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

Percentage Accurate: 99.9% → 99.9%

Time: 8.2s

Alternatives: 8

Speedup: 1.3×

• 24.7% 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: 99.9% 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 99.8%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
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 99.8

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

4. Applied egg-rr 99.8%

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

Alternative 2: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \cdot \left(0.253 + x \cdot 0.12\right) \leq -5000:\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.253, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 (* x (+ 0.253 (* x 0.12)))) -5000.0)
   (* -0.12 (* x x))
   (fma -0.253 x 1.0)))

C

double code(double x) {
	double tmp;
	if ((1.0 - (x * (0.253 + (x * 0.12)))) <= -5000.0) {
		tmp = -0.12 * (x * x);
	} else {
		tmp = fma(-0.253, x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - Float64(x * Float64(0.253 + Float64(x * 0.12)))) <= -5000.0)
		tmp = Float64(-0.12 * Float64(x * x));
	else
		tmp = fma(-0.253, x, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - N[(x * N[(0.253 + N[(x * 0.12), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5000.0], N[(-0.12 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(-0.253 * x + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 253/1000 binary64) (*.f64 x #s(literal 3/25 binary64))))) < -5e3

   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. 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

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

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

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

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

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

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

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

      13. metadata-eval N/A

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

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

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

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

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

      16. metadata-eval 99.7

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

   4. Applied egg-rr 99.7%

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. flip-+ N/A

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

      6. clear-num N/A

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

      7. /-lowering-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip-+ N/A

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

      10. /-lowering-/.f64 N/A

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

      11. associate-+r+ N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-25}{3}}{{x}^{2}}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-25}{3}}{{x}^{2}}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{\frac{\frac{-25}{3}}{\color{blue}{x \cdot x}}} \]

      3. *-lowering-*.f64 95.8

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

   9. Simplified 95.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{-8.333333333333334}{x \cdot x}}} \]
   10. Step-by-step derivation

       1. clear-num N/A

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

       2. div-inv N/A

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

       3. metadata-eval N/A

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

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

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

       5. *-lowering-*.f64 95.8

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

   11. Applied egg-rr 95.8%

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

   if -5e3 < (-.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 253/1000 binary64) (*.f64 x #s(literal 3/25 binary64)))))

   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 99.0

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

   5. Simplified 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.253, x, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \cdot \left(0.253 + x \cdot 0.12\right) \leq -5000:\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.253, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \cdot \left(0.253 + x \cdot 0.12\right) \leq -5000:\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.253, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 (* x (+ 0.253 (* x 0.12)))) -5000.0)
   (* x (* x -0.12))
   (fma -0.253 x 1.0)))

C

double code(double x) {
	double tmp;
	if ((1.0 - (x * (0.253 + (x * 0.12)))) <= -5000.0) {
		tmp = x * (x * -0.12);
	} else {
		tmp = fma(-0.253, x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - Float64(x * Float64(0.253 + Float64(x * 0.12)))) <= -5000.0)
		tmp = Float64(x * Float64(x * -0.12));
	else
		tmp = fma(-0.253, x, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - N[(x * N[(0.253 + N[(x * 0.12), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5000.0], N[(x * N[(x * -0.12), $MachinePrecision]), $MachinePrecision], N[(-0.253 * x + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 253/1000 binary64) (*.f64 x #s(literal 3/25 binary64))))) < -5e3

   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. 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

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

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

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

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

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

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

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

      13. metadata-eval N/A

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

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

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

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

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

      16. metadata-eval 99.7

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

   4. Applied egg-rr 99.7%

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. flip-+ N/A

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

      6. clear-num N/A

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

      7. /-lowering-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip-+ N/A

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

      10. /-lowering-/.f64 N/A

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

      11. associate-+r+ N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-25}{3}}{{x}^{2}}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-25}{3}}{{x}^{2}}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{\frac{\frac{-25}{3}}{\color{blue}{x \cdot x}}} \]

      3. *-lowering-*.f64 95.8

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

   9. Simplified 95.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{-8.333333333333334}{x \cdot x}}} \]
   10. Step-by-step derivation

       1. associate-/r/ N/A

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

       2. metadata-eval N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-lowering-*.f64 95.8

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

   11. Applied egg-rr 95.8%

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

   if -5e3 < (-.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 253/1000 binary64) (*.f64 x #s(literal 3/25 binary64)))))

   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 99.0

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

   5. Simplified 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.253, x, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \cdot \left(0.253 + x \cdot 0.12\right) \leq -5000:\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.253, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.5% 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. Simplified 58.9%

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

   if 2 < x

   1. Initial program 99.7%

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

   3. 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 7.3

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

   5. Simplified 7.3%

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

   6. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{-253}{1000} \cdot x + 0} \]

      2. accelerator-lowering-fma.f64 7.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.253, x, 0\right)} \]

   8. Simplified 7.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.253, x, 0\right)} \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 7.3

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

   10. Applied egg-rr 7.3%

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

Alternative 5: 97.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. distribute-neg-in N/A

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

   7. associate-+l+ N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

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

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

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

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

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

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

   13. metadata-eval N/A

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

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

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

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

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

   16. metadata-eval 99.8

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{-3}{25}, \color{blue}{1}\right) \]
6. Step-by-step derivation

7. Simplified 96.7%

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

Alternative 6: 97.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. distribute-neg-in N/A

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

   7. associate-+l+ N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

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

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

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

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

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

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

   13. metadata-eval N/A

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

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

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

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

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

   16. metadata-eval 99.8

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{-3}{25}, \color{blue}{1}\right) \]
6. Step-by-step derivation

7. Simplified 96.7%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

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

   4. *-lowering-*.f64 96.7

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

9. Applied egg-rr 96.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot -0.12, x, 1\right)} \]
10. 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 99.8%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
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 44.8

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

5. Simplified 44.8%

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

Alternative 8: 49.0% 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 99.8%

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

3. Taylor expanded in x around 0

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

5. Simplified 42.6%

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

Reproduce

herbie shell --seed 2024195 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))