0.12)));
}

Python

def code(x):
	return 1.0 - (x * (0.253 + (x * 0.12)))

Julia

function code(x)
	return Float64(1.0 - Float64(x * Float64(0.253 + Float64(x * 0.12))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - (x * (0.253 + (x * 0.12)));
end

Wolfram

code[x_] := N[(1.0 - N[(x * N[(0.253 + N[(x * 0.12), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 7: 48.7% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 99.8%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
2. Step-by-step derivation

   1. sub-neg 99.8%

      \[\leadsto \color{blue}{1 + \left(-x \cdot \left(0.253 + x \cdot 0.12\right)\right)} \]

   2. +-commutative 99.8%

      \[\leadsto \color{blue}{\left(-x \cdot \left(0.253 + x \cdot 0.12\right)\right) + 1} \]

   3. distribute-rgt-neg-in 99.8%

      \[\leadsto \color{blue}{x \cdot \left(-\left(0.253 + x \cdot 0.12\right)\right)} + 1 \]

   4. fma-define 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -\left(0.253 + x \cdot 0.12\right), 1\right)} \]

   5. +-commutative 99.8%

      \[\leadsto \mathsf{fma}\left(x, -\color{blue}{\left(x \cdot 0.12 + 0.253\right)}, 1\right) \]

   6. distribute-neg-in 99.8%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(-x \cdot 0.12\right) + \left(-0.253\right)}, 1\right) \]

   7. distribute-rgt-neg-in 99.8%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(-0.12\right)} + \left(-0.253\right), 1\right) \]

   8. metadata-eval 99.8%

      \[\leadsto \mathsf{fma}\left(x, x \cdot \left(-0.12\right) + \color{blue}{-0.253}, 1\right) \]

   9. metadata-eval 99.8%

      \[\leadsto \mathsf{fma}\left(x, x \cdot \left(-0.12\right) + \color{blue}{0.253 \cdot -1}, 1\right) \]

   10. fma-define 99.9%

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.12, 0.253 \cdot -1\right)}, 1\right) \]

   11. metadata-eval 99.9%

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{-0.12}, 0.253 \cdot -1\right), 1\right) \]

   12. metadata-eval 99.9%

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.12, \color{blue}{-0.253}\right), 1\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.12, -0.253\right), 1\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 53.5%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))