Rosa's Benchmark

Percentage Accurate: 99.7% → 99.8%

Time: 6.6s

Alternatives: 4

Speedup: 1.4×

• 32.5% 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: 99.7% 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, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Applied rewrites 99.8%

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

Alternative 2: 74.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(\frac{-6450306886639899}{50000000000000000} \cdot {x}^{2}\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 97.6%

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

   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. remove-double-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-out N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lft-mult-inverse N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. associate-*l/ N/A

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

      16. *-lft-identity N/A

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

      17. *-lft-identity N/A

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

      18. associate-*l/ N/A

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

      19. associate-*l* N/A

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

      20. *-commutative N/A

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

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

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

   5. Applied rewrites 60.8%

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

Alternative 3: 51.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. Initial program 99.7%

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

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 99.9

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

      12. lift-*.f64 N/A

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

      13. rem-square-sqrt N/A

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

      14. sqrt-prod N/A

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

      15. sqr-neg-rev N/A

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

      16. pow2 N/A

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

      17. sqrt-pow1 N/A

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

      18. metadata-eval N/A

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

      19. unpow1 N/A

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

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

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

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

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

      22. lower-*.f64 N/A

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

      23. metadata-eval 97.5

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 6.5

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

   7. Applied rewrites 6.5%

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

   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. remove-double-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-out N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lft-mult-inverse N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. associate-*l/ N/A

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

      16. *-lft-identity N/A

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

      17. *-lft-identity N/A

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

      18. associate-*l/ N/A

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

      19. associate-*l* N/A

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

      20. *-commutative N/A

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

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

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

   5. Applied rewrites 60.8%

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

Alternative 4: 5.1% 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. 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. lift-*.f64 N/A

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

   3. fp-cancel-sub-sign-inv N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 99.8

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

   12. lift-*.f64 N/A

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

   13. rem-square-sqrt N/A

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

   14. sqrt-prod N/A

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

   15. sqr-neg-rev N/A

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

   16. pow2 N/A

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

   17. sqrt-pow1 N/A

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

   18. metadata-eval N/A

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

   19. unpow1 N/A

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

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

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

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

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

   22. lower-*.f64 N/A

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

   23. metadata-eval 57.4

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

4. Applied rewrites 57.4%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 5.2

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

7. Applied rewrites 5.2%

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

Reproduce

herbie shell --seed 2024333 
(FPCore (x)
  :name "Rosa's Benchmark"
  :precision binary64
  (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))