Rosa's Benchmark

Percentage Accurate: 99.7% → 99.8%

Time: 5.1s

Alternatives: 3

Speedup: 1.2×

• 33.1% 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: 99.7% 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 99.9%

   \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
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 99.9%

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

   5. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

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 99.9%

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

   1. associate-*r* 99.9%

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

   2. distribute-rgt-out-- 99.8%

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

   3. pow2 99.8%

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

3. Applied egg-rr 99.8%

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

   1. add-sqr-sqrt 99.8%

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

   2. pow2 99.8%

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

   3. *-commutative 99.8%

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

   4. sqrt-prod 99.8%

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

   5. unpow2 99.8%

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

   6. sqrt-prod 48.3%

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

   7. add-sqr-sqrt 99.8%

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

5. Applied egg-rr 99.8%

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

   1. unpow2 99.8%

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

   2. *-commutative 99.8%

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

   3. *-commutative 99.8%

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

   4. swap-sqr 99.8%

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

   5. rem-square-sqrt 99.8%

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

   6. associate-*r* 99.9%

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

7. Applied egg-rr 99.9%

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

8. Final simplification 99.9%

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

Alternative 3: 49.8% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 0.954929658551372 * x

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(0.954929658551372 * 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. Taylor expanded in x around 0 48.1%

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

   1. *-commutative 48.1%

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

4. Simplified 48.1%

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

5. Final simplification 48.1%

   \[\leadsto 0.954929658551372 \cdot x \]

Reproduce

herbie shell --seed 2023331 
(FPCore (x)
  :name "Rosa's Benchmark"
  :precision binary64
  (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))