Rosa's Benchmark

Percentage Accurate: 99.8% → 99.8%

Time: 6.4s

Alternatives: 6

Speedup: 1.4×

• 33.5% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Final simplification 99.9%

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

Alternative 2: 74.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \mathbf{if}\;x \cdot 0.954929658551372 - t\_0 \cdot 0.12900613773279798 \leq -1 \cdot 10^{+21}:\\ \;\;\;\;-0.12900613773279798 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.954929658551372\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (if (<= (- (* x 0.954929658551372) (* t_0 0.12900613773279798)) -1e+21)
     (* -0.12900613773279798 t_0)
     (* x 0.954929658551372))))

C

double code(double x) {
	double t_0 = (x * x) * x;
	double tmp;
	if (((x * 0.954929658551372) - (t_0 * 0.12900613773279798)) <= -1e+21) {
		tmp = -0.12900613773279798 * t_0;
	} else {
		tmp = x * 0.954929658551372;
	}
	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 (((x * 0.954929658551372d0) - (t_0 * 0.12900613773279798d0)) <= (-1d+21)) then
        tmp = (-0.12900613773279798d0) * t_0
    else
        tmp = x * 0.954929658551372d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * x) * x;
	double tmp;
	if (((x * 0.954929658551372) - (t_0 * 0.12900613773279798)) <= -1e+21) {
		tmp = -0.12900613773279798 * t_0;
	} else {
		tmp = x * 0.954929658551372;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * x) * x
	tmp = 0
	if ((x * 0.954929658551372) - (t_0 * 0.12900613773279798)) <= -1e+21:
		tmp = -0.12900613773279798 * t_0
	else:
		tmp = x * 0.954929658551372
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * x)
	tmp = 0.0
	if (Float64(Float64(x * 0.954929658551372) - Float64(t_0 * 0.12900613773279798)) <= -1e+21)
		tmp = Float64(-0.12900613773279798 * t_0);
	else
		tmp = Float64(x * 0.954929658551372);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * x) * x;
	tmp = 0.0;
	if (((x * 0.954929658551372) - (t_0 * 0.12900613773279798)) <= -1e+21)
		tmp = -0.12900613773279798 * t_0;
	else
		tmp = x * 0.954929658551372;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(N[(x * 0.954929658551372), $MachinePrecision] - N[(t$95$0 * 0.12900613773279798), $MachinePrecision]), $MachinePrecision], -1e+21], N[(-0.12900613773279798 * t$95$0), $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))) < -1e21

   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. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.7%

      \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot -0.12900613773279798 \]

   if -1e21 < (-.f64 (*.f64 #s(literal 238732414637843/250000000000000 binary64) x) (*.f64 #s(literal 6450306886639899/50000000000000000 binary64) (*.f64 (*.f64 x x) x)))

   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. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 65.1

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

   5. Applied rewrites 65.1%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.954929658551372 - \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.12900613773279798 \leq -1 \cdot 10^{+21}:\\ \;\;\;\;-0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.954929658551372\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. lower-fma.f64 N/A

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

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

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 99.9

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 4: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. lower-fma.f64 N/A

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

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

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 99.9

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 99.9

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

4. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. metadata-eval N/A

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

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

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. distribute-rgt-out N/A

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

   10. +-commutative N/A

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

   11. sub-neg N/A

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

   12. lift--.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 99.9

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

   15. lift--.f64 N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. lift-*.f64 N/A

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

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

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

   20. metadata-eval N/A

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

   21. lower-fma.f64 99.9

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

6. Applied rewrites 99.9%

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

Alternative 5: 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 99.9%

   \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
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 99.8

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

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%

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

Alternative 6: 49.5% 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 99.9%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 52.2

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

5. Applied rewrites 52.2%

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

Reproduce

herbie shell --seed 2024271 
(FPCore (x)
  :name "Rosa's Benchmark"
  :precision binary64
  (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))