Rosa's Benchmark

Percentage Accurate: 99.8% → 99.8%

Time: 6.8s

Alternatives: 7

Speedup: 1.4×

• 33.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))

C

double code(double x) {
	return (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.954929658551372d0 * x) - (0.12900613773279798d0 * ((x * x) * x))
end function

Java

public static double code(double x) {
	return (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x));
}

Python

def code(x):
	return (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x))

Julia

function code(x)
	return Float64(Float64(0.954929658551372 * x) - Float64(0.12900613773279798 * Float64(Float64(x * x) * x)))
end

MATLAB

function tmp = code(x)
	tmp = (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x));
end

Wolfram

code[x_] := N[(N[(0.954929658551372 * x), $MachinePrecision] - N[(0.12900613773279798 * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))

C

double code(double x) {
	return (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.954929658551372d0 * x) - (0.12900613773279798d0 * ((x * x) * x))
end function

Java

public static double code(double x) {
	return (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x));
}

Python

def code(x):
	return (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x))

Julia

function code(x)
	return Float64(Float64(0.954929658551372 * x) - Float64(0.12900613773279798 * Float64(Float64(x * x) * x)))
end

MATLAB

function tmp = code(x)
	tmp = (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x));
end

Wolfram

code[x_] := N[(N[(0.954929658551372 * x), $MachinePrecision] - N[(0.12900613773279798 * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.12900613773279798, x \cdot x, 0.954929658551372\right) \cdot x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (fma -0.12900613773279798 (* x x) 0.954929658551372) x))

C

double code(double x) {
	return fma(-0.12900613773279798, (x * x), 0.954929658551372) * x;
}

Julia

function code(x)
	return Float64(fma(-0.12900613773279798, Float64(x * x), 0.954929658551372) * x)
end

Wolfram

code[x_] := N[(N[(-0.12900613773279798 * N[(x * x), $MachinePrecision] + 0.954929658551372), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\frac{238732414637843}{250000000000000} \cdot x - \frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{238732414637843}{250000000000000} \cdot x - \color{blue}{\frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)} \]

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

      \[\leadsto \color{blue}{\frac{238732414637843}{250000000000000} \cdot x + \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{238732414637843}{250000000000000} \cdot x} + \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

   5. rem-square-sqrt N/A

      \[\leadsto \frac{238732414637843}{250000000000000} \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} + \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

   6. sqrt-prod N/A

      \[\leadsto \frac{238732414637843}{250000000000000} \cdot \color{blue}{\sqrt{x \cdot x}} + \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

   7. sqr-neg-rev N/A

      \[\leadsto \frac{238732414637843}{250000000000000} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)}} + \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

   8. pow2 N/A

      \[\leadsto \frac{238732414637843}{250000000000000} \cdot \sqrt{\color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{2}}} + \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

   9. sqrt-pow1 N/A

      \[\leadsto \frac{238732414637843}{250000000000000} \cdot \color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{2}{2}\right)}} + \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

   10. metadata-eval N/A

       \[\leadsto \frac{238732414637843}{250000000000000} \cdot {\left(\mathsf{neg}\left(x\right)\right)}^{\color{blue}{1}} + \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

   11. unpow1 N/A

       \[\leadsto \frac{238732414637843}{250000000000000} \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

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

       \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot x\right)\right)} + \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

   13. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{238732414637843}{250000000000000} \cdot x}\right)\right) + \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

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

       \[\leadsto \left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)} \]

   15. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot x\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)}\right)\right) \]

   16. distribute-neg-in N/A

       \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\frac{238732414637843}{250000000000000} \cdot x + \frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)} \]

   17. lift-*.f64 N/A

       \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\frac{238732414637843}{250000000000000} \cdot x} + \frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right) \]

   18. lift-*.f64 N/A

       \[\leadsto \mathsf{neg}\left(\left(\frac{238732414637843}{250000000000000} \cdot x + \color{blue}{\frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)}\right)\right) \]

   19. lift-*.f64 N/A

       \[\leadsto \mathsf{neg}\left(\left(\frac{238732414637843}{250000000000000} \cdot x + \frac{6450306886639899}{50000000000000000} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}\right)\right) \]

   20. associate-*r* N/A

       \[\leadsto \mathsf{neg}\left(\left(\frac{238732414637843}{250000000000000} \cdot x + \color{blue}{\left(\frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right) \cdot x}\right)\right) \]

   21. distribute-rgt-out N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{x \cdot \left(\frac{238732414637843}{250000000000000} + \frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)}\right) \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.12900613773279798, x \cdot x, 0.954929658551372\right) \cdot x} \]
5. Add Preprocessing

Alternative 2: 75.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \leq -0.2:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot -0.12900613773279798\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.954929658551372 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<=
      (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x)))
      -0.2)
   (* (* (* x x) -0.12900613773279798) x)
   (* 0.954929658551372 x)))

C

double code(double x) {
	double tmp;
	if (((0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x))) <= -0.2) {
		tmp = ((x * x) * -0.12900613773279798) * x;
	} else {
		tmp = 0.954929658551372 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((0.954929658551372d0 * x) - (0.12900613773279798d0 * ((x * x) * x))) <= (-0.2d0)) then
        tmp = ((x * x) * (-0.12900613773279798d0)) * x
    else
        tmp = 0.954929658551372d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x))) <= -0.2) {
		tmp = ((x * x) * -0.12900613773279798) * x;
	} else {
		tmp = 0.954929658551372 * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x))) <= -0.2:
		tmp = ((x * x) * -0.12900613773279798) * x
	else:
		tmp = 0.954929658551372 * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(0.954929658551372 * x) - Float64(0.12900613773279798 * Float64(Float64(x * x) * x))) <= -0.2)
		tmp = Float64(Float64(Float64(x * x) * -0.12900613773279798) * x);
	else
		tmp = Float64(0.954929658551372 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x))) <= -0.2)
		tmp = ((x * x) * -0.12900613773279798) * x;
	else
		tmp = 0.954929658551372 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(0.954929658551372 * x), $MachinePrecision] - N[(0.12900613773279798 * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.2], N[(N[(N[(x * x), $MachinePrecision] * -0.12900613773279798), $MachinePrecision] * x), $MachinePrecision], N[(0.954929658551372 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 #s(literal 238732414637843/250000000000000 binary64) x) (*.f64 #s(literal 6450306886639899/50000000000000000 binary64) (*.f64 (*.f64 x x) x))) < -0.20000000000000001

   1. Initial program 99.7%

      \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\frac{238732414637843}{250000000000000} \cdot x - \frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{238732414637843}{250000000000000} \cdot x - \color{blue}{\frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)} \]

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

         \[\leadsto \color{blue}{\frac{238732414637843}{250000000000000} \cdot x + \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{238732414637843}{250000000000000} \cdot x} + \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \frac{238732414637843}{250000000000000} \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} + \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      6. sqrt-prod N/A

         \[\leadsto \frac{238732414637843}{250000000000000} \cdot \color{blue}{\sqrt{x \cdot x}} + \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      7. sqr-neg-rev N/A

         \[\leadsto \frac{238732414637843}{250000000000000} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)}} + \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      8. pow2 N/A

         \[\leadsto \frac{238732414637843}{250000000000000} \cdot \sqrt{\color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{2}}} + \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      9. sqrt-pow1 N/A

         \[\leadsto \frac{238732414637843}{250000000000000} \cdot \color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{2}{2}\right)}} + \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      10. metadata-eval N/A

          \[\leadsto \frac{238732414637843}{250000000000000} \cdot {\left(\mathsf{neg}\left(x\right)\right)}^{\color{blue}{1}} + \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      11. unpow1 N/A

          \[\leadsto \frac{238732414637843}{250000000000000} \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

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

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot x\right)\right)} + \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{238732414637843}{250000000000000} \cdot x}\right)\right) + \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

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

          \[\leadsto \left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot x\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)}\right)\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\frac{238732414637843}{250000000000000} \cdot x + \frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)} \]

      17. lift-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\frac{238732414637843}{250000000000000} \cdot x} + \frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\left(\frac{238732414637843}{250000000000000} \cdot x + \color{blue}{\frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)}\right)\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\left(\frac{238732414637843}{250000000000000} \cdot x + \frac{6450306886639899}{50000000000000000} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}\right)\right) \]

      20. associate-*r* N/A

          \[\leadsto \mathsf{neg}\left(\left(\frac{238732414637843}{250000000000000} \cdot x + \color{blue}{\left(\frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right) \cdot x}\right)\right) \]

      21. distribute-rgt-out N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{x \cdot \left(\frac{238732414637843}{250000000000000} + \frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)}\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.12900613773279798, x \cdot x, 0.954929658551372\right) \cdot x} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\left(\frac{-6450306886639899}{50000000000000000} \cdot {x}^{2}\right)} \cdot x \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \frac{-6450306886639899}{50000000000000000}\right)} \cdot x \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \frac{-6450306886639899}{50000000000000000}\right)} \cdot x \]

      3. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-6450306886639899}{50000000000000000}\right) \cdot x \]

      4. lower-*.f64 96.2

         \[\leadsto \left(\color{blue}{\left(x \cdot x\right)} \cdot -0.12900613773279798\right) \cdot x \]

   7. Applied rewrites 96.2%

      \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot -0.12900613773279798\right)} \cdot x \]

   if -0.20000000000000001 < (-.f64 (*.f64 #s(literal 238732414637843/250000000000000 binary64) x) (*.f64 #s(literal 6450306886639899/50000000000000000 binary64) (*.f64 (*.f64 x x) x)))

   1. Initial program 99.8%

      \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{238732414637843}{250000000000000} \cdot x} \]
   4. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto \frac{238732414637843}{250000000000000} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

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

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{238732414637843}{250000000000000}}\right) \]

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

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{238732414637843}{250000000000000}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{neg}\left(\color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot x}\right) \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{-238732414637843}{250000000000000}} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \color{blue}{\left(\frac{-238732414637843}{250000000000000} \cdot 1\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right)} \cdot 1\right) \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      11. lft-mult-inverse N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot \color{blue}{\left(\frac{1}{{x}^{2}} \cdot {x}^{2}\right)}\right) \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{{x}^{2}} \cdot \left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right)\right)} \cdot {x}^{2}\right) \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{{x}^{2}} \cdot \left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot {x}^{2}\right)\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      15. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{1 \cdot \left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot {x}^{2}\right)}{{x}^{2}}} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      16. *-lft-identity N/A

          \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot {x}^{2}}}{{x}^{2}} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      17. *-lft-identity N/A

          \[\leadsto \frac{\color{blue}{1 \cdot \left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot {x}^{2}\right)}}{{x}^{2}} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      18. associate-*l/ N/A

          \[\leadsto \color{blue}{\left(\frac{1}{{x}^{2}} \cdot \left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot {x}^{2}\right)\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      19. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{{x}^{2}} \cdot \left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right)\right) \cdot {x}^{2}\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot \frac{1}{{x}^{2}}\right)} \cdot {x}^{2}\right) \cdot \left(\mathsf{neg}\left(x\right)\right) \]

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

          \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}\right) \cdot x\right)} \]

   5. Applied rewrites 66.8%

      \[\leadsto \color{blue}{0.954929658551372 \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 75.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \leq -0.2:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(-0.12900613773279798 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0.954929658551372 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<=
      (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x)))
      -0.2)
   (* (* x x) (* -0.12900613773279798 x))
   (* 0.954929658551372 x)))

C

double code(double x) {
	double tmp;
	if (((0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x))) <= -0.2) {
		tmp = (x * x) * (-0.12900613773279798 * x);
	} else {
		tmp = 0.954929658551372 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((0.954929658551372d0 * x) - (0.12900613773279798d0 * ((x * x) * x))) <= (-0.2d0)) then
        tmp = (x * x) * ((-0.12900613773279798d0) * x)
    else
        tmp = 0.954929658551372d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x))) <= -0.2) {
		tmp = (x * x) * (-0.12900613773279798 * x);
	} else {
		tmp = 0.954929658551372 * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x))) <= -0.2:
		tmp = (x * x) * (-0.12900613773279798 * x)
	else:
		tmp = 0.954929658551372 * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(0.954929658551372 * x) - Float64(0.12900613773279798 * Float64(Float64(x * x) * x))) <= -0.2)
		tmp = Float64(Float64(x * x) * Float64(-0.12900613773279798 * x));
	else
		tmp = Float64(0.954929658551372 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x))) <= -0.2)
		tmp = (x * x) * (-0.12900613773279798 * x);
	else
		tmp = 0.954929658551372 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(0.954929658551372 * x), $MachinePrecision] - N[(0.12900613773279798 * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.2], N[(N[(x * x), $MachinePrecision] * N[(-0.12900613773279798 * x), $MachinePrecision]), $MachinePrecision], N[(0.954929658551372 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 #s(literal 238732414637843/250000000000000 binary64) x) (*.f64 #s(literal 6450306886639899/50000000000000000 binary64) (*.f64 (*.f64 x x) x))) < -0.20000000000000001

   1. Initial program 99.7%

      \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-6450306886639899}{50000000000000000} \cdot {x}^{3}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-6450306886639899}{50000000000000000} \cdot {x}^{3}} \]

      2. lower-pow.f64 96.2

         \[\leadsto -0.12900613773279798 \cdot \color{blue}{{x}^{3}} \]

   5. Applied rewrites 96.2%

      \[\leadsto \color{blue}{-0.12900613773279798 \cdot {x}^{3}} \]
   6. Step-by-step derivation

   7. Applied rewrites 96.2%

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(-0.12900613773279798 \cdot x\right)} \]

   if -0.20000000000000001 < (-.f64 (*.f64 #s(literal 238732414637843/250000000000000 binary64) x) (*.f64 #s(literal 6450306886639899/50000000000000000 binary64) (*.f64 (*.f64 x x) x)))

   1. Initial program 99.8%

      \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{238732414637843}{250000000000000} \cdot x} \]
   4. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto \frac{238732414637843}{250000000000000} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

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

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{238732414637843}{250000000000000}}\right) \]

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

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{238732414637843}{250000000000000}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{neg}\left(\color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot x}\right) \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{-238732414637843}{250000000000000}} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \color{blue}{\left(\frac{-238732414637843}{250000000000000} \cdot 1\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right)} \cdot 1\right) \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      11. lft-mult-inverse N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot \color{blue}{\left(\frac{1}{{x}^{2}} \cdot {x}^{2}\right)}\right) \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{{x}^{2}} \cdot \left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right)\right)} \cdot {x}^{2}\right) \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{{x}^{2}} \cdot \left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot {x}^{2}\right)\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      15. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{1 \cdot \left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot {x}^{2}\right)}{{x}^{2}}} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      16. *-lft-identity N/A

          \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot {x}^{2}}}{{x}^{2}} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      17. *-lft-identity N/A

          \[\leadsto \frac{\color{blue}{1 \cdot \left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot {x}^{2}\right)}}{{x}^{2}} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      18. associate-*l/ N/A

          \[\leadsto \color{blue}{\left(\frac{1}{{x}^{2}} \cdot \left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot {x}^{2}\right)\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      19. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{{x}^{2}} \cdot \left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right)\right) \cdot {x}^{2}\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot \frac{1}{{x}^{2}}\right)} \cdot {x}^{2}\right) \cdot \left(\mathsf{neg}\left(x\right)\right) \]

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

          \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}\right) \cdot x\right)} \]

   5. Applied rewrites 66.8%

      \[\leadsto \color{blue}{0.954929658551372 \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 75.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \mathbf{if}\;0.954929658551372 \cdot x - 0.12900613773279798 \cdot t\_0 \leq -0.2:\\ \;\;\;\;-0.12900613773279798 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;0.954929658551372 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (if (<= (- (* 0.954929658551372 x) (* 0.12900613773279798 t_0)) -0.2)
     (* -0.12900613773279798 t_0)
     (* 0.954929658551372 x))))

C

double code(double x) {
	double t_0 = (x * x) * x;
	double tmp;
	if (((0.954929658551372 * x) - (0.12900613773279798 * t_0)) <= -0.2) {
		tmp = -0.12900613773279798 * t_0;
	} else {
		tmp = 0.954929658551372 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * x) * x
    if (((0.954929658551372d0 * x) - (0.12900613773279798d0 * t_0)) <= (-0.2d0)) then
        tmp = (-0.12900613773279798d0) * t_0
    else
        tmp = 0.954929658551372d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * x) * x;
	double tmp;
	if (((0.954929658551372 * x) - (0.12900613773279798 * t_0)) <= -0.2) {
		tmp = -0.12900613773279798 * t_0;
	} else {
		tmp = 0.954929658551372 * x;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * x) * x
	tmp = 0
	if ((0.954929658551372 * x) - (0.12900613773279798 * t_0)) <= -0.2:
		tmp = -0.12900613773279798 * t_0
	else:
		tmp = 0.954929658551372 * x
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * x)
	tmp = 0.0
	if (Float64(Float64(0.954929658551372 * x) - Float64(0.12900613773279798 * t_0)) <= -0.2)
		tmp = Float64(-0.12900613773279798 * t_0);
	else
		tmp = Float64(0.954929658551372 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * x) * x;
	tmp = 0.0;
	if (((0.954929658551372 * x) - (0.12900613773279798 * t_0)) <= -0.2)
		tmp = -0.12900613773279798 * t_0;
	else
		tmp = 0.954929658551372 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(N[(0.954929658551372 * x), $MachinePrecision] - N[(0.12900613773279798 * t$95$0), $MachinePrecision]), $MachinePrecision], -0.2], N[(-0.12900613773279798 * t$95$0), $MachinePrecision], N[(0.954929658551372 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 #s(literal 238732414637843/250000000000000 binary64) x) (*.f64 #s(literal 6450306886639899/50000000000000000 binary64) (*.f64 (*.f64 x x) x))) < -0.20000000000000001

   1. Initial program 99.7%

      \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-6450306886639899}{50000000000000000} \cdot {x}^{3}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-6450306886639899}{50000000000000000} \cdot {x}^{3}} \]

      2. lower-pow.f64 96.2

         \[\leadsto -0.12900613773279798 \cdot \color{blue}{{x}^{3}} \]

   5. Applied rewrites 96.2%

      \[\leadsto \color{blue}{-0.12900613773279798 \cdot {x}^{3}} \]
   6. Step-by-step derivation

   7. Applied rewrites 96.2%

      \[\leadsto -0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{x}\right) \]

   if -0.20000000000000001 < (-.f64 (*.f64 #s(literal 238732414637843/250000000000000 binary64) x) (*.f64 #s(literal 6450306886639899/50000000000000000 binary64) (*.f64 (*.f64 x x) x)))

   1. Initial program 99.8%

      \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{238732414637843}{250000000000000} \cdot x} \]
   4. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto \frac{238732414637843}{250000000000000} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

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

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{238732414637843}{250000000000000}}\right) \]

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

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{238732414637843}{250000000000000}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{neg}\left(\color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot x}\right) \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{-238732414637843}{250000000000000}} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \color{blue}{\left(\frac{-238732414637843}{250000000000000} \cdot 1\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right)} \cdot 1\right) \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      11. lft-mult-inverse N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot \color{blue}{\left(\frac{1}{{x}^{2}} \cdot {x}^{2}\right)}\right) \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{{x}^{2}} \cdot \left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right)\right)} \cdot {x}^{2}\right) \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{{x}^{2}} \cdot \left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot {x}^{2}\right)\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      15. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{1 \cdot \left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot {x}^{2}\right)}{{x}^{2}}} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      16. *-lft-identity N/A

          \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot {x}^{2}}}{{x}^{2}} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      17. *-lft-identity N/A

          \[\leadsto \frac{\color{blue}{1 \cdot \left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot {x}^{2}\right)}}{{x}^{2}} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      18. associate-*l/ N/A

          \[\leadsto \color{blue}{\left(\frac{1}{{x}^{2}} \cdot \left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot {x}^{2}\right)\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      19. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{{x}^{2}} \cdot \left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right)\right) \cdot {x}^{2}\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot \frac{1}{{x}^{2}}\right)} \cdot {x}^{2}\right) \cdot \left(\mathsf{neg}\left(x\right)\right) \]

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

          \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}\right) \cdot x\right)} \]

   5. Applied rewrites 66.8%

      \[\leadsto \color{blue}{0.954929658551372 \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 51.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \leq -0.2:\\ \;\;\;\;-0.954929658551372 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.954929658551372 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<=
      (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x)))
      -0.2)
   (* -0.954929658551372 x)
   (* 0.954929658551372 x)))

C

double code(double x) {
	double tmp;
	if (((0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x))) <= -0.2) {
		tmp = -0.954929658551372 * x;
	} else {
		tmp = 0.954929658551372 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((0.954929658551372d0 * x) - (0.12900613773279798d0 * ((x * x) * x))) <= (-0.2d0)) then
        tmp = (-0.954929658551372d0) * x
    else
        tmp = 0.954929658551372d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x))) <= -0.2) {
		tmp = -0.954929658551372 * x;
	} else {
		tmp = 0.954929658551372 * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x))) <= -0.2:
		tmp = -0.954929658551372 * x
	else:
		tmp = 0.954929658551372 * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(0.954929658551372 * x) - Float64(0.12900613773279798 * Float64(Float64(x * x) * x))) <= -0.2)
		tmp = Float64(-0.954929658551372 * x);
	else
		tmp = Float64(0.954929658551372 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x))) <= -0.2)
		tmp = -0.954929658551372 * x;
	else
		tmp = 0.954929658551372 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(0.954929658551372 * x), $MachinePrecision] - N[(0.12900613773279798 * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.2], N[(-0.954929658551372 * x), $MachinePrecision], N[(0.954929658551372 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 #s(literal 238732414637843/250000000000000 binary64) x) (*.f64 #s(literal 6450306886639899/50000000000000000 binary64) (*.f64 (*.f64 x x) x))) < -0.20000000000000001

   1. Initial program 99.7%

      \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\frac{238732414637843}{250000000000000} \cdot x - \frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{238732414637843}{250000000000000} \cdot x - \color{blue}{\frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)} \]

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

         \[\leadsto \color{blue}{\frac{238732414637843}{250000000000000} \cdot x + \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) + \frac{238732414637843}{250000000000000} \cdot x} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} + \frac{238732414637843}{250000000000000} \cdot x \]

      6. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} + \frac{238732414637843}{250000000000000} \cdot x \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot x\right) \cdot \left(x \cdot x\right)} + \frac{238732414637843}{250000000000000} \cdot x \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot x\right)} + \frac{238732414637843}{250000000000000} \cdot x \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot x, \frac{238732414637843}{250000000000000} \cdot x\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot x}, \frac{238732414637843}{250000000000000} \cdot x\right) \]

      11. metadata-eval 99.8

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{-0.12900613773279798} \cdot x, 0.954929658551372 \cdot x\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{-6450306886639899}{50000000000000000} \cdot x, \color{blue}{\frac{238732414637843}{250000000000000} \cdot x}\right) \]

      13. rem-square-sqrt N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{-6450306886639899}{50000000000000000} \cdot x, \frac{238732414637843}{250000000000000} \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) \]

      14. sqrt-prod N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{-6450306886639899}{50000000000000000} \cdot x, \frac{238732414637843}{250000000000000} \cdot \color{blue}{\sqrt{x \cdot x}}\right) \]

      15. sqr-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{-6450306886639899}{50000000000000000} \cdot x, \frac{238732414637843}{250000000000000} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)}}\right) \]

      16. pow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{-6450306886639899}{50000000000000000} \cdot x, \frac{238732414637843}{250000000000000} \cdot \sqrt{\color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{2}}}\right) \]

      17. sqrt-pow1 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{-6450306886639899}{50000000000000000} \cdot x, \frac{238732414637843}{250000000000000} \cdot \color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{2}{2}\right)}}\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{-6450306886639899}{50000000000000000} \cdot x, \frac{238732414637843}{250000000000000} \cdot {\left(\mathsf{neg}\left(x\right)\right)}^{\color{blue}{1}}\right) \]

      19. unpow1 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{-6450306886639899}{50000000000000000} \cdot x, \frac{238732414637843}{250000000000000} \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

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

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{-6450306886639899}{50000000000000000} \cdot x, \color{blue}{\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot x\right)}\right) \]

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

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{-6450306886639899}{50000000000000000} \cdot x, \color{blue}{\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot x}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{-6450306886639899}{50000000000000000} \cdot x, \color{blue}{\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot x}\right) \]

      23. metadata-eval 96.1

          \[\leadsto \mathsf{fma}\left(x \cdot x, -0.12900613773279798 \cdot x, \color{blue}{-0.954929658551372} \cdot x\right) \]

   4. Applied rewrites 96.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, -0.12900613773279798 \cdot x, -0.954929658551372 \cdot x\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-238732414637843}{250000000000000} \cdot x} \]
   6. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto \frac{-238732414637843}{250000000000000} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{-238732414637843}{250000000000000} \cdot \left(\mathsf{neg}\left(\color{blue}{-1 \cdot x}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{-238732414637843}{250000000000000} \cdot \left(\mathsf{neg}\left(\color{blue}{x \cdot -1}\right)\right) \]

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

         \[\leadsto \frac{-238732414637843}{250000000000000} \cdot \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(-1\right)\right)\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{-238732414637843}{250000000000000} \cdot \left(x \cdot \color{blue}{1}\right) \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{-238732414637843}{250000000000000} \cdot \left(x \cdot \color{blue}{\left(\frac{1}{{x}^{2}} \cdot {x}^{2}\right)}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \frac{-238732414637843}{250000000000000} \cdot \left(x \cdot \color{blue}{\frac{1 \cdot {x}^{2}}{{x}^{2}}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \frac{-238732414637843}{250000000000000} \cdot \left(x \cdot \frac{\color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot {x}^{2}}{{x}^{2}}\right) \]

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

         \[\leadsto \frac{-238732414637843}{250000000000000} \cdot \left(x \cdot \frac{\color{blue}{\mathsf{neg}\left(-1 \cdot {x}^{2}\right)}}{{x}^{2}}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \frac{-238732414637843}{250000000000000} \cdot \left(x \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({x}^{2}\right)\right)}\right)}{{x}^{2}}\right) \]

      11. remove-double-neg N/A

          \[\leadsto \frac{-238732414637843}{250000000000000} \cdot \left(x \cdot \frac{\color{blue}{{x}^{2}}}{{x}^{2}}\right) \]

      12. associate-/l* N/A

          \[\leadsto \frac{-238732414637843}{250000000000000} \cdot \color{blue}{\frac{x \cdot {x}^{2}}{{x}^{2}}} \]

      13. unpow2 N/A

          \[\leadsto \frac{-238732414637843}{250000000000000} \cdot \frac{x \cdot \color{blue}{\left(x \cdot x\right)}}{{x}^{2}} \]

      14. cube-mult N/A

          \[\leadsto \frac{-238732414637843}{250000000000000} \cdot \frac{\color{blue}{{x}^{3}}}{{x}^{2}} \]

      15. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{-238732414637843}{250000000000000} \cdot {x}^{3}}{{x}^{2}}} \]

      16. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\frac{-238732414637843}{250000000000000}}{{x}^{2}} \cdot {x}^{3}} \]

      17. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{\frac{-238732414637843}{250000000000000} \cdot 1}}{{x}^{2}} \cdot {x}^{3} \]

      18. associate-*r/ N/A

          \[\leadsto \color{blue}{\left(\frac{-238732414637843}{250000000000000} \cdot \frac{1}{{x}^{2}}\right)} \cdot {x}^{3} \]

      19. metadata-eval N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right)} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{3} \]

      20. unpow3 N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot \frac{1}{{x}^{2}}\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \]

      21. unpow2 N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot \frac{1}{{x}^{2}}\right) \cdot \left(\color{blue}{{x}^{2}} \cdot x\right) \]

      22. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}\right) \cdot x} \]

      23. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}\right) \cdot x} \]

   7. Applied rewrites 6.3%

      \[\leadsto \color{blue}{-0.954929658551372 \cdot x} \]

   if -0.20000000000000001 < (-.f64 (*.f64 #s(literal 238732414637843/250000000000000 binary64) x) (*.f64 #s(literal 6450306886639899/50000000000000000 binary64) (*.f64 (*.f64 x x) x)))

   1. Initial program 99.8%

      \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{238732414637843}{250000000000000} \cdot x} \]
   4. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto \frac{238732414637843}{250000000000000} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

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

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{238732414637843}{250000000000000}}\right) \]

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

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{238732414637843}{250000000000000}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{neg}\left(\color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot x}\right) \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{-238732414637843}{250000000000000}} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \color{blue}{\left(\frac{-238732414637843}{250000000000000} \cdot 1\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right)} \cdot 1\right) \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      11. lft-mult-inverse N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot \color{blue}{\left(\frac{1}{{x}^{2}} \cdot {x}^{2}\right)}\right) \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{{x}^{2}} \cdot \left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right)\right)} \cdot {x}^{2}\right) \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{{x}^{2}} \cdot \left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot {x}^{2}\right)\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      15. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{1 \cdot \left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot {x}^{2}\right)}{{x}^{2}}} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      16. *-lft-identity N/A

          \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot {x}^{2}}}{{x}^{2}} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      17. *-lft-identity N/A

          \[\leadsto \frac{\color{blue}{1 \cdot \left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot {x}^{2}\right)}}{{x}^{2}} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      18. associate-*l/ N/A

          \[\leadsto \color{blue}{\left(\frac{1}{{x}^{2}} \cdot \left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot {x}^{2}\right)\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      19. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{{x}^{2}} \cdot \left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right)\right) \cdot {x}^{2}\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot \frac{1}{{x}^{2}}\right)} \cdot {x}^{2}\right) \cdot \left(\mathsf{neg}\left(x\right)\right) \]

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

          \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}\right) \cdot x\right)} \]

   5. Applied rewrites 66.8%

      \[\leadsto \color{blue}{0.954929658551372 \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.12900613773279798 \cdot x, x, 0.954929658551372\right) \cdot x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (fma (* -0.12900613773279798 x) x 0.954929658551372) x))

C

double code(double x) {
	return fma((-0.12900613773279798 * x), x, 0.954929658551372) * x;
}

Julia

function code(x)
	return Float64(fma(Float64(-0.12900613773279798 * x), x, 0.954929658551372) * x)
end

Wolfram

code[x_] := N[(N[(N[(-0.12900613773279798 * x), $MachinePrecision] * x + 0.954929658551372), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(\frac{238732414637843}{250000000000000} + \frac{-6450306886639899}{50000000000000000} \cdot {x}^{2}\right)} \]

5. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.12900613773279798 \cdot x, x, 0.954929658551372\right) \cdot x} \]
6. Add Preprocessing

Alternative 7: 5.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ -0.954929658551372 \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* -0.954929658551372 x))

C

double code(double x) {
	return -0.954929658551372 * x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-0.954929658551372d0) * x
end function

Java

public static double code(double x) {
	return -0.954929658551372 * x;
}

Python

def code(x):
	return -0.954929658551372 * x

Julia

function code(x)
	return Float64(-0.954929658551372 * x)
end

MATLAB

function tmp = code(x)
	tmp = -0.954929658551372 * x;
end

Wolfram

code[x_] := N[(-0.954929658551372 * x), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\frac{238732414637843}{250000000000000} \cdot x - \frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{238732414637843}{250000000000000} \cdot x - \color{blue}{\frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)} \]

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

      \[\leadsto \color{blue}{\frac{238732414637843}{250000000000000} \cdot x + \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} \]

   4. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) + \frac{238732414637843}{250000000000000} \cdot x} \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} + \frac{238732414637843}{250000000000000} \cdot x \]

   6. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} + \frac{238732414637843}{250000000000000} \cdot x \]

   7. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot x\right) \cdot \left(x \cdot x\right)} + \frac{238732414637843}{250000000000000} \cdot x \]

   8. *-commutative N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot x\right)} + \frac{238732414637843}{250000000000000} \cdot x \]

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot x, \frac{238732414637843}{250000000000000} \cdot x\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot x}, \frac{238732414637843}{250000000000000} \cdot x\right) \]

   11. metadata-eval 99.8

       \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{-0.12900613773279798} \cdot x, 0.954929658551372 \cdot x\right) \]

   12. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{-6450306886639899}{50000000000000000} \cdot x, \color{blue}{\frac{238732414637843}{250000000000000} \cdot x}\right) \]

   13. rem-square-sqrt N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{-6450306886639899}{50000000000000000} \cdot x, \frac{238732414637843}{250000000000000} \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) \]

   14. sqrt-prod N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{-6450306886639899}{50000000000000000} \cdot x, \frac{238732414637843}{250000000000000} \cdot \color{blue}{\sqrt{x \cdot x}}\right) \]

   15. sqr-neg-rev N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{-6450306886639899}{50000000000000000} \cdot x, \frac{238732414637843}{250000000000000} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)}}\right) \]

   16. pow2 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{-6450306886639899}{50000000000000000} \cdot x, \frac{238732414637843}{250000000000000} \cdot \sqrt{\color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{2}}}\right) \]

   17. sqrt-pow1 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{-6450306886639899}{50000000000000000} \cdot x, \frac{238732414637843}{250000000000000} \cdot \color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{2}{2}\right)}}\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{-6450306886639899}{50000000000000000} \cdot x, \frac{238732414637843}{250000000000000} \cdot {\left(\mathsf{neg}\left(x\right)\right)}^{\color{blue}{1}}\right) \]

   19. unpow1 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{-6450306886639899}{50000000000000000} \cdot x, \frac{238732414637843}{250000000000000} \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

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

       \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{-6450306886639899}{50000000000000000} \cdot x, \color{blue}{\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot x\right)}\right) \]

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

       \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{-6450306886639899}{50000000000000000} \cdot x, \color{blue}{\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot x}\right) \]

   22. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{-6450306886639899}{50000000000000000} \cdot x, \color{blue}{\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot x}\right) \]

   23. metadata-eval 48.4

       \[\leadsto \mathsf{fma}\left(x \cdot x, -0.12900613773279798 \cdot x, \color{blue}{-0.954929658551372} \cdot x\right) \]

4. Applied rewrites 48.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, -0.12900613773279798 \cdot x, -0.954929658551372 \cdot x\right)} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-238732414637843}{250000000000000} \cdot x} \]
6. Step-by-step derivation

   1. remove-double-neg N/A

      \[\leadsto \frac{-238732414637843}{250000000000000} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

   2. mul-1-neg N/A

      \[\leadsto \frac{-238732414637843}{250000000000000} \cdot \left(\mathsf{neg}\left(\color{blue}{-1 \cdot x}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \frac{-238732414637843}{250000000000000} \cdot \left(\mathsf{neg}\left(\color{blue}{x \cdot -1}\right)\right) \]

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

      \[\leadsto \frac{-238732414637843}{250000000000000} \cdot \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(-1\right)\right)\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{-238732414637843}{250000000000000} \cdot \left(x \cdot \color{blue}{1}\right) \]

   6. lft-mult-inverse N/A

      \[\leadsto \frac{-238732414637843}{250000000000000} \cdot \left(x \cdot \color{blue}{\left(\frac{1}{{x}^{2}} \cdot {x}^{2}\right)}\right) \]

   7. associate-*l/ N/A

      \[\leadsto \frac{-238732414637843}{250000000000000} \cdot \left(x \cdot \color{blue}{\frac{1 \cdot {x}^{2}}{{x}^{2}}}\right) \]

   8. metadata-eval N/A

      \[\leadsto \frac{-238732414637843}{250000000000000} \cdot \left(x \cdot \frac{\color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot {x}^{2}}{{x}^{2}}\right) \]

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

      \[\leadsto \frac{-238732414637843}{250000000000000} \cdot \left(x \cdot \frac{\color{blue}{\mathsf{neg}\left(-1 \cdot {x}^{2}\right)}}{{x}^{2}}\right) \]

   10. mul-1-neg N/A

       \[\leadsto \frac{-238732414637843}{250000000000000} \cdot \left(x \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({x}^{2}\right)\right)}\right)}{{x}^{2}}\right) \]

   11. remove-double-neg N/A

       \[\leadsto \frac{-238732414637843}{250000000000000} \cdot \left(x \cdot \frac{\color{blue}{{x}^{2}}}{{x}^{2}}\right) \]

   12. associate-/l* N/A

       \[\leadsto \frac{-238732414637843}{250000000000000} \cdot \color{blue}{\frac{x \cdot {x}^{2}}{{x}^{2}}} \]

   13. unpow2 N/A

       \[\leadsto \frac{-238732414637843}{250000000000000} \cdot \frac{x \cdot \color{blue}{\left(x \cdot x\right)}}{{x}^{2}} \]

   14. cube-mult N/A

       \[\leadsto \frac{-238732414637843}{250000000000000} \cdot \frac{\color{blue}{{x}^{3}}}{{x}^{2}} \]

   15. associate-*r/ N/A

       \[\leadsto \color{blue}{\frac{\frac{-238732414637843}{250000000000000} \cdot {x}^{3}}{{x}^{2}}} \]

   16. associate-*l/ N/A

       \[\leadsto \color{blue}{\frac{\frac{-238732414637843}{250000000000000}}{{x}^{2}} \cdot {x}^{3}} \]

   17. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{\frac{-238732414637843}{250000000000000} \cdot 1}}{{x}^{2}} \cdot {x}^{3} \]

   18. associate-*r/ N/A

       \[\leadsto \color{blue}{\left(\frac{-238732414637843}{250000000000000} \cdot \frac{1}{{x}^{2}}\right)} \cdot {x}^{3} \]

   19. metadata-eval N/A

       \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right)} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{3} \]

   20. unpow3 N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot \frac{1}{{x}^{2}}\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \]

   21. unpow2 N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot \frac{1}{{x}^{2}}\right) \cdot \left(\color{blue}{{x}^{2}} \cdot x\right) \]

   22. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}\right) \cdot x} \]

   23. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000}\right)\right) \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}\right) \cdot x} \]

7. Applied rewrites 5.0%

   \[\leadsto \color{blue}{-0.954929658551372 \cdot x} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024326 
(FPCore (x)
  :name "Rosa's Benchmark"
  :precision binary64
  (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))