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

Percentage Accurate: 99.9% → 99.7%

Time: 7.4s

Alternatives: 6

Speedup: 1.3×

• 24.4% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. sub-neg N/A

      \[\leadsto \color{blue}{1 + \left(\mathsf{neg}\left(x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right)\right) + 1} \]

   3. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\frac{253}{1000} + x \cdot \frac{3}{25}\right) \cdot x}\right)\right) + 1 \]

   4. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right)\right) \cdot x} + 1 \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right), x, 1\right)} \]

   6. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(x \cdot \frac{3}{25} + \frac{253}{1000}\right)}\right), x, 1\right) \]

   7. distribute-neg-in N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{3}{25}\right)\right) + \left(\mathsf{neg}\left(\frac{253}{1000}\right)\right)}, x, 1\right) \]

   8. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{3}{25}\right)\right)} + \left(\mathsf{neg}\left(\frac{253}{1000}\right)\right), x, 1\right) \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, \mathsf{neg}\left(\frac{3}{25}\right), \mathsf{neg}\left(\frac{253}{1000}\right)\right)}, x, 1\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, \color{blue}{\frac{-3}{25}}, \mathsf{neg}\left(\frac{253}{1000}\right)\right), x, 1\right) \]

   11. metadata-eval 99.9

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

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, -0.12, -0.253\right), x, 1\right)} \]
5. Step-by-step derivation

   1. flip-+ N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\left(x \cdot \frac{-3}{25}\right) \cdot \left(x \cdot \frac{-3}{25}\right) - \frac{-253}{1000} \cdot \frac{-253}{1000}}{x \cdot \frac{-3}{25} - \frac{-253}{1000}}}, x, 1\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{\left(x \cdot \frac{-3}{25}\right) \cdot \left(x \cdot \frac{-3}{25}\right) - \color{blue}{\frac{64009}{1000000}}}{x \cdot \frac{-3}{25} - \frac{-253}{1000}}, x, 1\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{\left(x \cdot \frac{-3}{25}\right) \cdot \left(x \cdot \frac{-3}{25}\right) - \color{blue}{\frac{253}{1000} \cdot \frac{253}{1000}}}{x \cdot \frac{-3}{25} - \frac{-253}{1000}}, x, 1\right) \]

   4. swap-sqr N/A

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(x \cdot x\right) \cdot \left(\frac{-3}{25} \cdot \frac{-3}{25}\right)} - \frac{253}{1000} \cdot \frac{253}{1000}}{x \cdot \frac{-3}{25} - \frac{-253}{1000}}, x, 1\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot \color{blue}{\frac{9}{625}} - \frac{253}{1000} \cdot \frac{253}{1000}}{x \cdot \frac{-3}{25} - \frac{-253}{1000}}, x, 1\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot \color{blue}{\left(\frac{3}{25} \cdot \frac{3}{25}\right)} - \frac{253}{1000} \cdot \frac{253}{1000}}{x \cdot \frac{-3}{25} - \frac{-253}{1000}}, x, 1\right) \]

   7. swap-sqr N/A

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(x \cdot \frac{3}{25}\right) \cdot \left(x \cdot \frac{3}{25}\right)} - \frac{253}{1000} \cdot \frac{253}{1000}}{x \cdot \frac{-3}{25} - \frac{-253}{1000}}, x, 1\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\left(x \cdot \frac{3}{25}\right) \cdot \left(x \cdot \frac{3}{25}\right) - \frac{253}{1000} \cdot \frac{253}{1000}}{x \cdot \frac{-3}{25} - \frac{-253}{1000}}}, x, 1\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(x \cdot \frac{3}{25}\right) \cdot \left(x \cdot \frac{3}{25}\right) + \left(\mathsf{neg}\left(\frac{253}{1000} \cdot \frac{253}{1000}\right)\right)}}{x \cdot \frac{-3}{25} - \frac{-253}{1000}}, x, 1\right) \]

   10. swap-sqr N/A

       \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(x \cdot x\right) \cdot \left(\frac{3}{25} \cdot \frac{3}{25}\right)} + \left(\mathsf{neg}\left(\frac{253}{1000} \cdot \frac{253}{1000}\right)\right)}{x \cdot \frac{-3}{25} - \frac{-253}{1000}}, x, 1\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot \color{blue}{\frac{9}{625}} + \left(\mathsf{neg}\left(\frac{253}{1000} \cdot \frac{253}{1000}\right)\right)}{x \cdot \frac{-3}{25} - \frac{-253}{1000}}, x, 1\right) \]

   12. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x \cdot \left(x \cdot \frac{9}{625}\right)} + \left(\mathsf{neg}\left(\frac{253}{1000} \cdot \frac{253}{1000}\right)\right)}{x \cdot \frac{-3}{25} - \frac{-253}{1000}}, x, 1\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{9}{625}\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{64009}{1000000}}\right)\right)}{x \cdot \frac{-3}{25} - \frac{-253}{1000}}, x, 1\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{9}{625}\right) + \color{blue}{\frac{-64009}{1000000}}}{x \cdot \frac{-3}{25} - \frac{-253}{1000}}, x, 1\right) \]

   15. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \frac{9}{625}, \frac{-64009}{1000000}\right)}}{x \cdot \frac{-3}{25} - \frac{-253}{1000}}, x, 1\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{9}{625}}, \frac{-64009}{1000000}\right)}{x \cdot \frac{-3}{25} - \frac{-253}{1000}}, x, 1\right) \]

   17. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, x \cdot \frac{9}{625}, \frac{-64009}{1000000}\right)}{\color{blue}{x \cdot \frac{-3}{25} + \left(\mathsf{neg}\left(\frac{-253}{1000}\right)\right)}}, x, 1\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, x \cdot \frac{9}{625}, \frac{-64009}{1000000}\right)}{x \cdot \frac{-3}{25} + \color{blue}{\frac{253}{1000}}}, x, 1\right) \]

   19. accelerator-lowering-fma.f64 99.8

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

6. Applied egg-rr 99.8%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{fma}\left(x, x \cdot 0.0144, -0.064009\right)}{\mathsf{fma}\left(x, -0.12, 0.253\right)}}, x, 1\right) \]
7. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(x \cdot x\right) \cdot \frac{9}{625}} + \frac{-64009}{1000000}}{\mathsf{fma}\left(x, \frac{-3}{25}, \frac{253}{1000}\right)}, x, 1\right) \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{9}{625}, \frac{-64009}{1000000}\right)}}{\mathsf{fma}\left(x, \frac{-3}{25}, \frac{253}{1000}\right)}, x, 1\right) \]

   3. *-lowering-*.f64 99.9

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

8. Applied egg-rr 99.9%

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

Alternative 2: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot \left(0.253 + x \cdot 0.12\right) \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-0.253, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, -0.12, -0.253\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x (+ 0.253 (* x 0.12))) 2e-6)
   (fma -0.253 x 1.0)
   (* x (fma x -0.12 -0.253))))

C

double code(double x) {
	double tmp;
	if ((x * (0.253 + (x * 0.12))) <= 2e-6) {
		tmp = fma(-0.253, x, 1.0);
	} else {
		tmp = x * fma(x, -0.12, -0.253);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * Float64(0.253 + Float64(x * 0.12))) <= 2e-6)
		tmp = fma(-0.253, x, 1.0);
	else
		tmp = Float64(x * fma(x, -0.12, -0.253));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x * N[(0.253 + N[(x * 0.12), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-6], N[(-0.253 * x + 1.0), $MachinePrecision], N[(x * N[(x * -0.12 + -0.253), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (+.f64 #s(literal 253/1000 binary64) (*.f64 x #s(literal 3/25 binary64)))) < 1.99999999999999991e-6

   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

      \[\leadsto \color{blue}{1 + \frac{-253}{1000} \cdot x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-253}{1000} \cdot x + 1} \]

      2. accelerator-lowering-fma.f64 99.5

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

   5. Simplified 99.5%

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

   if 1.99999999999999991e-6 < (*.f64 x (+.f64 #s(literal 253/1000 binary64) (*.f64 x #s(literal 3/25 binary64))))

   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

      \[\leadsto \color{blue}{-1 \cdot \left({x}^{2} \cdot \left(\frac{3}{25} + \frac{253}{1000} \cdot \frac{1}{x}\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({x}^{2} \cdot \left(\frac{3}{25} + \frac{253}{1000} \cdot \frac{1}{x}\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{3}{25} + \frac{253}{1000} \cdot \frac{1}{x}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{x \cdot \left(x \cdot \left(\frac{3}{25} + \frac{253}{1000} \cdot \frac{1}{x}\right)\right)}\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(x \cdot \left(\frac{3}{25} + \frac{253}{1000} \cdot \frac{1}{x}\right)\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(x \cdot \left(\frac{3}{25} + \frac{253}{1000} \cdot \frac{1}{x}\right)\right)\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(\left(\frac{3}{25} + \frac{253}{1000} \cdot \frac{1}{x}\right)\right)\right)\right)} \]

      7. distribute-neg-in N/A

         \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{25}\right)\right) + \left(\mathsf{neg}\left(\frac{253}{1000} \cdot \frac{1}{x}\right)\right)\right)}\right) \]

      8. metadata-eval N/A

         \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{\frac{-3}{25}} + \left(\mathsf{neg}\left(\frac{253}{1000} \cdot \frac{1}{x}\right)\right)\right)\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{-3}{25} \cdot x + \left(\mathsf{neg}\left(\frac{253}{1000} \cdot \frac{1}{x}\right)\right) \cdot x\right)} \]

      10. distribute-lft-neg-in N/A

          \[\leadsto x \cdot \left(\frac{-3}{25} \cdot x + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{253}{1000}\right)\right) \cdot \frac{1}{x}\right)} \cdot x\right) \]

      11. metadata-eval N/A

          \[\leadsto x \cdot \left(\frac{-3}{25} \cdot x + \left(\color{blue}{\frac{-253}{1000}} \cdot \frac{1}{x}\right) \cdot x\right) \]

      12. associate-*l* N/A

          \[\leadsto x \cdot \left(\frac{-3}{25} \cdot x + \color{blue}{\frac{-253}{1000} \cdot \left(\frac{1}{x} \cdot x\right)}\right) \]

      13. lft-mult-inverse N/A

          \[\leadsto x \cdot \left(\frac{-3}{25} \cdot x + \frac{-253}{1000} \cdot \color{blue}{1}\right) \]

      14. metadata-eval N/A

          \[\leadsto x \cdot \left(\frac{-3}{25} \cdot x + \color{blue}{\frac{-253}{1000}}\right) \]

      15. metadata-eval N/A

          \[\leadsto x \cdot \left(\frac{-3}{25} \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{253}{1000}\right)\right)}\right) \]

      16. metadata-eval N/A

          \[\leadsto x \cdot \left(\frac{-3}{25} \cdot x + \color{blue}{\frac{-253}{1000}}\right) \]

      17. *-commutative N/A

          \[\leadsto x \cdot \left(\color{blue}{x \cdot \frac{-3}{25}} + \frac{-253}{1000}\right) \]

      18. accelerator-lowering-fma.f64 99.3

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

   5. Simplified 99.3%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, -0.12, -0.253\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot \left(0.253 + x \cdot 0.12\right) \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-0.253, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x (+ 0.253 (* x 0.12))) 2e-6)
   (fma -0.253 x 1.0)
   (* x (* x -0.12))))

C

double code(double x) {
	double tmp;
	if ((x * (0.253 + (x * 0.12))) <= 2e-6) {
		tmp = fma(-0.253, x, 1.0);
	} else {
		tmp = x * (x * -0.12);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * Float64(0.253 + Float64(x * 0.12))) <= 2e-6)
		tmp = fma(-0.253, x, 1.0);
	else
		tmp = Float64(x * Float64(x * -0.12));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x * N[(0.253 + N[(x * 0.12), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-6], N[(-0.253 * x + 1.0), $MachinePrecision], N[(x * N[(x * -0.12), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (+.f64 #s(literal 253/1000 binary64) (*.f64 x #s(literal 3/25 binary64)))) < 1.99999999999999991e-6

   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

      \[\leadsto \color{blue}{1 + \frac{-253}{1000} \cdot x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-253}{1000} \cdot x + 1} \]

      2. accelerator-lowering-fma.f64 99.5

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

   5. Simplified 99.5%

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

   if 1.99999999999999991e-6 < (*.f64 x (+.f64 #s(literal 253/1000 binary64) (*.f64 x #s(literal 3/25 binary64))))

   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

      \[\leadsto \color{blue}{\frac{-3}{25} \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{-3}{25} \cdot \color{blue}{\left(x \cdot x\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-3}{25} \cdot x\right) \cdot x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-3}{25} \cdot x\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-3}{25} \cdot x\right)} \]

      5. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(x \cdot \frac{-3}{25}\right)} \]

      6. *-lowering-*.f64 98.4

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

   5. Simplified 98.4%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot -0.12\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. sub-neg N/A

      \[\leadsto \color{blue}{1 + \left(\mathsf{neg}\left(x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right)\right) + 1} \]

   3. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\frac{253}{1000} + x \cdot \frac{3}{25}\right) \cdot x}\right)\right) + 1 \]

   4. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right)\right) \cdot x} + 1 \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right), x, 1\right)} \]

   6. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(x \cdot \frac{3}{25} + \frac{253}{1000}\right)}\right), x, 1\right) \]

   7. distribute-neg-in N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{3}{25}\right)\right) + \left(\mathsf{neg}\left(\frac{253}{1000}\right)\right)}, x, 1\right) \]

   8. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{3}{25}\right)\right)} + \left(\mathsf{neg}\left(\frac{253}{1000}\right)\right), x, 1\right) \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, \mathsf{neg}\left(\frac{3}{25}\right), \mathsf{neg}\left(\frac{253}{1000}\right)\right)}, x, 1\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, \color{blue}{\frac{-3}{25}}, \mathsf{neg}\left(\frac{253}{1000}\right)\right), x, 1\right) \]

   11. metadata-eval 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 5: 51.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + \frac{-253}{1000} \cdot x} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{-253}{1000} \cdot x + 1} \]

   2. accelerator-lowering-fma.f64 46.8

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

5. Simplified 46.8%

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

Alternative 6: 50.0% accurate, 17.0× 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 99.9%

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

3. Taylor expanded in x around 0

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

5. Simplified 45.2%

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

Reproduce

herbie shell --seed 2024198 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))