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

Percentage Accurate: 99.9% → 99.9%

Time: 7.8s

Alternatives: 7

Speedup: 1.3×

• 24.5% 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, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. distribute-neg-in N/A

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

   13. lift-*.f64 N/A

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

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

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

   15. lower-fma.f64 N/A

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

   16. metadata-eval N/A

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

   17. 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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. cancel-sign-sub-inv N/A

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

   7. *-commutative N/A

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

   8. lower--.f64 N/A

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

   9. lower-*.f64 99.9

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

6. Applied rewrites 99.9%

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

Alternative 2: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot \left(x \cdot 0.12 + 0.253\right) \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(-0.253, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, -0.12, -0.253\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x (+ (* x 0.12) 0.253)) 5e-9)
   (fma -0.253 x 1.0)
   (* x (fma x -0.12 -0.253))))

C

double code(double x) {
	double tmp;
	if ((x * ((x * 0.12) + 0.253)) <= 5e-9) {
		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(Float64(x * 0.12) + 0.253)) <= 5e-9)
		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[(N[(x * 0.12), $MachinePrecision] + 0.253), $MachinePrecision]), $MachinePrecision], 5e-9], 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)))) < 5.0000000000000001e-9

   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 99.4

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

   5. Applied rewrites 99.4%

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

   if 5.0000000000000001e-9 < (*.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. 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 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(x \cdot 0.12 + 0.253\right) \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(-0.253, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, -0.12, -0.253\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot \left(x \cdot 0.12 + 0.253\right) \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(-0.253, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x (+ (* x 0.12) 0.253)) 5e-9)
   (fma -0.253 x 1.0)
   (* -0.12 (* x x))))

C

double code(double x) {
	double tmp;
	if ((x * ((x * 0.12) + 0.253)) <= 5e-9) {
		tmp = fma(-0.253, x, 1.0);
	} else {
		tmp = -0.12 * (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * Float64(Float64(x * 0.12) + 0.253)) <= 5e-9)
		tmp = fma(-0.253, x, 1.0);
	else
		tmp = Float64(-0.12 * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x * N[(N[(x * 0.12), $MachinePrecision] + 0.253), $MachinePrecision]), $MachinePrecision], 5e-9], N[(-0.253 * x + 1.0), $MachinePrecision], N[(-0.12 * N[(x * x), $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)))) < 5.0000000000000001e-9

   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 99.4

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

   5. Applied rewrites 99.4%

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

   if 5.0000000000000001e-9 < (*.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. 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 96.9

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

   5. Applied rewrites 96.9%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lower-*.f64 96.9

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

   7. Applied rewrites 96.9%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(x \cdot 0.12 + 0.253\right) \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(-0.253, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot \left(x \cdot 0.12 + 0.253\right) \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(-0.253, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x (+ (* x 0.12) 0.253)) 5e-9)
   (fma -0.253 x 1.0)
   (* x (* x -0.12))))

C

double code(double x) {
	double tmp;
	if ((x * ((x * 0.12) + 0.253)) <= 5e-9) {
		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(Float64(x * 0.12) + 0.253)) <= 5e-9)
		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[(N[(x * 0.12), $MachinePrecision] + 0.253), $MachinePrecision]), $MachinePrecision], 5e-9], 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)))) < 5.0000000000000001e-9

   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 99.4

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

   5. Applied rewrites 99.4%

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

   if 5.0000000000000001e-9 < (*.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. 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 96.9

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

   5. Applied rewrites 96.9%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(x \cdot 0.12 + 0.253\right) \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(-0.253, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \end{array} \]
5. 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

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. distribute-neg-in N/A

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

   13. lift-*.f64 N/A

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

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

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

   15. lower-fma.f64 N/A

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

   16. metadata-eval N/A

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

   17. 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: 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. lower-fma.f64 53.1

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

5. Applied rewrites 53.1%

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

Alternative 7: 49.9% 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 51.1%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))