Rosa's Benchmark

Percentage Accurate: 99.7% → 99.7%

Time: 9.4s

Alternatives: 4

Speedup: 1.4×

• 34.1% 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: 99.7% 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 2: 74.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;0.954929658551372 \cdot x - 0.12900613773279798 \cdot t\_0 \leq -1000:\\ \;\;\;\;t\_0 \cdot -0.12900613773279798\\ \mathbf{else}:\\ \;\;\;\;0.954929658551372 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= (- (* 0.954929658551372 x) (* 0.12900613773279798 t_0)) -1000.0)
     (* t_0 -0.12900613773279798)
     (* 0.954929658551372 x))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double tmp;
	if (((0.954929658551372 * x) - (0.12900613773279798 * t_0)) <= -1000.0) {
		tmp = t_0 * -0.12900613773279798;
	} else {
		tmp = 0.954929658551372 * x;
	}
	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 (((0.954929658551372d0 * x) - (0.12900613773279798d0 * t_0)) <= (-1000.0d0)) then
        tmp = t_0 * (-0.12900613773279798d0)
    else
        tmp = 0.954929658551372d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (x * x);
	double tmp;
	if (((0.954929658551372 * x) - (0.12900613773279798 * t_0)) <= -1000.0) {
		tmp = t_0 * -0.12900613773279798;
	} else {
		tmp = 0.954929658551372 * x;
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * x)
	tmp = 0
	if ((0.954929658551372 * x) - (0.12900613773279798 * t_0)) <= -1000.0:
		tmp = t_0 * -0.12900613773279798
	else:
		tmp = 0.954929658551372 * x
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (Float64(Float64(0.954929658551372 * x) - Float64(0.12900613773279798 * t_0)) <= -1000.0)
		tmp = Float64(t_0 * -0.12900613773279798);
	else
		tmp = Float64(0.954929658551372 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * x);
	tmp = 0.0;
	if (((0.954929658551372 * x) - (0.12900613773279798 * t_0)) <= -1000.0)
		tmp = t_0 * -0.12900613773279798;
	else
		tmp = 0.954929658551372 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(0.954929658551372 * x), $MachinePrecision] - N[(0.12900613773279798 * t$95$0), $MachinePrecision]), $MachinePrecision], -1000.0], N[(t$95$0 * -0.12900613773279798), $MachinePrecision], N[(0.954929658551372 * x), $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))) < -1e3

   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. *-lowering-*.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

         \[\leadsto \frac{-6450306886639899}{50000000000000000} \cdot \left(x \cdot \color{blue}{{x}^{2}}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{-6450306886639899}{50000000000000000} \cdot \color{blue}{\left(x \cdot {x}^{2}\right)} \]

      5. unpow2 N/A

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

      6. *-lowering-*.f64 98.6

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

   5. Simplified 98.6%

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

   if -1e3 < (-.f64 (*.f64 #s(literal 238732414637843/250000000000000 binary64) x) (*.f64 #s(literal 6450306886639899/50000000000000000 binary64) (*.f64 (*.f64 x x) 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

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

      1. *-lowering-*.f64 57.4

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

   5. Simplified 57.4%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(x \cdot \left(x \cdot x\right)\right) \leq -1000:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot -0.12900613773279798\\ \mathbf{else}:\\ \;\;\;\;0.954929658551372 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.4× 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(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. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(\frac{238732414637843}{250000000000000} + \frac{-6450306886639899}{50000000000000000} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto x \cdot \left(\frac{238732414637843}{250000000000000} + \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right)} \cdot {x}^{2}\right) \]

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

      \[\leadsto x \cdot \left(\frac{238732414637843}{250000000000000} + \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot {x}^{2}\right)\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto x \cdot \left(\frac{238732414637843}{250000000000000} + \left(\mathsf{neg}\left(\color{blue}{{x}^{2} \cdot \frac{6450306886639899}{50000000000000000}}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto x \cdot \left(\color{blue}{\frac{238732414637843}{250000000000000} \cdot 1} + \left(\mathsf{neg}\left({x}^{2} \cdot \frac{6450306886639899}{50000000000000000}\right)\right)\right) \]

   5. lft-mult-inverse N/A

      \[\leadsto x \cdot \left(\frac{238732414637843}{250000000000000} \cdot \color{blue}{\left(\frac{1}{\mathsf{neg}\left({x}^{2}\right)} \cdot \left(\mathsf{neg}\left({x}^{2}\right)\right)\right)} + \left(\mathsf{neg}\left({x}^{2} \cdot \frac{6450306886639899}{50000000000000000}\right)\right)\right) \]

   6. distribute-neg-frac2 N/A

      \[\leadsto x \cdot \left(\frac{238732414637843}{250000000000000} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)} \cdot \left(\mathsf{neg}\left({x}^{2}\right)\right)\right) + \left(\mathsf{neg}\left({x}^{2} \cdot \frac{6450306886639899}{50000000000000000}\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(\frac{238732414637843}{250000000000000} \cdot \left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)\right) \cdot \left(\mathsf{neg}\left({x}^{2}\right)\right)} + \left(\mathsf{neg}\left({x}^{2} \cdot \frac{6450306886639899}{50000000000000000}\right)\right)\right) \]

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

      \[\leadsto x \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot \frac{1}{{x}^{2}}\right)\right)} \cdot \left(\mathsf{neg}\left({x}^{2}\right)\right) + \left(\mathsf{neg}\left({x}^{2} \cdot \frac{6450306886639899}{50000000000000000}\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot \left(\mathsf{neg}\left({x}^{2}\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{6450306886639899}{50000000000000000} \cdot {x}^{2}}\right)\right)\right) \]

   10. distribute-rgt-neg-out N/A

       \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot \left(\mathsf{neg}\left({x}^{2}\right)\right) + \color{blue}{\frac{6450306886639899}{50000000000000000} \cdot \left(\mathsf{neg}\left({x}^{2}\right)\right)}\right) \]

   11. distribute-rgt-in N/A

       \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left({x}^{2}\right)\right) \cdot \left(\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot \frac{1}{{x}^{2}}\right)\right) + \frac{6450306886639899}{50000000000000000}\right)\right)} \]

   12. +-commutative N/A

       \[\leadsto x \cdot \left(\left(\mathsf{neg}\left({x}^{2}\right)\right) \cdot \color{blue}{\left(\frac{6450306886639899}{50000000000000000} + \left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)}\right) \]

   13. sub-neg N/A

       \[\leadsto x \cdot \left(\left(\mathsf{neg}\left({x}^{2}\right)\right) \cdot \color{blue}{\left(\frac{6450306886639899}{50000000000000000} - \frac{238732414637843}{250000000000000} \cdot \frac{1}{{x}^{2}}\right)}\right) \]

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

       \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left({x}^{2} \cdot \left(\frac{6450306886639899}{50000000000000000} - \frac{238732414637843}{250000000000000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \]

5. Simplified 99.8%

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

Alternative 4: 49.0% accurate, 4.0× 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 99.8%

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

3. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 45.4

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

5. Simplified 45.4%

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

Reproduce

herbie shell --seed 2024204 
(FPCore (x)
  :name "Rosa's Benchmark"
  :precision binary64
  (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))