Rosa's Benchmark

Percentage Accurate: 99.8% → 99.8%

Time: 9.9s

Alternatives: 3

Speedup: 1.4×

• 33.3% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

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

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

   5. distribute-rgt-out N/A

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

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

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

   7. *-commutative N/A

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

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

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

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

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

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

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

   11. metadata-eval 99.8

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

4. Applied egg-rr 99.8%

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

Alternative 2: 74.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.954929658551372 - 0.12900613773279798 \cdot \left(x \cdot \left(x \cdot x\right)\right) \leq -2000000:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot -0.12900613773279798\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.954929658551372\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<=
      (- (* x 0.954929658551372) (* 0.12900613773279798 (* x (* x x))))
      -2000000.0)
   (* x (* x (* x -0.12900613773279798)))
   (* x 0.954929658551372)))

C

double code(double x) {
	double tmp;
	if (((x * 0.954929658551372) - (0.12900613773279798 * (x * (x * x)))) <= -2000000.0) {
		tmp = x * (x * (x * -0.12900613773279798));
	} else {
		tmp = x * 0.954929658551372;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((x * 0.954929658551372d0) - (0.12900613773279798d0 * (x * (x * x)))) <= (-2000000.0d0)) then
        tmp = x * (x * (x * (-0.12900613773279798d0)))
    else
        tmp = x * 0.954929658551372d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((x * 0.954929658551372) - (0.12900613773279798 * (x * (x * x)))) <= -2000000.0) {
		tmp = x * (x * (x * -0.12900613773279798));
	} else {
		tmp = x * 0.954929658551372;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((x * 0.954929658551372) - (0.12900613773279798 * (x * (x * x)))) <= -2000000.0:
		tmp = x * (x * (x * -0.12900613773279798))
	else:
		tmp = x * 0.954929658551372
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x * 0.954929658551372) - Float64(0.12900613773279798 * Float64(x * Float64(x * x)))) <= -2000000.0)
		tmp = Float64(x * Float64(x * Float64(x * -0.12900613773279798)));
	else
		tmp = Float64(x * 0.954929658551372);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((x * 0.954929658551372) - (0.12900613773279798 * (x * (x * x)))) <= -2000000.0)
		tmp = x * (x * (x * -0.12900613773279798));
	else
		tmp = x * 0.954929658551372;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x * 0.954929658551372), $MachinePrecision] - N[(0.12900613773279798 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2000000.0], N[(x * N[(x * N[(x * -0.12900613773279798), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * 0.954929658551372), $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))) < -2e6

   1. Initial program 98.7%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

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

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

      5. distribute-rgt-out N/A

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

      11. metadata-eval 99.8

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

   4. Applied egg-rr 99.8%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. swap-sqr N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{x}{\frac{\left(x \cdot x\right) \cdot \frac{-6450306886639899}{50000000000000000} - \frac{238732414637843}{250000000000000}}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\frac{41606458931834106848498498730201}{2500000000000000000000000000000000}} - \frac{238732414637843}{250000000000000} \cdot \frac{238732414637843}{250000000000000}}} \]

      7. metadata-eval N/A

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

      8. swap-sqr N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. clear-num N/A

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

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{x}{\frac{1}{\mathsf{fma}\left(x, x \cdot -0.12900613773279798, 0.954929658551372\right)}}} \]

   7. Taylor expanded in x around inf

      \[\leadsto \frac{x}{\color{blue}{\frac{\frac{-50000000000000000}{6450306886639899}}{{x}^{2}}}} \]
   8. Step-by-step derivation

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

         \[\leadsto \frac{x}{\color{blue}{\frac{\frac{-50000000000000000}{6450306886639899}}{{x}^{2}}}} \]

      2. remove-double-neg N/A

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

      3. +-rgt-identity N/A

         \[\leadsto \frac{x}{\frac{\frac{-50000000000000000}{6450306886639899}}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left({x}^{2}\right)\right) + 0\right)}\right)}} \]

      4. distribute-neg-in N/A

         \[\leadsto \frac{x}{\frac{\frac{-50000000000000000}{6450306886639899}}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right) + \left(\mathsf{neg}\left(0\right)\right)}}} \]

      5. remove-double-neg N/A

         \[\leadsto \frac{x}{\frac{\frac{-50000000000000000}{6450306886639899}}{\color{blue}{{x}^{2}} + \left(\mathsf{neg}\left(0\right)\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{x}{\frac{\frac{-50000000000000000}{6450306886639899}}{\color{blue}{x \cdot x} + \left(\mathsf{neg}\left(0\right)\right)}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x}{\frac{\frac{-50000000000000000}{6450306886639899}}{x \cdot x + \color{blue}{0}}} \]

      8. accelerator-lowering-fma.f64 96.6

         \[\leadsto \frac{x}{\frac{-7.751569170074954}{\color{blue}{\mathsf{fma}\left(x, x, 0\right)}}} \]

   9. Simplified 96.6%

      \[\leadsto \frac{x}{\color{blue}{\frac{-7.751569170074954}{\mathsf{fma}\left(x, x, 0\right)}}} \]

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

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

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

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

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

       9. *-lowering-*.f64 96.6

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

   12. Simplified 96.6%

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

   if -2e6 < (-.f64 (*.f64 #s(literal 238732414637843/250000000000000 binary64) x) (*.f64 #s(literal 6450306886639899/50000000000000000 binary64) (*.f64 (*.f64 x x) x)))

   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 0

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

      1. metadata-eval N/A

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

      2. lft-mult-inverse N/A

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

      3. distribute-neg-frac2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. associate-*r* N/A

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

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

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

      8. unpow2 N/A

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

      9. unpow3 N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. +-rgt-identity N/A

          \[\leadsto \color{blue}{\left(-1 \cdot {x}^{3}\right) \cdot \left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot \frac{1}{{x}^{2}}\right)\right) + 0} \]

      13. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot \left(-1 \cdot {x}^{3}\right)} + 0 \]

   5. Simplified 59.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.954929658551372, x, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 59.3

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

   7. Applied egg-rr 59.3%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.954929658551372 - 0.12900613773279798 \cdot \left(x \cdot \left(x \cdot x\right)\right) \leq -2000000:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot -0.12900613773279798\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.954929658551372\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 49.2% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x * 0.954929658551372

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. lft-mult-inverse N/A

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

   3. distribute-neg-frac2 N/A

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

   4. associate-*l* N/A

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

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

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

   6. associate-*r* N/A

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

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

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

   8. unpow2 N/A

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

   9. unpow3 N/A

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

   10. mul-1-neg N/A

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

   11. *-commutative N/A

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

   12. +-rgt-identity N/A

       \[\leadsto \color{blue}{\left(-1 \cdot {x}^{3}\right) \cdot \left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot \frac{1}{{x}^{2}}\right)\right) + 0} \]

   13. *-commutative N/A

       \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot \left(-1 \cdot {x}^{3}\right)} + 0 \]

5. Simplified 42.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.954929658551372, x, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 42.7

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

7. Applied egg-rr 42.7%

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

Reproduce

herbie shell --seed 2024195 
(FPCore (x)
  :name "Rosa's Benchmark"
  :precision binary64
  (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))