Rosa's Benchmark

Percentage Accurate: 99.8% → 99.8%

Time: 13.8s

Alternatives: 5

Speedup: 1.2×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.954929658551372, x, {x}^{3} \cdot -0.12900613773279798\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma 0.954929658551372 x (* (pow x 3.0) -0.12900613773279798)))

C

double code(double x) {
	return fma(0.954929658551372, x, (pow(x, 3.0) * -0.12900613773279798));
}

Julia

function code(x)
	return fma(0.954929658551372, x, Float64((x ^ 3.0) * -0.12900613773279798))
end

Wolfram

code[x_] := N[(0.954929658551372 * x + N[(N[Power[x, 3.0], $MachinePrecision] * -0.12900613773279798), $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. Step-by-step derivation

   1. fmm-def 99.8%

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

   2. *-commutative 99.8%

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

   3. distribute-rgt-neg-in 99.8%

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

   4. unpow3 99.8%

      \[\leadsto \mathsf{fma}\left(0.954929658551372, x, \color{blue}{{x}^{3}} \cdot \left(-0.12900613773279798\right)\right) \]

   5. metadata-eval 99.8%

      \[\leadsto \mathsf{fma}\left(0.954929658551372, x, {x}^{3} \cdot \color{blue}{-0.12900613773279798}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.954929658551372, x, {x}^{3} \cdot -0.12900613773279798\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(0.954929658551372 - \frac{0.12900613773279798}{{x}^{-2}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (- 0.954929658551372 (/ 0.12900613773279798 (pow x -2.0)))))

C

double code(double x) {
	return x * (0.954929658551372 - (0.12900613773279798 / pow(x, -2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (0.954929658551372d0 - (0.12900613773279798d0 / (x ** (-2.0d0))))
end function

Java

public static double code(double x) {
	return x * (0.954929658551372 - (0.12900613773279798 / Math.pow(x, -2.0)));
}

Python

def code(x):
	return x * (0.954929658551372 - (0.12900613773279798 / math.pow(x, -2.0)))

Julia

function code(x)
	return Float64(x * Float64(0.954929658551372 - Float64(0.12900613773279798 / (x ^ -2.0))))
end

MATLAB

function tmp = code(x)
	tmp = x * (0.954929658551372 - (0.12900613773279798 / (x ^ -2.0)));
end

Wolfram

code[x_] := N[(x * N[(0.954929658551372 - N[(0.12900613773279798 / N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision]), $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. Step-by-step derivation

   1. fmm-def 99.8%

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

   2. *-commutative 99.8%

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

   3. distribute-rgt-neg-in 99.8%

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

   4. unpow3 99.8%

      \[\leadsto \mathsf{fma}\left(0.954929658551372, x, \color{blue}{{x}^{3}} \cdot \left(-0.12900613773279798\right)\right) \]

   5. metadata-eval 99.8%

      \[\leadsto \mathsf{fma}\left(0.954929658551372, x, {x}^{3} \cdot \color{blue}{-0.12900613773279798}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.954929658551372, x, {x}^{3} \cdot -0.12900613773279798\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 99.8%

      \[\leadsto \color{blue}{0.954929658551372 \cdot x + {x}^{3} \cdot -0.12900613773279798} \]

   2. pow3 99.8%

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

   3. *-commutative 99.8%

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

   4. metadata-eval 99.8%

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

   5. cancel-sign-sub-inv 99.8%

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

   6. associate-*r* 99.8%

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

   7. distribute-rgt-out-- 99.8%

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

   8. *-commutative 99.8%

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

   9. pow2 99.8%

      \[\leadsto x \cdot \left(0.954929658551372 - \color{blue}{{x}^{2}} \cdot 0.12900613773279798\right) \]

6. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{x \cdot \left(0.954929658551372 - {x}^{2} \cdot 0.12900613773279798\right)} \]
7. Step-by-step derivation

   1. /-rgt-identity 99.8%

      \[\leadsto x \cdot \left(0.954929658551372 - \color{blue}{\frac{{x}^{2}}{1}} \cdot 0.12900613773279798\right) \]

   2. clear-num 99.8%

      \[\leadsto x \cdot \left(0.954929658551372 - \color{blue}{\frac{1}{\frac{1}{{x}^{2}}}} \cdot 0.12900613773279798\right) \]

   3. pow-flip 99.8%

      \[\leadsto x \cdot \left(0.954929658551372 - \frac{1}{\color{blue}{{x}^{\left(-2\right)}}} \cdot 0.12900613773279798\right) \]

   4. metadata-eval 99.8%

      \[\leadsto x \cdot \left(0.954929658551372 - \frac{1}{{x}^{\color{blue}{-2}}} \cdot 0.12900613773279798\right) \]

8. Applied egg-rr 99.8%

   \[\leadsto x \cdot \left(0.954929658551372 - \color{blue}{\frac{1}{{x}^{-2}}} \cdot 0.12900613773279798\right) \]
9. Step-by-step derivation

   1. associate-*l/ 99.8%

      \[\leadsto x \cdot \left(0.954929658551372 - \color{blue}{\frac{1 \cdot 0.12900613773279798}{{x}^{-2}}}\right) \]

   2. metadata-eval 99.8%

      \[\leadsto x \cdot \left(0.954929658551372 - \frac{\color{blue}{0.12900613773279798}}{{x}^{-2}}\right) \]

10. Applied egg-rr 99.8%

    \[\leadsto x \cdot \left(0.954929658551372 - \color{blue}{\frac{0.12900613773279798}{{x}^{-2}}}\right) \]
11. Add Preprocessing

Alternative 3: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(x * N[(0.954929658551372 - N[(0.12900613773279798 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $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. Step-by-step derivation

   1. fmm-def 99.8%

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

   2. *-commutative 99.8%

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

   3. distribute-rgt-neg-in 99.8%

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

   4. unpow3 99.8%

      \[\leadsto \mathsf{fma}\left(0.954929658551372, x, \color{blue}{{x}^{3}} \cdot \left(-0.12900613773279798\right)\right) \]

   5. metadata-eval 99.8%

      \[\leadsto \mathsf{fma}\left(0.954929658551372, x, {x}^{3} \cdot \color{blue}{-0.12900613773279798}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.954929658551372, x, {x}^{3} \cdot -0.12900613773279798\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 99.8%

      \[\leadsto \color{blue}{0.954929658551372 \cdot x + {x}^{3} \cdot -0.12900613773279798} \]

   2. pow3 99.8%

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

   3. *-commutative 99.8%

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

   4. metadata-eval 99.8%

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

   5. cancel-sign-sub-inv 99.8%

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

   6. associate-*r* 99.8%

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

   7. distribute-rgt-out-- 99.8%

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

   8. *-commutative 99.8%

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

   9. pow2 99.8%

      \[\leadsto x \cdot \left(0.954929658551372 - \color{blue}{{x}^{2}} \cdot 0.12900613773279798\right) \]

6. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{x \cdot \left(0.954929658551372 - {x}^{2} \cdot 0.12900613773279798\right)} \]
7. Step-by-step derivation

   1. pow2 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 4: 51.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8:\\ \;\;\;\;0.954929658551372 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.954929658551372\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.8) (* 0.954929658551372 x) (* x -0.954929658551372)))

C

double code(double x) {
	double tmp;
	if (x <= 2.8) {
		tmp = 0.954929658551372 * x;
	} else {
		tmp = x * -0.954929658551372;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.8d0) then
        tmp = 0.954929658551372d0 * x
    else
        tmp = x * (-0.954929658551372d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.8) {
		tmp = 0.954929658551372 * x;
	} else {
		tmp = x * -0.954929658551372;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.8:
		tmp = 0.954929658551372 * x
	else:
		tmp = x * -0.954929658551372
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.8)
		tmp = Float64(0.954929658551372 * x);
	else
		tmp = Float64(x * -0.954929658551372);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.8)
		tmp = 0.954929658551372 * x;
	else
		tmp = x * -0.954929658551372;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.8], N[(0.954929658551372 * x), $MachinePrecision], N[(x * -0.954929658551372), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.7999999999999998

   1. Initial program 99.8%

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

      1. fmm-def 99.8%

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

      2. *-commutative 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. unpow3 99.8%

         \[\leadsto \mathsf{fma}\left(0.954929658551372, x, \color{blue}{{x}^{3}} \cdot \left(-0.12900613773279798\right)\right) \]

      5. metadata-eval 99.8%

         \[\leadsto \mathsf{fma}\left(0.954929658551372, x, {x}^{3} \cdot \color{blue}{-0.12900613773279798}\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.954929658551372, x, {x}^{3} \cdot -0.12900613773279798\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 71.9%

      \[\leadsto \color{blue}{0.954929658551372 \cdot x} \]
   6. Step-by-step derivation

      1. *-commutative 71.9%

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

   7. Simplified 71.9%

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

   if 2.7999999999999998 < x

   1. Initial program 99.8%

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

      1. fmm-def 99.8%

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

      2. *-commutative 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. unpow3 100.0%

         \[\leadsto \mathsf{fma}\left(0.954929658551372, x, \color{blue}{{x}^{3}} \cdot \left(-0.12900613773279798\right)\right) \]

      5. metadata-eval 100.0%

         \[\leadsto \mathsf{fma}\left(0.954929658551372, x, {x}^{3} \cdot \color{blue}{-0.12900613773279798}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.954929658551372, x, {x}^{3} \cdot -0.12900613773279798\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 0.5%

      \[\leadsto \color{blue}{0.954929658551372 \cdot x} \]
   6. Step-by-step derivation

      1. *-commutative 0.5%

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

   7. Simplified 0.5%

      \[\leadsto \color{blue}{x \cdot 0.954929658551372} \]
   8. Step-by-step derivation

      1. *-commutative 0.5%

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

      2. add-sqr-sqrt 0.5%

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

      3. sqrt-unprod 0.4%

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

      4. rem-cube-cbrt 0.4%

         \[\leadsto \sqrt{\color{blue}{{\left(\sqrt[3]{0.954929658551372 \cdot x}\right)}^{3}} \cdot \left(0.954929658551372 \cdot x\right)} \]

      5. rem-cube-cbrt 0.4%

         \[\leadsto \sqrt{{\left(\sqrt[3]{0.954929658551372 \cdot x}\right)}^{3} \cdot \color{blue}{{\left(\sqrt[3]{0.954929658551372 \cdot x}\right)}^{3}}} \]

      6. rem-cube-cbrt 0.4%

         \[\leadsto \sqrt{\color{blue}{\left(0.954929658551372 \cdot x\right)} \cdot {\left(\sqrt[3]{0.954929658551372 \cdot x}\right)}^{3}} \]

      7. *-commutative 0.4%

         \[\leadsto \sqrt{\color{blue}{\left(x \cdot 0.954929658551372\right)} \cdot {\left(\sqrt[3]{0.954929658551372 \cdot x}\right)}^{3}} \]

      8. rem-cube-cbrt 0.4%

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

      9. *-commutative 0.4%

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

      10. swap-sqr 0.4%

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

      11. pow2 0.4%

          \[\leadsto \sqrt{\color{blue}{{x}^{2}} \cdot \left(0.954929658551372 \cdot 0.954929658551372\right)} \]

      12. metadata-eval 0.4%

          \[\leadsto \sqrt{{x}^{2} \cdot \color{blue}{0.9118906527810399}} \]

   9. Applied egg-rr 0.4%

      \[\leadsto \color{blue}{\sqrt{{x}^{2} \cdot 0.9118906527810399}} \]

   10. Taylor expanded in x around -inf 6.4%

       \[\leadsto \color{blue}{-0.954929658551372 \cdot x} \]
   11. Step-by-step derivation

       1. *-commutative 6.4%

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

   12. Simplified 6.4%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8:\\ \;\;\;\;0.954929658551372 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.954929658551372\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 5.1% accurate, 3.7× 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 99.8%

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

   1. fmm-def 99.8%

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

   2. *-commutative 99.8%

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

   3. distribute-rgt-neg-in 99.8%

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

   4. unpow3 99.8%

      \[\leadsto \mathsf{fma}\left(0.954929658551372, x, \color{blue}{{x}^{3}} \cdot \left(-0.12900613773279798\right)\right) \]

   5. metadata-eval 99.8%

      \[\leadsto \mathsf{fma}\left(0.954929658551372, x, {x}^{3} \cdot \color{blue}{-0.12900613773279798}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.954929658551372, x, {x}^{3} \cdot -0.12900613773279798\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 55.1%

   \[\leadsto \color{blue}{0.954929658551372 \cdot x} \]
6. Step-by-step derivation

   1. *-commutative 55.1%

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

7. Simplified 55.1%

   \[\leadsto \color{blue}{x \cdot 0.954929658551372} \]
8. Step-by-step derivation

   1. *-commutative 55.1%

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

   2. add-sqr-sqrt 28.6%

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

   3. sqrt-unprod 29.9%

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

   4. rem-cube-cbrt 29.7%

      \[\leadsto \sqrt{\color{blue}{{\left(\sqrt[3]{0.954929658551372 \cdot x}\right)}^{3}} \cdot \left(0.954929658551372 \cdot x\right)} \]

   5. rem-cube-cbrt 29.5%

      \[\leadsto \sqrt{{\left(\sqrt[3]{0.954929658551372 \cdot x}\right)}^{3} \cdot \color{blue}{{\left(\sqrt[3]{0.954929658551372 \cdot x}\right)}^{3}}} \]

   6. rem-cube-cbrt 29.7%

      \[\leadsto \sqrt{\color{blue}{\left(0.954929658551372 \cdot x\right)} \cdot {\left(\sqrt[3]{0.954929658551372 \cdot x}\right)}^{3}} \]

   7. *-commutative 29.7%

      \[\leadsto \sqrt{\color{blue}{\left(x \cdot 0.954929658551372\right)} \cdot {\left(\sqrt[3]{0.954929658551372 \cdot x}\right)}^{3}} \]

   8. rem-cube-cbrt 29.9%

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

   9. *-commutative 29.9%

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

   10. swap-sqr 29.8%

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

   11. pow2 29.8%

       \[\leadsto \sqrt{\color{blue}{{x}^{2}} \cdot \left(0.954929658551372 \cdot 0.954929658551372\right)} \]

   12. metadata-eval 29.9%

       \[\leadsto \sqrt{{x}^{2} \cdot \color{blue}{0.9118906527810399}} \]

9. Applied egg-rr 29.9%

   \[\leadsto \color{blue}{\sqrt{{x}^{2} \cdot 0.9118906527810399}} \]

10. Taylor expanded in x around -inf 4.9%

    \[\leadsto \color{blue}{-0.954929658551372 \cdot x} \]
11. Step-by-step derivation

    1. *-commutative 4.9%

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

12. Simplified 4.9%

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

Reproduce

herbie shell --seed 2024180 
(FPCore (x)
  :name "Rosa's Benchmark"
  :precision binary64
  (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))