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

Percentage Accurate: 99.9% → 99.8%

Time: 7.6s

Alternatives: 8

Speedup: 1.3×

• 24.8% 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, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * -0.12 + N[(x * -0.253 + 1.0), $MachinePrecision]), $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. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. distribute-lft-in N/A

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

   8. *-commutative N/A

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

   9. distribute-neg-in N/A

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

   10. associate-+l+ N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r* N/A

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

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

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

   14. lower-fma.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. metadata-eval N/A

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

   17. *-commutative N/A

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

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

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

   19. lower-fma.f64 N/A

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

   20. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot \left(0.253 + x \cdot 0.12\right) \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(-0.253, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, -0.12, -0.253\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x (+ 0.253 (* x 0.12))) 0.5)
   (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))) <= 0.5) {
		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))) <= 0.5)
		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], 0.5], 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)))) < 0.5

   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. lower-fma.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 0.5 < (*.f64 x (+.f64 #s(literal 253/1000 binary64) (*.f64 x #s(literal 3/25 binary64))))

   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

      \[\leadsto \color{blue}{-1 \cdot \left({x}^{2} \cdot \left(\frac{3}{25} + \frac{253}{1000} \cdot \frac{1}{x}\right)\right)} \]
   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. lower-*.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. lower-fma.f64 98.6

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

   5. Applied rewrites 98.6%

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

Alternative 3: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot \left(0.253 + x \cdot 0.12\right) \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(-0.253, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.12\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x (+ 0.253 (* x 0.12))) 0.5) (fma -0.253 x 1.0) (* (* x x) -0.12)))

C

double code(double x) {
	double tmp;
	if ((x * (0.253 + (x * 0.12))) <= 0.5) {
		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))) <= 0.5)
		tmp = fma(-0.253, x, 1.0);
	else
		tmp = Float64(Float64(x * x) * -0.12);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x * N[(0.253 + N[(x * 0.12), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], N[(-0.253 * x + 1.0), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * -0.12), $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)))) < 0.5

   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. lower-fma.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 0.5 < (*.f64 x (+.f64 #s(literal 253/1000 binary64) (*.f64 x #s(literal 3/25 binary64))))

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sub-neg N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 N/A

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

      20. metadata-eval 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{25} \cdot {x}^{2}} \]

      2. unpow2 N/A

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

      3. lower-*.f64 97.9

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

   7. Applied rewrites 97.9%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(0.253 + x \cdot 0.12\right) \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(-0.253, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.12\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot \left(0.253 + x \cdot 0.12\right) \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(-0.253, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x (+ 0.253 (* x 0.12))) 0.5) (fma -0.253 x 1.0) (* x (* x -0.12))))

C

double code(double x) {
	double tmp;
	if ((x * (0.253 + (x * 0.12))) <= 0.5) {
		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))) <= 0.5)
		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], 0.5], 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)))) < 0.5

   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. lower-fma.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 0.5 < (*.f64 x (+.f64 #s(literal 253/1000 binary64) (*.f64 x #s(literal 3/25 binary64))))

   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

      \[\leadsto \color{blue}{\frac{-3}{25} \cdot {x}^{2}} \]
   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. lower-*.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. lower-*.f64 97.9

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

   5. Applied rewrites 97.9%

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

Alternative 5: 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. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

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

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

   7. lower-fma.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. distribute-neg-in N/A

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

   11. lift-*.f64 N/A

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

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

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

   13. lower-fma.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

Alternative 6: 50.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.253\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2

   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. Applied rewrites 62.7%

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

   if 2 < x

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sub-neg N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 N/A

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

      20. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \left({x}^{2} \cdot \left(\frac{3}{25} + \frac{253}{1000} \cdot \frac{1}{x}\right)\right)} \]
   6. 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. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-neg-in N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

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

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

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

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

      16. associate-*l* N/A

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

      17. distribute-lft-neg-out N/A

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

      18. lft-mult-inverse N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto x \cdot \frac{-253}{1000} \]
   9. Step-by-step derivation

   10. Applied rewrites 6.8%

       \[\leadsto x \cdot -0.253 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 51.0% 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. lower-fma.f64 47.9

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

5. Applied rewrites 47.9%

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

Alternative 8: 49.1% 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. Applied rewrites 45.3%

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

Reproduce

herbie shell --seed 2024220 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))