Rosa's Benchmark

Percentage Accurate: 99.8% → 99.8%

Time: 9.7s

Alternatives: 6

Speedup: 1.4×

• 33.8% 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: 99.8% 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, 1.3× 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 99.5%

   \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
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. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

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

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

   8. lift-*.f64 N/A

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

   9. distribute-rgt-out N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

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

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

   13. lower-fma.f64 N/A

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

   14. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. metadata-eval N/A

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

   5. cancel-sign-sub-inv N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

Alternative 2: 73.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.954929658551372 - 0.12900613773279798 \cdot \left(x \cdot \left(x \cdot x\right)\right) \leq -400000:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot -0.12900613773279798\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.954929658551372\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<=
      (- (* x 0.954929658551372) (* 0.12900613773279798 (* x (* x x))))
      -400000.0)
   (* (* x x) (* x -0.12900613773279798))
   (* x 0.954929658551372)))

C

double code(double x) {
	double tmp;
	if (((x * 0.954929658551372) - (0.12900613773279798 * (x * (x * x)))) <= -400000.0) {
		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)))) <= (-400000.0d0)) 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)))) <= -400000.0) {
		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)))) <= -400000.0:
		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)))) <= -400000.0)
		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)))) <= -400000.0)
		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], -400000.0], 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))) < -4e5

   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. 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 98.9

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

   5. Applied rewrites 98.9%

      \[\leadsto \color{blue}{-0.12900613773279798 \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]
   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 -4e5 < (-.f64 (*.f64 #s(literal 238732414637843/250000000000000 binary64) x) (*.f64 #s(literal 6450306886639899/50000000000000000 binary64) (*.f64 (*.f64 x x) x)))

   1. Initial program 99.3%

      \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
   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 69.7

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

   5. Applied rewrites 69.7%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.954929658551372 - 0.12900613773279798 \cdot \left(x \cdot \left(x \cdot x\right)\right) \leq -400000:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot -0.12900613773279798\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.954929658551372\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \cdot 0.954929658551372 - 0.12900613773279798 \cdot t\_0 \leq -400000:\\ \;\;\;\;-0.12900613773279798 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.954929658551372\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= (- (* x 0.954929658551372) (* 0.12900613773279798 t_0)) -400000.0)
     (* -0.12900613773279798 t_0)
     (* x 0.954929658551372))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double tmp;
	if (((x * 0.954929658551372) - (0.12900613773279798 * t_0)) <= -400000.0) {
		tmp = -0.12900613773279798 * t_0;
	} else {
		tmp = x * 0.954929658551372;
	}
	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 (((x * 0.954929658551372d0) - (0.12900613773279798d0 * t_0)) <= (-400000.0d0)) then
        tmp = (-0.12900613773279798d0) * t_0
    else
        tmp = x * 0.954929658551372d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (x * x);
	double tmp;
	if (((x * 0.954929658551372) - (0.12900613773279798 * t_0)) <= -400000.0) {
		tmp = -0.12900613773279798 * t_0;
	} else {
		tmp = x * 0.954929658551372;
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * x)
	tmp = 0
	if ((x * 0.954929658551372) - (0.12900613773279798 * t_0)) <= -400000.0:
		tmp = -0.12900613773279798 * t_0
	else:
		tmp = x * 0.954929658551372
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (Float64(Float64(x * 0.954929658551372) - Float64(0.12900613773279798 * t_0)) <= -400000.0)
		tmp = Float64(-0.12900613773279798 * t_0);
	else
		tmp = Float64(x * 0.954929658551372);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * x);
	tmp = 0.0;
	if (((x * 0.954929658551372) - (0.12900613773279798 * t_0)) <= -400000.0)
		tmp = -0.12900613773279798 * t_0;
	else
		tmp = x * 0.954929658551372;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(x * 0.954929658551372), $MachinePrecision] - N[(0.12900613773279798 * t$95$0), $MachinePrecision]), $MachinePrecision], -400000.0], N[(-0.12900613773279798 * t$95$0), $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))) < -4e5

   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. 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 98.9

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

   5. Applied rewrites 98.9%

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

   if -4e5 < (-.f64 (*.f64 #s(literal 238732414637843/250000000000000 binary64) x) (*.f64 #s(literal 6450306886639899/50000000000000000 binary64) (*.f64 (*.f64 x x) x)))

   1. Initial program 99.3%

      \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
   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 69.7

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

   5. Applied rewrites 69.7%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.954929658551372 - 0.12900613773279798 \cdot \left(x \cdot \left(x \cdot x\right)\right) \leq -400000:\\ \;\;\;\;-0.12900613773279798 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.954929658551372\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 1.4× speedup

Math

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

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 * x), -0.12900613773279798, 0.954929658551372))
end

Wolfram

code[x_] := N[(x * N[(N[(x * x), $MachinePrecision] * -0.12900613773279798 + 0.954929658551372), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
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. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

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

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

   8. lift-*.f64 N/A

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

   9. distribute-rgt-out N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

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

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

   13. lower-fma.f64 N/A

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

   14. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

Alternative 5: 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 99.5%

   \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
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. Applied rewrites 99.8%

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

Alternative 6: 48.8% accurate, 4.0× 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 99.5%

   \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
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 51.6

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

5. Applied rewrites 51.6%

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

6. Final simplification 51.6%

   \[\leadsto x \cdot 0.954929658551372 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(FPCore (x)
  :name "Rosa's Benchmark"
  :precision binary64
  (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))