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

Percentage Accurate: 99.9% → 99.8%

Time: 5.5s

Alternatives: 8

Speedup: 1.3×

• 26.2% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot \left(0.253 + x \cdot 0.12\right) \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(-0.253, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.12, x, -0.253\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x (+ 0.253 (* x 0.12))) 1e-8)
   (fma -0.253 x 1.0)
   (* (fma -0.12 x -0.253) x)))

C

double code(double x) {
	double tmp;
	if ((x * (0.253 + (x * 0.12))) <= 1e-8) {
		tmp = fma(-0.253, x, 1.0);
	} else {
		tmp = fma(-0.12, x, -0.253) * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * Float64(0.253 + Float64(x * 0.12))) <= 1e-8)
		tmp = fma(-0.253, x, 1.0);
	else
		tmp = Float64(fma(-0.12, x, -0.253) * x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x * N[(0.253 + N[(x * 0.12), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-8], N[(-0.253 * x + 1.0), $MachinePrecision], N[(N[(-0.12 * x + -0.253), $MachinePrecision] * x), $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)))) < 1e-8

   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 99.3

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

   5. Applied rewrites 99.3%

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

   if 1e-8 < (*.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. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-in N/A

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

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

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

      12. associate-*l* N/A

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

      13. lft-mult-inverse N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 99.2

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

   5. Applied rewrites 99.2%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(0.253 + x \cdot 0.12\right) \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(-0.253, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.12, x, -0.253\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot \left(0.253 + x \cdot 0.12\right) \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(-0.253, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.12\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x (+ 0.253 (* x 0.12))) 1e-8)
   (fma -0.253 x 1.0)
   (* (* x x) -0.12)))

C

double code(double x) {
	double tmp;
	if ((x * (0.253 + (x * 0.12))) <= 1e-8) {
		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))) <= 1e-8)
		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], 1e-8], 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)))) < 1e-8

   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 99.3

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

   5. Applied rewrites 99.3%

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

   if 1e-8 < (*.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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(0.253 + x \cdot 0.12\right) \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(-0.253, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.12\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot \left(0.253 + x \cdot 0.12\right) \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(-0.253, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.12 \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x (+ 0.253 (* x 0.12))) 1e-8)
   (fma -0.253 x 1.0)
   (* (* -0.12 x) x)))

C

double code(double x) {
	double tmp;
	if ((x * (0.253 + (x * 0.12))) <= 1e-8) {
		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(0.253 + Float64(x * 0.12))) <= 1e-8)
		tmp = fma(-0.253, x, 1.0);
	else
		tmp = Float64(Float64(-0.12 * x) * x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x * N[(0.253 + N[(x * 0.12), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-8], N[(-0.253 * x + 1.0), $MachinePrecision], N[(N[(-0.12 * x), $MachinePrecision] * x), $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)))) < 1e-8

   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 99.3

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

   5. Applied rewrites 99.3%

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

   if 1e-8 < (*.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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

   7. Applied rewrites 98.8%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(0.253 + x \cdot 0.12\right) \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(-0.253, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.12 \cdot x\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2:\\ \;\;\;\;x \cdot 0.253\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.253, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.2) (* x 0.253) (fma -0.253 x 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = x * 0.253;
	} else {
		tmp = fma(-0.253, x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.2)
		tmp = Float64(x * 0.253);
	else
		tmp = fma(-0.253, x, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -4.2], N[(x * 0.253), $MachinePrecision], N[(-0.253 * x + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.20000000000000018

   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

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   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. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. distribute-lft-in N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. associate-*l* N/A

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

      12. lft-mult-inverse N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 0.4%

       \[\leadsto -0.253 \cdot \color{blue}{x} \]
   11. Step-by-step derivation

   12. Applied rewrites 6.9%

       \[\leadsto x \cdot 0.253 \]

   if -4.20000000000000018 < x

   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 66.9

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

   5. Applied rewrites 66.9%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2:\\ \;\;\;\;x \cdot 0.253\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.253, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(-0.12 * x + -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. lift-*.f64 N/A

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

   3. fp-cancel-sub-sign-inv N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-neg-out 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. distribute-rgt-neg-in N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. distribute-neg-in N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

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

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

   15. lower-fma.f64 N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

Alternative 7: 52.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 99.9

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

4. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. metadata-eval N/A

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

   3. unpow1 N/A

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

   4. metadata-eval N/A

      \[\leadsto 1 - \mathsf{fma}\left(x \cdot x, \frac{3}{25}, \left|\frac{253}{1000}\right| \cdot {x}^{\color{blue}{\left(\frac{2}{2}\right)}}\right) \]

   5. sqrt-pow1 N/A

      \[\leadsto 1 - \mathsf{fma}\left(x \cdot x, \frac{3}{25}, \left|\frac{253}{1000}\right| \cdot \color{blue}{\sqrt{{x}^{2}}}\right) \]

   6. pow2 N/A

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

   7. rem-sqrt-square-rev N/A

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

   8. fabs-mul N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fabs.f64 99.3

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

6. Applied rewrites 99.3%

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

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 - \left|\frac{253}{1000} \cdot x\right|} \]
8. Step-by-step derivation

   1. fabs-mul N/A

      \[\leadsto 1 - \color{blue}{\left|\frac{253}{1000}\right| \cdot \left|x\right|} \]

   2. metadata-eval N/A

      \[\leadsto 1 - \color{blue}{\frac{253}{1000}} \cdot \left|x\right| \]

   3. fp-cancel-sub-sign-inv N/A

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

   4. +-commutative N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lft-mult-inverse N/A

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

   9. distribute-rgt-neg-out N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. mul-1-neg N/A

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

   13. lower-fma.f64 N/A

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

   14. mul-1-neg N/A

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

   15. *-commutative N/A

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

   16. associate-*l* N/A

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

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

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

   18. lft-mult-inverse N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. lower-fabs.f64 51.7

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

9. Applied rewrites 51.7%

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

Alternative 8: 48.8% 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 48.3%

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

6. Final simplification 48.3%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024320 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))