Rosa's Benchmark

Percentage Accurate: 99.8% → 99.8%

Time: 6.2s

Alternatives: 5

Speedup: 1.4×

• 34.4% 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, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(x * -0.12900613773279798), $MachinePrecision] * x), $MachinePrecision] * x + N[(0.954929658551372 * x), $MachinePrecision]), $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. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

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

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

   8. lift-*.f64 N/A

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

   9. distribute-rgt-out N/A

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

   10. lower-*.f64 N/A

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

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

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

   12. lift-*.f64 N/A

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

   13. associate-*r* N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{1}{{\left(\mathsf{fma}\left(x \cdot -0.12900613773279798, x, 0.954929658551372\right) \cdot x\right)}^{-1}}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow-1 N/A

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

   4. remove-double-div 99.8

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. distribute-rgt-in N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lower-*.f64 99.8

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.8

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

8. Applied rewrites 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 74.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.954929658551372 \cdot x - \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.12900613773279798 \leq -5:\\ \;\;\;\;\left(\left(x \cdot -0.12900613773279798\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.954929658551372 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<=
      (- (* 0.954929658551372 x) (* (* (* x x) x) 0.12900613773279798))
      -5.0)
   (* (* (* x -0.12900613773279798) x) x)
   (* 0.954929658551372 x)))

C

double code(double x) {
	double tmp;
	if (((0.954929658551372 * x) - (((x * x) * x) * 0.12900613773279798)) <= -5.0) {
		tmp = ((x * -0.12900613773279798) * x) * 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) - (((x * x) * x) * 0.12900613773279798d0)) <= (-5.0d0)) then
        tmp = ((x * (-0.12900613773279798d0)) * x) * 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) - (((x * x) * x) * 0.12900613773279798)) <= -5.0) {
		tmp = ((x * -0.12900613773279798) * x) * x;
	} else {
		tmp = 0.954929658551372 * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((0.954929658551372 * x) - (((x * x) * x) * 0.12900613773279798)) <= -5.0:
		tmp = ((x * -0.12900613773279798) * x) * x
	else:
		tmp = 0.954929658551372 * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(0.954929658551372 * x) - Float64(Float64(Float64(x * x) * x) * 0.12900613773279798)) <= -5.0)
		tmp = Float64(Float64(Float64(x * -0.12900613773279798) * x) * x);
	else
		tmp = Float64(0.954929658551372 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((0.954929658551372 * x) - (((x * x) * x) * 0.12900613773279798)) <= -5.0)
		tmp = ((x * -0.12900613773279798) * x) * x;
	else
		tmp = 0.954929658551372 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(0.954929658551372 * x), $MachinePrecision] - N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * 0.12900613773279798), $MachinePrecision]), $MachinePrecision], -5.0], N[(N[(N[(x * -0.12900613773279798), $MachinePrecision] * x), $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))) < -5

   1. Initial program 99.9%

      \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
   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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 98.3

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

   7. Applied rewrites 98.3%

      \[\leadsto x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot -0.12900613773279798\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 98.3

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

   9. Applied rewrites 98.4%

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

   if -5 < (-.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. lower-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.954929658551372 \cdot x - \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.12900613773279798 \leq -5:\\ \;\;\;\;\left(\left(x \cdot -0.12900613773279798\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.954929658551372 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.954929658551372 \cdot x - \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.12900613773279798 \leq -5:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot -0.12900613773279798\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.954929658551372 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<=
      (- (* 0.954929658551372 x) (* (* (* x x) x) 0.12900613773279798))
      -5.0)
   (* (* (* x x) -0.12900613773279798) x)
   (* 0.954929658551372 x)))

C

double code(double x) {
	double tmp;
	if (((0.954929658551372 * x) - (((x * x) * x) * 0.12900613773279798)) <= -5.0) {
		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) - (((x * x) * x) * 0.12900613773279798d0)) <= (-5.0d0)) 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) - (((x * x) * x) * 0.12900613773279798)) <= -5.0) {
		tmp = ((x * x) * -0.12900613773279798) * x;
	} else {
		tmp = 0.954929658551372 * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((0.954929658551372 * x) - (((x * x) * x) * 0.12900613773279798)) <= -5.0:
		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(Float64(Float64(x * x) * x) * 0.12900613773279798)) <= -5.0)
		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) - (((x * x) * x) * 0.12900613773279798)) <= -5.0)
		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[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * 0.12900613773279798), $MachinePrecision]), $MachinePrecision], -5.0], 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))) < -5

   1. Initial program 99.9%

      \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
   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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 98.3

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

   7. Applied rewrites 98.3%

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

   if -5 < (-.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. lower-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.954929658551372 \cdot x - \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.12900613773279798 \leq -5:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot -0.12900613773279798\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.954929658551372 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(x * -0.12900613773279798), $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. 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. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

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

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

   8. lift-*.f64 N/A

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

   9. distribute-rgt-out N/A

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

   10. lower-*.f64 N/A

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

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

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

   12. lift-*.f64 N/A

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

   13. associate-*r* N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 5: 48.9% 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. Taylor expanded in x around 0

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

   1. lower-*.f64 44.5

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

5. Applied rewrites 44.5%

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

Reproduce

herbie shell --seed 2024288 
(FPCore (x)
  :name "Rosa's Benchmark"
  :precision binary64
  (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))