Rosa's Benchmark

Percentage Accurate: 99.8% → 99.8%

Time: 8.2s

Alternatives: 6

Speedup: 1.0×

• 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, 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. fma-neg 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. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(x * N[(-0.12900613773279798 * N[Power[x, 2.0], $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. Add Preprocessing

3. Taylor expanded in x around 0 99.8%

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

   1. metadata-eval 99.8%

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

   2. distribute-lft-neg-in 99.8%

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

   3. +-commutative 99.8%

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

   4. distribute-lft-neg-in 99.8%

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

   5. metadata-eval 99.8%

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

   6. fma-define 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

   \[\leadsto x \cdot \mathsf{fma}\left(-0.12900613773279798, {x}^{2}, 0.954929658551372\right) \]
7. Add Preprocessing

Alternative 3: 99.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. add-cbrt-cube 90.0%

      \[\leadsto 0.954929658551372 \cdot x - \color{blue}{\sqrt[3]{\left(\left(0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right) \cdot \left(0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)}} \]

   2. pow1/3 63.3%

      \[\leadsto 0.954929658551372 \cdot x - \color{blue}{{\left(\left(\left(0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right) \cdot \left(0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)}^{0.3333333333333333}} \]

   3. pow3 63.3%

      \[\leadsto 0.954929658551372 \cdot x - {\color{blue}{\left({\left(0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)}^{3}\right)}}^{0.3333333333333333} \]

   4. *-commutative 63.3%

      \[\leadsto 0.954929658551372 \cdot x - {\left({\color{blue}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.12900613773279798\right)}}^{3}\right)}^{0.3333333333333333} \]

   5. unpow-prod-down 63.3%

      \[\leadsto 0.954929658551372 \cdot x - {\color{blue}{\left({\left(\left(x \cdot x\right) \cdot x\right)}^{3} \cdot {0.12900613773279798}^{3}\right)}}^{0.3333333333333333} \]

   6. pow3 63.3%

      \[\leadsto 0.954929658551372 \cdot x - {\left({\color{blue}{\left({x}^{3}\right)}}^{3} \cdot {0.12900613773279798}^{3}\right)}^{0.3333333333333333} \]

   7. pow-pow 63.3%

      \[\leadsto 0.954929658551372 \cdot x - {\left(\color{blue}{{x}^{\left(3 \cdot 3\right)}} \cdot {0.12900613773279798}^{3}\right)}^{0.3333333333333333} \]

   8. metadata-eval 63.3%

      \[\leadsto 0.954929658551372 \cdot x - {\left({x}^{\color{blue}{9}} \cdot {0.12900613773279798}^{3}\right)}^{0.3333333333333333} \]

   9. metadata-eval 63.3%

      \[\leadsto 0.954929658551372 \cdot x - {\left({x}^{9} \cdot \color{blue}{0.0021469954286136776}\right)}^{0.3333333333333333} \]

4. Applied egg-rr 63.3%

   \[\leadsto 0.954929658551372 \cdot x - \color{blue}{{\left({x}^{9} \cdot 0.0021469954286136776\right)}^{0.3333333333333333}} \]
5. Step-by-step derivation

   1. unpow1/3 90.0%

      \[\leadsto 0.954929658551372 \cdot x - \color{blue}{\sqrt[3]{{x}^{9} \cdot 0.0021469954286136776}} \]

6. Simplified 90.0%

   \[\leadsto 0.954929658551372 \cdot x - \color{blue}{\sqrt[3]{{x}^{9} \cdot 0.0021469954286136776}} \]
7. Step-by-step derivation

   1. *-commutative 90.0%

      \[\leadsto 0.954929658551372 \cdot x - \sqrt[3]{\color{blue}{0.0021469954286136776 \cdot {x}^{9}}} \]

   2. metadata-eval 90.0%

      \[\leadsto 0.954929658551372 \cdot x - \sqrt[3]{\color{blue}{{0.12900613773279798}^{3}} \cdot {x}^{9}} \]

   3. sqr-pow 40.0%

      \[\leadsto 0.954929658551372 \cdot x - \sqrt[3]{{0.12900613773279798}^{3} \cdot \color{blue}{\left({x}^{\left(\frac{9}{2}\right)} \cdot {x}^{\left(\frac{9}{2}\right)}\right)}} \]

   4. metadata-eval 40.0%

      \[\leadsto 0.954929658551372 \cdot x - \sqrt[3]{{0.12900613773279798}^{3} \cdot \left({x}^{\color{blue}{4.5}} \cdot {x}^{\left(\frac{9}{2}\right)}\right)} \]

   5. metadata-eval 40.0%

      \[\leadsto 0.954929658551372 \cdot x - \sqrt[3]{{0.12900613773279798}^{3} \cdot \left({x}^{\color{blue}{\left(3 + 1.5\right)}} \cdot {x}^{\left(\frac{9}{2}\right)}\right)} \]

   6. pow-prod-up 40.0%

      \[\leadsto 0.954929658551372 \cdot x - \sqrt[3]{{0.12900613773279798}^{3} \cdot \left(\color{blue}{\left({x}^{3} \cdot {x}^{1.5}\right)} \cdot {x}^{\left(\frac{9}{2}\right)}\right)} \]

   7. metadata-eval 40.0%

      \[\leadsto 0.954929658551372 \cdot x - \sqrt[3]{{0.12900613773279798}^{3} \cdot \left(\left({x}^{\color{blue}{\left(1.5 \cdot 2\right)}} \cdot {x}^{1.5}\right) \cdot {x}^{\left(\frac{9}{2}\right)}\right)} \]

   8. pow-pow 40.0%

      \[\leadsto 0.954929658551372 \cdot x - \sqrt[3]{{0.12900613773279798}^{3} \cdot \left(\left(\color{blue}{{\left({x}^{1.5}\right)}^{2}} \cdot {x}^{1.5}\right) \cdot {x}^{\left(\frac{9}{2}\right)}\right)} \]

   9. metadata-eval 40.0%

      \[\leadsto 0.954929658551372 \cdot x - \sqrt[3]{{0.12900613773279798}^{3} \cdot \left(\left({\left({x}^{1.5}\right)}^{2} \cdot {x}^{1.5}\right) \cdot {x}^{\color{blue}{4.5}}\right)} \]

   10. metadata-eval 40.0%

       \[\leadsto 0.954929658551372 \cdot x - \sqrt[3]{{0.12900613773279798}^{3} \cdot \left(\left({\left({x}^{1.5}\right)}^{2} \cdot {x}^{1.5}\right) \cdot {x}^{\color{blue}{\left(3 + 1.5\right)}}\right)} \]

   11. pow-prod-up 40.0%

       \[\leadsto 0.954929658551372 \cdot x - \sqrt[3]{{0.12900613773279798}^{3} \cdot \left(\left({\left({x}^{1.5}\right)}^{2} \cdot {x}^{1.5}\right) \cdot \color{blue}{\left({x}^{3} \cdot {x}^{1.5}\right)}\right)} \]

   12. metadata-eval 40.0%

       \[\leadsto 0.954929658551372 \cdot x - \sqrt[3]{{0.12900613773279798}^{3} \cdot \left(\left({\left({x}^{1.5}\right)}^{2} \cdot {x}^{1.5}\right) \cdot \left({x}^{\color{blue}{\left(1.5 \cdot 2\right)}} \cdot {x}^{1.5}\right)\right)} \]

   13. pow-pow 40.0%

       \[\leadsto 0.954929658551372 \cdot x - \sqrt[3]{{0.12900613773279798}^{3} \cdot \left(\left({\left({x}^{1.5}\right)}^{2} \cdot {x}^{1.5}\right) \cdot \left(\color{blue}{{\left({x}^{1.5}\right)}^{2}} \cdot {x}^{1.5}\right)\right)} \]

   14. unswap-sqr 40.0%

       \[\leadsto 0.954929658551372 \cdot x - \sqrt[3]{{0.12900613773279798}^{3} \cdot \color{blue}{\left(\left({\left({x}^{1.5}\right)}^{2} \cdot {\left({x}^{1.5}\right)}^{2}\right) \cdot \left({x}^{1.5} \cdot {x}^{1.5}\right)\right)}} \]

   15. unpow2 40.0%

       \[\leadsto 0.954929658551372 \cdot x - \sqrt[3]{{0.12900613773279798}^{3} \cdot \left(\left({\left({x}^{1.5}\right)}^{2} \cdot {\left({x}^{1.5}\right)}^{2}\right) \cdot \color{blue}{{\left({x}^{1.5}\right)}^{2}}\right)} \]

   16. pow3 40.0%

       \[\leadsto 0.954929658551372 \cdot x - \sqrt[3]{{0.12900613773279798}^{3} \cdot \color{blue}{{\left({\left({x}^{1.5}\right)}^{2}\right)}^{3}}} \]

   17. cube-prod 40.0%

       \[\leadsto 0.954929658551372 \cdot x - \sqrt[3]{\color{blue}{{\left(0.12900613773279798 \cdot {\left({x}^{1.5}\right)}^{2}\right)}^{3}}} \]

   18. pow3 40.0%

       \[\leadsto 0.954929658551372 \cdot x - \sqrt[3]{\color{blue}{\left(\left(0.12900613773279798 \cdot {\left({x}^{1.5}\right)}^{2}\right) \cdot \left(0.12900613773279798 \cdot {\left({x}^{1.5}\right)}^{2}\right)\right) \cdot \left(0.12900613773279798 \cdot {\left({x}^{1.5}\right)}^{2}\right)}} \]

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

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(x * Float64(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[(x * 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. Add Preprocessing

3. Final simplification 99.8%

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

Alternative 5: 50.9% 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. Add Preprocessing

   3. Taylor expanded in x around 0 61.2%

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

      1. *-commutative 61.2%

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

   5. Simplified 61.2%

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

   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. Add Preprocessing

   3. Taylor expanded in x around 0 0.5%

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

      1. *-commutative 0.5%

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

   5. Simplified 0.5%

      \[\leadsto \color{blue}{x \cdot 0.954929658551372} \]
   6. 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. *-commutative 0.4%

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

      5. *-commutative 0.4%

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

      6. swap-sqr 0.4%

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

      7. unpow2 0.4%

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

      8. metadata-eval 0.4%

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

   7. Applied egg-rr 0.4%

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

   8. Taylor expanded in x around -inf 6.6%

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

      1. *-commutative 6.6%

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

   10. Simplified 6.6%

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

4. Final simplification 50.3%

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

Alternative 6: 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. Add Preprocessing

3. Taylor expanded in x around 0 49.1%

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

   1. *-commutative 49.1%

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

5. Simplified 49.1%

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

   1. *-commutative 49.1%

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

   2. add-sqr-sqrt 22.6%

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

   3. sqrt-unprod 29.1%

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

   4. *-commutative 29.1%

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

   5. *-commutative 29.1%

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

   6. swap-sqr 29.5%

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

   7. unpow2 29.5%

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

   8. metadata-eval 29.4%

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

7. Applied egg-rr 29.4%

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

8. Taylor expanded in x around -inf 5.1%

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

   1. *-commutative 5.1%

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

10. Simplified 5.1%

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

11. Final simplification 5.1%

    \[\leadsto x \cdot -0.954929658551372 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024060 
(FPCore (x)
  :name "Rosa's Benchmark"
  :precision binary64
  (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))