Rosa's Benchmark

Percentage Accurate: 99.8% → 99.8%

Time: 10.2s

Alternatives: 4

Speedup: 1.4×

• 34.4% 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, 1.0× 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: 73.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(x \cdot \left(x \cdot x\right)\right) \leq -5000000000000:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot -0.12900613773279798\right)\\ \mathbf{else}:\\ \;\;\;\;0.954929658551372 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<=
      (- (* 0.954929658551372 x) (* 0.12900613773279798 (* x (* x x))))
      -5000000000000.0)
   (* (* x x) (* x -0.12900613773279798))
   (* 0.954929658551372 x)))

C

double code(double x) {
	double tmp;
	if (((0.954929658551372 * x) - (0.12900613773279798 * (x * (x * x)))) <= -5000000000000.0) {
		tmp = (x * x) * (x * -0.12900613773279798);
	} 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)))) <= (-5000000000000.0d0)) then
        tmp = (x * x) * (x * (-0.12900613773279798d0))
    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)))) <= -5000000000000.0) {
		tmp = (x * x) * (x * -0.12900613773279798);
	} else {
		tmp = 0.954929658551372 * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((0.954929658551372 * x) - (0.12900613773279798 * (x * (x * x)))) <= -5000000000000.0:
		tmp = (x * x) * (x * -0.12900613773279798)
	else:
		tmp = 0.954929658551372 * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(0.954929658551372 * x) - Float64(0.12900613773279798 * Float64(x * Float64(x * x)))) <= -5000000000000.0)
		tmp = Float64(Float64(x * x) * Float64(x * -0.12900613773279798));
	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)))) <= -5000000000000.0)
		tmp = (x * x) * (x * -0.12900613773279798);
	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[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5000000000000.0], N[(N[(x * x), $MachinePrecision] * N[(x * -0.12900613773279798), $MachinePrecision]), $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))) < -5e12

   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 inf

      \[\leadsto \color{blue}{\frac{-6450306886639899}{50000000000000000} \cdot {x}^{3}} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{-6450306886639899}{50000000000000000} \cdot {x}^{3} + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-6450306886639899}{50000000000000000}, {x}^{3}, 0\right)} \]

      3. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{-6450306886639899}{50000000000000000}, \color{blue}{{x}^{3} + 0}, 0\right) \]

      4. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(\frac{-6450306886639899}{50000000000000000}, \color{blue}{x \cdot \left(x \cdot x\right)} + 0, 0\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-6450306886639899}{50000000000000000}, x \cdot \color{blue}{{x}^{2}} + 0, 0\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-6450306886639899}{50000000000000000}, \color{blue}{\mathsf{fma}\left(x, {x}^{2}, 0\right)}, 0\right) \]

      7. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{-6450306886639899}{50000000000000000}, \mathsf{fma}\left(x, \color{blue}{{x}^{2} + 0}, 0\right), 0\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-6450306886639899}{50000000000000000}, \mathsf{fma}\left(x, \color{blue}{x \cdot x} + 0, 0\right), 0\right) \]

      9. accelerator-lowering-fma.f64 98.4

         \[\leadsto \mathsf{fma}\left(-0.12900613773279798, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, x, 0\right)}, 0\right), 0\right) \]

   5. Simplified 98.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.12900613773279798, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x, 0\right), 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{-6450306886639899}{50000000000000000} \cdot \left(x \cdot \left(x \cdot x + 0\right) + 0\right)} \]

      2. +-rgt-identity N/A

         \[\leadsto \frac{-6450306886639899}{50000000000000000} \cdot \color{blue}{\left(x \cdot \left(x \cdot x + 0\right)\right)} \]

      3. +-rgt-identity N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. +-rgt-identity N/A

         \[\leadsto \color{blue}{\left(x \cdot x + 0\right)} \cdot \left(x \cdot \frac{-6450306886639899}{50000000000000000}\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \cdot \left(x \cdot \frac{-6450306886639899}{50000000000000000}\right) \]

      10. *-lowering-*.f64 98.5

          \[\leadsto \mathsf{fma}\left(x, x, 0\right) \cdot \color{blue}{\left(x \cdot -0.12900613773279798\right)} \]

   7. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right) \cdot \left(x \cdot -0.12900613773279798\right)} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 98.5

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

   9. Applied egg-rr 98.5%

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

   if -5e12 < (-.f64 (*.f64 #s(literal 238732414637843/250000000000000 binary64) x) (*.f64 #s(literal 6450306886639899/50000000000000000 binary64) (*.f64 (*.f64 x x) x)))

   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 0

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

      1. metadata-eval N/A

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

      2. lft-mult-inverse N/A

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

      3. distribute-neg-frac2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. associate-*r* N/A

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

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

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

      8. unpow2 N/A

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

      9. unpow3 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{{x}^{3}}\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot \color{blue}{\left(-1 \cdot {x}^{3}\right)} \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\left(-1 \cdot {x}^{3}\right) \cdot \left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot \frac{1}{{x}^{2}}\right)\right)} \]

      12. +-rgt-identity N/A

          \[\leadsto \color{blue}{\left(-1 \cdot {x}^{3}\right) \cdot \left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot \frac{1}{{x}^{2}}\right)\right) + 0} \]

      13. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot \left(-1 \cdot {x}^{3}\right)} + 0 \]

   5. Simplified 67.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.954929658551372, x, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{238732414637843}{250000000000000} \cdot x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{238732414637843}{250000000000000}} \]

      3. *-lowering-*.f64 67.5

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

   7. Applied egg-rr 67.5%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(x \cdot \left(x \cdot x\right)\right) \leq -5000000000000:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot -0.12900613773279798\right)\\ \mathbf{else}:\\ \;\;\;\;0.954929658551372 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.4× speedup

Math

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

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 99.8%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

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

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

   5. distribute-rgt-out N/A

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

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

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

   7. *-commutative N/A

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

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

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

   9. accelerator-lowering-fma.f64 N/A

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

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

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

   11. metadata-eval 99.8

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

4. Applied egg-rr 99.8%

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

Alternative 4: 49.2% 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. metadata-eval N/A

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

   2. lft-mult-inverse N/A

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

   3. distribute-neg-frac2 N/A

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

   4. associate-*l* N/A

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

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

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

   6. associate-*r* N/A

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

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

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

   8. unpow2 N/A

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

   9. unpow3 N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{{x}^{3}}\right)\right) \]

   10. mul-1-neg N/A

       \[\leadsto \left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot \color{blue}{\left(-1 \cdot {x}^{3}\right)} \]

   11. *-commutative N/A

       \[\leadsto \color{blue}{\left(-1 \cdot {x}^{3}\right) \cdot \left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot \frac{1}{{x}^{2}}\right)\right)} \]

   12. +-rgt-identity N/A

       \[\leadsto \color{blue}{\left(-1 \cdot {x}^{3}\right) \cdot \left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot \frac{1}{{x}^{2}}\right)\right) + 0} \]

   13. *-commutative N/A

       \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{238732414637843}{250000000000000} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot \left(-1 \cdot {x}^{3}\right)} + 0 \]

5. Simplified 52.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.954929658551372, x, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{\frac{238732414637843}{250000000000000} \cdot x} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{x \cdot \frac{238732414637843}{250000000000000}} \]

   3. *-lowering-*.f64 52.6

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

7. Applied egg-rr 52.6%

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

8. Final simplification 52.6%

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

Reproduce

herbie shell --seed 2024196 
(FPCore (x)
  :name "Rosa's Benchmark"
  :precision binary64
  (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))