Rosa's Benchmark

Percentage Accurate: 99.7% → 99.8%

Time: 8.8s

Alternatives: 8

Speedup: 1.4×

• 32.8% 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: 99.7% 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, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

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

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot x\right)}, \frac{-6450306886639899}{50000000000000000}, \frac{238732414637843}{250000000000000} \cdot x\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \color{blue}{\left(x \cdot x\right)}, \frac{-6450306886639899}{50000000000000000}, \frac{238732414637843}{250000000000000} \cdot x\right) \]

   3. cube-unmult N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{-6450306886639899}{50000000000000000}, \frac{238732414637843}{250000000000000} \cdot x\right) \]

   4. lift-pow.f64 99.8

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 74.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.954929658551372 - 0.12900613773279798 \cdot \left(x \cdot \left(x \cdot x\right)\right) \leq -40:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot -0.12900613773279798\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.954929658551372\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<=
      (- (* x 0.954929658551372) (* 0.12900613773279798 (* x (* x x))))
      -40.0)
   (* (* x x) (* x -0.12900613773279798))
   (* x 0.954929658551372)))

C

double code(double x) {
	double tmp;
	if (((x * 0.954929658551372) - (0.12900613773279798 * (x * (x * x)))) <= -40.0) {
		tmp = (x * x) * (x * -0.12900613773279798);
	} else {
		tmp = x * 0.954929658551372;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((x * 0.954929658551372d0) - (0.12900613773279798d0 * (x * (x * x)))) <= (-40.0d0)) then
        tmp = (x * x) * (x * (-0.12900613773279798d0))
    else
        tmp = x * 0.954929658551372d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((x * 0.954929658551372) - (0.12900613773279798 * (x * (x * x)))) <= -40.0) {
		tmp = (x * x) * (x * -0.12900613773279798);
	} else {
		tmp = x * 0.954929658551372;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((x * 0.954929658551372) - (0.12900613773279798 * (x * (x * x)))) <= -40.0:
		tmp = (x * x) * (x * -0.12900613773279798)
	else:
		tmp = x * 0.954929658551372
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x * 0.954929658551372) - Float64(0.12900613773279798 * Float64(x * Float64(x * x)))) <= -40.0)
		tmp = Float64(Float64(x * x) * Float64(x * -0.12900613773279798));
	else
		tmp = Float64(x * 0.954929658551372);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((x * 0.954929658551372) - (0.12900613773279798 * (x * (x * x)))) <= -40.0)
		tmp = (x * x) * (x * -0.12900613773279798);
	else
		tmp = x * 0.954929658551372;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x * 0.954929658551372), $MachinePrecision] - N[(0.12900613773279798 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -40.0], N[(N[(x * x), $MachinePrecision] * N[(x * -0.12900613773279798), $MachinePrecision]), $MachinePrecision], N[(x * 0.954929658551372), $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))) < -40

   1. Initial program 99.6%

      \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
   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. lower-*.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. lower-*.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. lower-*.f64 98.6

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

   5. Applied rewrites 98.6%

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

   7. Applied rewrites 98.8%

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

   if -40 < (-.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. lower-*.f64 65.9

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

   5. Applied rewrites 65.9%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.954929658551372 - 0.12900613773279798 \cdot \left(x \cdot \left(x \cdot x\right)\right) \leq -40:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot -0.12900613773279798\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.954929658551372\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(FPCore (x)
  :name "Rosa's Benchmark"
  :precision binary64
  (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))