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

Percentage Accurate: 99.9% → 99.8%

Time: 7.9s

Alternatives: 6

Speedup: 1.0×

• 25.1% 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: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \left(1 + -0.253 \cdot x\right) + -0.12 \cdot {x}^{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (+ 1.0 (* -0.253 x)) (* -0.12 (pow x 2.0))))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 + ((-0.253d0) * x)) + ((-0.12d0) * (x ** 2.0d0))
end function

Java

public static double code(double x) {
	return (1.0 + (-0.253 * x)) + (-0.12 * Math.pow(x, 2.0));
}

Python

def code(x):
	return (1.0 + (-0.253 * x)) + (-0.12 * math.pow(x, 2.0))

Julia

function code(x)
	return Float64(Float64(1.0 + Float64(-0.253 * x)) + Float64(-0.12 * (x ^ 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 + (-0.253 * x)) + (-0.12 * (x ^ 2.0));
end

Wolfram

code[x_] := N[(N[(1.0 + N[(-0.253 * x), $MachinePrecision]), $MachinePrecision] + N[(-0.12 * N[Power[x, 2.0], $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. distribute-rgt-in 99.8%

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

   2. *-commutative 99.8%

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

   3. *-commutative 99.8%

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

   4. associate-*l* 99.9%

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

   5. pow2 99.9%

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

4. Applied egg-rr 99.9%

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

   1. associate--r+ 99.9%

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

   2. cancel-sign-sub-inv 99.9%

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

   3. *-commutative 99.9%

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

   4. cancel-sign-sub-inv 99.9%

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

   5. metadata-eval 99.9%

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

   6. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\left(1 + -0.253 \cdot x\right) + -0.12 \cdot {x}^{2}} \]
7. Add Preprocessing

Alternative 2: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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 99.9%

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

   5. +-commutative 99.9%

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

   6. distribute-neg-in 99.9%

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

   7. distribute-rgt-neg-in 99.9%

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

   8. metadata-eval 99.9%

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

   9. metadata-eval 99.9%

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

   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 99.9%

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

6. Final simplification 99.9%

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

Alternative 3: 95.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{+24} \lor \neg \left(x \leq 6.4 \cdot 10^{-12}\right):\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -3.3e+24) (not (<= x 6.4e-12))) (* x (* x -0.12)) 1.0))

C

double code(double x) {
	double tmp;
	if ((x <= -3.3e+24) || !(x <= 6.4e-12)) {
		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 <= (-3.3d+24)) .or. (.not. (x <= 6.4d-12))) 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 <= -3.3e+24) || !(x <= 6.4e-12)) {
		tmp = x * (x * -0.12);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -3.3e+24) or not (x <= 6.4e-12):
		tmp = x * (x * -0.12)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -3.3e+24) || !(x <= 6.4e-12))
		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 <= -3.3e+24) || ~((x <= 6.4e-12)))
		tmp = x * (x * -0.12);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -3.3e+24], N[Not[LessEqual[x, 6.4e-12]], $MachinePrecision]], N[(x * N[(x * -0.12), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -3.2999999999999999e24 or 6.4000000000000002e-12 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 96.9%

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

      1. *-commutative 96.9%

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

   5. Simplified 96.9%

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

      1. unpow2 96.9%

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

      2. associate-*l* 96.9%

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

      3. add-sqr-sqrt 43.1%

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

      4. sqrt-unprod 43.5%

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

      5. swap-sqr 43.5%

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

      6. unpow2 43.5%

         \[\leadsto x \cdot \sqrt{\color{blue}{{x}^{2}} \cdot \left(-0.12 \cdot -0.12\right)} \]

      7. metadata-eval 43.5%

         \[\leadsto x \cdot \sqrt{{x}^{2} \cdot \color{blue}{0.0144}} \]

      8. sqrt-prod 43.5%

         \[\leadsto x \cdot \color{blue}{\left(\sqrt{{x}^{2}} \cdot \sqrt{0.0144}\right)} \]

      9. sqrt-pow1 0.4%

         \[\leadsto x \cdot \left(\color{blue}{{x}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{0.0144}\right) \]

      10. metadata-eval 0.4%

          \[\leadsto x \cdot \left({x}^{\color{blue}{1}} \cdot \sqrt{0.0144}\right) \]

      11. pow1 0.4%

          \[\leadsto x \cdot \left(\color{blue}{x} \cdot \sqrt{0.0144}\right) \]

      12. metadata-eval 0.4%

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

      13. *-commutative 0.4%

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

      14. metadata-eval 0.4%

          \[\leadsto x \cdot \left(\color{blue}{\frac{1}{8.333333333333334}} \cdot x\right) \]

      15. associate-/r/ 0.4%

          \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{8.333333333333334}{x}}} \]

      16. un-div-inv 0.4%

          \[\leadsto \color{blue}{\frac{x}{\frac{8.333333333333334}{x}}} \]

      17. div-inv 0.4%

          \[\leadsto \frac{x}{\color{blue}{8.333333333333334 \cdot \frac{1}{x}}} \]

      18. associate-/r* 0.4%

          \[\leadsto \color{blue}{\frac{\frac{x}{8.333333333333334}}{\frac{1}{x}}} \]

   7. Applied egg-rr 96.7%

      \[\leadsto \color{blue}{\frac{x \cdot -0.12}{\frac{1}{x}}} \]
   8. Step-by-step derivation

      1. associate-/r/ 96.9%

         \[\leadsto \color{blue}{\frac{x \cdot -0.12}{1} \cdot x} \]

      2. /-rgt-identity 96.9%

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

   9. Applied egg-rr 96.9%

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

   if -3.2999999999999999e24 < x < 6.4000000000000002e-12

   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 96.2%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{+24} \lor \neg \left(x \leq 6.4 \cdot 10^{-12}\right):\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 5: 97.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.0 - N[(x * N[(x * 0.12), $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 99.5%

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

4. Taylor expanded in x around inf 98.0%

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

   1. *-commutative 98.0%

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

6. Simplified 98.0%

   \[\leadsto 1 - x \cdot \color{blue}{\left(x \cdot 0.12\right)} \]
7. Add Preprocessing

Alternative 6: 49.5% 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. Add Preprocessing

3. Taylor expanded in x around 0 49.2%

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

Reproduce

herbie shell --seed 2024107 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))