Rosa's Benchmark

Percentage Accurate: 99.8% → 99.8%

Time: 14.2s

Alternatives: 4

Speedup: 1.2×

• 32.3% 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, 0.1× speedup

Math

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

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), Float64(x * 0.12900613773279798), 0.954929658551372))
end

Wolfram

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

   1. *-commutative 99.8%

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

   2. metadata-eval 99.8%

      \[\leadsto x \cdot \left(0.954929658551372 - \color{blue}{\sqrt{0.016642583572733644}} \cdot \left(x \cdot x\right)\right) \]

   3. pow2 99.8%

      \[\leadsto x \cdot \left(0.954929658551372 - \sqrt{0.016642583572733644} \cdot \color{blue}{{x}^{2}}\right) \]

   4. metadata-eval 99.8%

      \[\leadsto x \cdot \left(0.954929658551372 - \sqrt{0.016642583572733644} \cdot {x}^{\color{blue}{\left(\frac{4}{2}\right)}}\right) \]

   5. sqrt-pow1 95.9%

      \[\leadsto x \cdot \left(0.954929658551372 - \sqrt{0.016642583572733644} \cdot \color{blue}{\sqrt{{x}^{4}}}\right) \]

   6. sqrt-prod 95.9%

      \[\leadsto x \cdot \left(0.954929658551372 - \color{blue}{\sqrt{0.016642583572733644 \cdot {x}^{4}}}\right) \]

   7. *-commutative 95.9%

      \[\leadsto x \cdot \left(0.954929658551372 - \sqrt{\color{blue}{{x}^{4} \cdot 0.016642583572733644}}\right) \]

   8. sub-neg 95.9%

      \[\leadsto x \cdot \color{blue}{\left(0.954929658551372 + \left(-\sqrt{{x}^{4} \cdot 0.016642583572733644}\right)\right)} \]

   9. +-commutative 95.9%

      \[\leadsto x \cdot \color{blue}{\left(\left(-\sqrt{{x}^{4} \cdot 0.016642583572733644}\right) + 0.954929658551372\right)} \]

   10. sqrt-prod 95.9%

       \[\leadsto x \cdot \left(\left(-\color{blue}{\sqrt{{x}^{4}} \cdot \sqrt{0.016642583572733644}}\right) + 0.954929658551372\right) \]

   11. metadata-eval 95.9%

       \[\leadsto x \cdot \left(\left(-\sqrt{{x}^{4}} \cdot \color{blue}{0.12900613773279798}\right) + 0.954929658551372\right) \]

   12. sqrt-pow1 99.8%

       \[\leadsto x \cdot \left(\left(-\color{blue}{{x}^{\left(\frac{4}{2}\right)}} \cdot 0.12900613773279798\right) + 0.954929658551372\right) \]

   13. metadata-eval 99.8%

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

   14. pow2 99.8%

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

   15. associate-*l* 99.8%

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

   16. distribute-lft-neg-in 99.8%

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

   17. fma-define 99.8%

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

10. Applied egg-rr 99.8%

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

Alternative 2: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(x * N[(0.954929658551372 - N[(x * N[(x * 0.12900613773279798), $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. add-sqr-sqrt 99.7%

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

   2. sqrt-unprod 95.9%

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

   3. swap-sqr 95.9%

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

   4. pow-prod-up 95.9%

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

   5. metadata-eval 95.9%

      \[\leadsto x \cdot \left(0.954929658551372 - \sqrt{{x}^{\color{blue}{4}} \cdot \left(0.12900613773279798 \cdot 0.12900613773279798\right)}\right) \]

   6. metadata-eval 95.9%

      \[\leadsto x \cdot \left(0.954929658551372 - \sqrt{{x}^{4} \cdot \color{blue}{0.016642583572733644}}\right) \]

8. Applied egg-rr 95.9%

   \[\leadsto x \cdot \left(0.954929658551372 - \color{blue}{\sqrt{{x}^{4} \cdot 0.016642583572733644}}\right) \]
9. Step-by-step derivation

   1. sqrt-prod 95.9%

      \[\leadsto x \cdot \left(0.954929658551372 - \color{blue}{\sqrt{{x}^{4}} \cdot \sqrt{0.016642583572733644}}\right) \]

   2. metadata-eval 95.9%

      \[\leadsto x \cdot \left(0.954929658551372 - \sqrt{{x}^{4}} \cdot \color{blue}{0.12900613773279798}\right) \]

   3. sqrt-pow1 99.8%

      \[\leadsto x \cdot \left(0.954929658551372 - \color{blue}{{x}^{\left(\frac{4}{2}\right)}} \cdot 0.12900613773279798\right) \]

   4. metadata-eval 99.8%

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

   5. pow2 99.8%

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

   6. associate-*l* 99.8%

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

   7. *-commutative 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 3: 51.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7:\\ \;\;\;\;x \cdot 0.954929658551372\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.954929658551372\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.7) (* x 0.954929658551372) (* x -0.954929658551372)))

C

double code(double x) {
	double tmp;
	if (x <= 2.7) {
		tmp = x * 0.954929658551372;
	} else {
		tmp = x * -0.954929658551372;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.7d0) then
        tmp = x * 0.954929658551372d0
    else
        tmp = x * (-0.954929658551372d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.7) {
		tmp = x * 0.954929658551372;
	} else {
		tmp = x * -0.954929658551372;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.7:
		tmp = x * 0.954929658551372
	else:
		tmp = x * -0.954929658551372
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.7)
		tmp = Float64(x * 0.954929658551372);
	else
		tmp = Float64(x * -0.954929658551372);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.7)
		tmp = x * 0.954929658551372;
	else
		tmp = x * -0.954929658551372;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.7], N[(x * 0.954929658551372), $MachinePrecision], N[(x * -0.954929658551372), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.7000000000000002

   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 68.9%

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

      1. *-commutative 68.9%

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

   7. Simplified 68.9%

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

   if 2.7000000000000002 < x

   1. Initial program 99.7%

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

      1. fmm-def 99.7%

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

      2. *-commutative 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. unpow3 99.7%

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

      5. metadata-eval 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.954929658551372, x, {x}^{3} \cdot -0.12900613773279798\right)} \]
   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. rem-exp-log 0.5%

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

      3. add-sqr-sqrt 0.5%

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

      4. sqrt-unprod 0.4%

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

      5. rem-exp-log 0.4%

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

      6. *-commutative 0.4%

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

      7. rem-exp-log 0.4%

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

      8. *-commutative 0.4%

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

      9. swap-sqr 0.4%

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

      10. pow2 0.4%

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

      11. 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. Add Preprocessing

Alternative 4: 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 49.7%

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

   1. *-commutative 49.7%

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

7. Simplified 49.7%

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

   1. *-commutative 49.7%

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

   2. rem-exp-log 19.0%

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

   3. add-sqr-sqrt 19.0%

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

   4. sqrt-unprod 8.0%

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

   5. rem-exp-log 8.1%

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

   6. *-commutative 8.1%

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

   7. rem-exp-log 20.6%

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

   8. *-commutative 20.6%

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

   9. swap-sqr 20.6%

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

   10. pow2 20.6%

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

   11. metadata-eval 20.6%

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

9. Applied egg-rr 20.6%

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

10. Taylor expanded in x around -inf 5.1%

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

    1. *-commutative 5.1%

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

12. Simplified 5.1%

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

Reproduce

herbie shell --seed 2024170 
(FPCore (x)
  :name "Rosa's Benchmark"
  :precision binary64
  (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))