Rosa's Benchmark

Percentage Accurate: 99.8% → 99.8%

Time: 8.7s

Alternatives: 7

Speedup: 1.2×

• 33.5% 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: 99.8% 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, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (0.954929658551372d0 + ((-0.12900613773279798d0) * (x * x)))
end function

Java

public static double code(double x) {
	return x * (0.954929658551372 + (-0.12900613773279798 * (x * x)));
}

Python

def code(x):
	return x * (0.954929658551372 + (-0.12900613773279798 * (x * x)))

Julia

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

MATLAB

function tmp = code(x)
	tmp = x * (0.954929658551372 + (-0.12900613773279798 * (x * x)));
end

Wolfram

code[x_] := N[(x * N[(0.954929658551372 + N[(-0.12900613773279798 * 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. Step-by-step derivation

   1. associate-*r* N/A

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

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

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

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{238732414637843}{250000000000000} - \frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{238732414637843}{250000000000000} + \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(\mathsf{neg}\left(\left(\frac{6450306886639899}{50000000000000000} \cdot x\right) \cdot x\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(\mathsf{neg}\left(x \cdot \left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right)}\right)\right)\right)\right) \]

   13. metadata-eval 99.8%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-6450306886639899}{50000000000000000}\right)\right)\right)\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{x \cdot \left(0.954929658551372 + x \cdot \left(x \cdot -0.12900613773279798\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{238732414637843}{250000000000000} + x \cdot \left(x \cdot \frac{-6450306886639899}{50000000000000000}\right)\right), \color{blue}{x}\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(x \cdot \left(x \cdot \frac{-6450306886639899}{50000000000000000}\right)\right)\right), x\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(\left(x \cdot x\right) \cdot \frac{-6450306886639899}{50000000000000000}\right)\right), x\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(\frac{-6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)\right), x\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(\frac{-6450306886639899}{50000000000000000}, \left(x \cdot x\right)\right)\right), x\right) \]

   7. *-lowering-*.f64 99.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(\frac{-6450306886639899}{50000000000000000}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right) \]

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

   \[\leadsto x \cdot \left(0.954929658551372 + -0.12900613773279798 \cdot \left(x \cdot x\right)\right) \]
8. Add Preprocessing

Alternative 2: 74.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7:\\ \;\;\;\;0.954929658551372 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(-0.12900613773279798 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.7)
   (* 0.954929658551372 x)
   (* (* x x) (* -0.12900613773279798 x))))

C

double code(double x) {
	double tmp;
	if (x <= 2.7) {
		tmp = 0.954929658551372 * x;
	} else {
		tmp = (x * x) * (-0.12900613773279798 * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.7d0) then
        tmp = 0.954929658551372d0 * x
    else
        tmp = (x * x) * ((-0.12900613773279798d0) * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.7) {
		tmp = 0.954929658551372 * x;
	} else {
		tmp = (x * x) * (-0.12900613773279798 * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.7:
		tmp = 0.954929658551372 * x
	else:
		tmp = (x * x) * (-0.12900613773279798 * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.7)
		tmp = Float64(0.954929658551372 * x);
	else
		tmp = Float64(Float64(x * x) * Float64(-0.12900613773279798 * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.7)
		tmp = 0.954929658551372 * x;
	else
		tmp = (x * x) * (-0.12900613773279798 * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.7], N[(0.954929658551372 * x), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(-0.12900613773279798 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.7000000000000002

   1. Initial program 99.8%

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

      1. associate-*r* N/A

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

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

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{238732414637843}{250000000000000} - \frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{238732414637843}{250000000000000} + \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(\mathsf{neg}\left(\left(\frac{6450306886639899}{50000000000000000} \cdot x\right) \cdot x\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(\mathsf{neg}\left(x \cdot \left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right)}\right)\right)\right)\right) \]

      13. metadata-eval 99.8%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-6450306886639899}{50000000000000000}\right)\right)\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\frac{238732414637843}{250000000000000}}\right) \]
   6. Step-by-step derivation

   7. Simplified 64.8%

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

   if 2.7000000000000002 < x

   1. Initial program 99.6%

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

      1. associate-*r* N/A

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

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

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{238732414637843}{250000000000000} - \frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{238732414637843}{250000000000000} + \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(\mathsf{neg}\left(\left(\frac{6450306886639899}{50000000000000000} \cdot x\right) \cdot x\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(\mathsf{neg}\left(x \cdot \left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right)}\right)\right)\right)\right) \]

      13. metadata-eval 99.8%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-6450306886639899}{50000000000000000}\right)\right)\right)\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \left(0.954929658551372 + x \cdot \left(x \cdot -0.12900613773279798\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{238732414637843}{250000000000000} + x \cdot \left(x \cdot \frac{-6450306886639899}{50000000000000000}\right)\right), \color{blue}{x}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(x \cdot \left(x \cdot \frac{-6450306886639899}{50000000000000000}\right)\right)\right), x\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(\left(x \cdot x\right) \cdot \frac{-6450306886639899}{50000000000000000}\right)\right), x\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(\frac{-6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(\frac{-6450306886639899}{50000000000000000}, \left(x \cdot x\right)\right)\right), x\right) \]

      7. *-lowering-*.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(\frac{-6450306886639899}{50000000000000000}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{-6450306886639899}{50000000000000000} \cdot {x}^{2}\right)}, x\right) \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-6450306886639899}{50000000000000000}, \left({x}^{2}\right)\right), x\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-6450306886639899}{50000000000000000}, \left(x \cdot x\right)\right), x\right) \]

      3. *-lowering-*.f64 97.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-6450306886639899}{50000000000000000}, \mathsf{*.f64}\left(x, x\right)\right), x\right) \]

   9. Simplified 97.8%

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\frac{-6450306886639899}{50000000000000000} \cdot x\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{-6450306886639899}{50000000000000000}} \cdot x\right)\right) \]

       5. *-lowering-*.f64 97.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\frac{-6450306886639899}{50000000000000000}, \color{blue}{x}\right)\right) \]

   11. Applied egg-rr 97.8%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(-0.12900613773279798 \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7:\\ \;\;\;\;0.954929658551372 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(-0.12900613773279798 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7:\\ \;\;\;\;0.954929658551372 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.7)
   (* 0.954929658551372 x)
   (* x (* -0.12900613773279798 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= 2.7) {
		tmp = 0.954929658551372 * x;
	} else {
		tmp = x * (-0.12900613773279798 * (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.7d0) then
        tmp = 0.954929658551372d0 * x
    else
        tmp = x * ((-0.12900613773279798d0) * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.7) {
		tmp = 0.954929658551372 * x;
	} else {
		tmp = x * (-0.12900613773279798 * (x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.7:
		tmp = 0.954929658551372 * x
	else:
		tmp = x * (-0.12900613773279798 * (x * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.7)
		tmp = Float64(0.954929658551372 * x);
	else
		tmp = Float64(x * Float64(-0.12900613773279798 * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.7)
		tmp = 0.954929658551372 * x;
	else
		tmp = x * (-0.12900613773279798 * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.7], N[(0.954929658551372 * x), $MachinePrecision], N[(x * N[(-0.12900613773279798 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.7000000000000002

   1. Initial program 99.8%

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

      1. associate-*r* N/A

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

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

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{238732414637843}{250000000000000} - \frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{238732414637843}{250000000000000} + \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(\mathsf{neg}\left(\left(\frac{6450306886639899}{50000000000000000} \cdot x\right) \cdot x\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(\mathsf{neg}\left(x \cdot \left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right)}\right)\right)\right)\right) \]

      13. metadata-eval 99.8%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-6450306886639899}{50000000000000000}\right)\right)\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\frac{238732414637843}{250000000000000}}\right) \]
   6. Step-by-step derivation

   7. Simplified 64.8%

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

   if 2.7000000000000002 < x

   1. Initial program 99.6%

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

      1. associate-*r* N/A

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

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

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{238732414637843}{250000000000000} - \frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{238732414637843}{250000000000000} + \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(\mathsf{neg}\left(\left(\frac{6450306886639899}{50000000000000000} \cdot x\right) \cdot x\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(\mathsf{neg}\left(x \cdot \left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right)}\right)\right)\right)\right) \]

      13. metadata-eval 99.8%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-6450306886639899}{50000000000000000}\right)\right)\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-6450306886639899}{50000000000000000} \cdot {x}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{-6450306886639899}{50000000000000000}, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{-6450306886639899}{50000000000000000}, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      8. *-lowering-*.f64 97.8%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{-6450306886639899}{50000000000000000}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   7. Simplified 97.8%

      \[\leadsto \color{blue}{x \cdot \left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7:\\ \;\;\;\;0.954929658551372 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7:\\ \;\;\;\;0.954929658551372 \cdot x\\ \mathbf{else}:\\ \;\;\;\;-0.12900613773279798 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.7)
   (* 0.954929658551372 x)
   (* -0.12900613773279798 (* x (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= 2.7) {
		tmp = 0.954929658551372 * x;
	} else {
		tmp = -0.12900613773279798 * (x * (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.7d0) then
        tmp = 0.954929658551372d0 * x
    else
        tmp = (-0.12900613773279798d0) * (x * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.7) {
		tmp = 0.954929658551372 * x;
	} else {
		tmp = -0.12900613773279798 * (x * (x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.7:
		tmp = 0.954929658551372 * x
	else:
		tmp = -0.12900613773279798 * (x * (x * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.7)
		tmp = Float64(0.954929658551372 * x);
	else
		tmp = Float64(-0.12900613773279798 * Float64(x * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.7)
		tmp = 0.954929658551372 * x;
	else
		tmp = -0.12900613773279798 * (x * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.7], N[(0.954929658551372 * x), $MachinePrecision], N[(-0.12900613773279798 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.7000000000000002

   1. Initial program 99.8%

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

      1. associate-*r* N/A

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

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

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{238732414637843}{250000000000000} - \frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{238732414637843}{250000000000000} + \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(\mathsf{neg}\left(\left(\frac{6450306886639899}{50000000000000000} \cdot x\right) \cdot x\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(\mathsf{neg}\left(x \cdot \left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right)}\right)\right)\right)\right) \]

      13. metadata-eval 99.8%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-6450306886639899}{50000000000000000}\right)\right)\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\frac{238732414637843}{250000000000000}}\right) \]
   6. Step-by-step derivation

   7. Simplified 64.8%

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

   if 2.7000000000000002 < x

   1. Initial program 99.6%

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

      1. associate-*r* N/A

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

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

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{238732414637843}{250000000000000} - \frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{238732414637843}{250000000000000} + \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(\mathsf{neg}\left(\left(\frac{6450306886639899}{50000000000000000} \cdot x\right) \cdot x\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(\mathsf{neg}\left(x \cdot \left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right)}\right)\right)\right)\right) \]

      13. metadata-eval 99.8%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-6450306886639899}{50000000000000000}\right)\right)\right)\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \left(0.954929658551372 + x \cdot \left(x \cdot -0.12900613773279798\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{238732414637843}{250000000000000} + x \cdot \left(x \cdot \frac{-6450306886639899}{50000000000000000}\right)\right), \color{blue}{x}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(x \cdot \left(x \cdot \frac{-6450306886639899}{50000000000000000}\right)\right)\right), x\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(\left(x \cdot x\right) \cdot \frac{-6450306886639899}{50000000000000000}\right)\right), x\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(\frac{-6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(\frac{-6450306886639899}{50000000000000000}, \left(x \cdot x\right)\right)\right), x\right) \]

      7. *-lowering-*.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(\frac{-6450306886639899}{50000000000000000}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(\frac{-6450306886639899}{50000000000000000}, \color{blue}{\left({x}^{3}\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-6450306886639899}{50000000000000000}, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-6450306886639899}{50000000000000000}, \left(x \cdot {x}^{\color{blue}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{-6450306886639899}{50000000000000000}, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-6450306886639899}{50000000000000000}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      6. *-lowering-*.f64 97.7%

         \[\leadsto \mathsf{*.f64}\left(\frac{-6450306886639899}{50000000000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   9. Simplified 97.7%

      \[\leadsto \color{blue}{-0.12900613773279798 \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7:\\ \;\;\;\;0.954929658551372 \cdot x\\ \mathbf{else}:\\ \;\;\;\;-0.12900613773279798 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (0.954929658551372d0 + (x * ((-0.12900613773279798d0) * x)))
end function

Java

public static double code(double x) {
	return x * (0.954929658551372 + (x * (-0.12900613773279798 * x)));
}

Python

def code(x):
	return x * (0.954929658551372 + (x * (-0.12900613773279798 * x)))

Julia

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

MATLAB

function tmp = code(x)
	tmp = x * (0.954929658551372 + (x * (-0.12900613773279798 * x)));
end

Wolfram

code[x_] := N[(x * N[(0.954929658551372 + N[(x * N[(-0.12900613773279798 * 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. Step-by-step derivation

   1. associate-*r* N/A

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

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

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

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{238732414637843}{250000000000000} - \frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{238732414637843}{250000000000000} + \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(\mathsf{neg}\left(\left(\frac{6450306886639899}{50000000000000000} \cdot x\right) \cdot x\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(\mathsf{neg}\left(x \cdot \left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right)}\right)\right)\right)\right) \]

   13. metadata-eval 99.8%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-6450306886639899}{50000000000000000}\right)\right)\right)\right) \]

3. Simplified 99.8%

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

5. Final simplification 99.8%

   \[\leadsto x \cdot \left(0.954929658551372 + x \cdot \left(-0.12900613773279798 \cdot x\right)\right) \]
6. Add Preprocessing

Alternative 6: 51.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7:\\ \;\;\;\;0.954929658551372 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.954929658551372\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.7) (* 0.954929658551372 x) (* x -0.954929658551372)))

C

double code(double x) {
	double tmp;
	if (x <= 2.7) {
		tmp = 0.954929658551372 * x;
	} else {
		tmp = x * -0.954929658551372;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.7d0) then
        tmp = 0.954929658551372d0 * x
    else
        tmp = x * (-0.954929658551372d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.7) {
		tmp = 0.954929658551372 * x;
	} else {
		tmp = x * -0.954929658551372;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.7:
		tmp = 0.954929658551372 * x
	else:
		tmp = x * -0.954929658551372
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.7)
		tmp = Float64(0.954929658551372 * x);
	else
		tmp = Float64(x * -0.954929658551372);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.7)
		tmp = 0.954929658551372 * x;
	else
		tmp = x * -0.954929658551372;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.7], N[(0.954929658551372 * x), $MachinePrecision], N[(x * -0.954929658551372), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.7000000000000002

   1. Initial program 99.8%

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

      1. associate-*r* N/A

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

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

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{238732414637843}{250000000000000} - \frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{238732414637843}{250000000000000} + \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(\mathsf{neg}\left(\left(\frac{6450306886639899}{50000000000000000} \cdot x\right) \cdot x\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(\mathsf{neg}\left(x \cdot \left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right)}\right)\right)\right)\right) \]

      13. metadata-eval 99.8%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-6450306886639899}{50000000000000000}\right)\right)\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\frac{238732414637843}{250000000000000}}\right) \]
   6. Step-by-step derivation

   7. Simplified 64.8%

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

   if 2.7000000000000002 < x

   1. Initial program 99.6%

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

      1. associate-*r* N/A

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

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

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{238732414637843}{250000000000000} - \frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{238732414637843}{250000000000000} + \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(\mathsf{neg}\left(\left(\frac{6450306886639899}{50000000000000000} \cdot x\right) \cdot x\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(\mathsf{neg}\left(x \cdot \left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right)}\right)\right)\right)\right) \]

      13. metadata-eval 99.8%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-6450306886639899}{50000000000000000}\right)\right)\right)\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \left(0.954929658551372 + x \cdot \left(x \cdot -0.12900613773279798\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. distribute-rgt-out-- N/A

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

      6. associate-*r* N/A

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

      7. flip-- N/A

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

      8. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{238732414637843}{250000000000000} \cdot x + \frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)}{\left(\frac{238732414637843}{250000000000000} \cdot x\right) \cdot \left(\frac{238732414637843}{250000000000000} \cdot x\right) - \left(\frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)}\right)}\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{\left(\frac{238732414637843}{250000000000000} \cdot x\right) \cdot \left(\frac{238732414637843}{250000000000000} \cdot x\right) - \left(\frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)}{\frac{238732414637843}{250000000000000} \cdot x + \frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)}}}\right)\right) \]

      11. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\frac{238732414637843}{250000000000000} \cdot x - \color{blue}{\frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)}}\right)\right) \]

      12. fmm-def N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\mathsf{fma}\left(\frac{238732414637843}{250000000000000}, \color{blue}{x}, \mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\mathsf{fma}\left(\frac{238732414637843}{250000000000000}, x, \mathsf{neg}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{6450306886639899}{50000000000000000}\right)\right)}\right)\right) \]

   6. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{x \cdot \left(0.954929658551372 + -0.12900613773279798 \cdot \left(x \cdot x\right)\right)}}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{250000000000000}{238732414637843}}{x}\right)}\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 0.5%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\frac{250000000000000}{238732414637843}, \color{blue}{x}\right)\right) \]

   9. Simplified 0.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{1.0471975511965979}{x}}} \]
   10. Step-by-step derivation

       1. clear-num N/A

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

       2. /-lowering-/.f64 0.5%

          \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\frac{250000000000000}{238732414637843}}\right) \]

   11. Applied egg-rr 0.5%

       \[\leadsto \color{blue}{\frac{x}{1.0471975511965979}} \]
   12. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\frac{250000000000000}{238732414637843}}{x}}} \]

       2. div-inv N/A

          \[\leadsto \frac{1}{\frac{250000000000000}{238732414637843} \cdot \color{blue}{\frac{1}{x}}} \]

       3. associate-/r* N/A

          \[\leadsto \frac{\frac{1}{\frac{250000000000000}{238732414637843}}}{\color{blue}{\frac{1}{x}}} \]

       4. frac-2neg N/A

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

       5. metadata-eval N/A

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

       6. metadata-eval N/A

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

       7. metadata-eval N/A

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

       8. metadata-eval N/A

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

       9. div-inv N/A

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

       10. inv-pow N/A

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

       11. sqr-pow N/A

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

       12. unpow-prod-down N/A

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

       13. sqr-neg N/A

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

       14. unpow-prod-down N/A

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

       15. sqr-pow N/A

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

       16. inv-pow N/A

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

       17. remove-double-div N/A

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

       18. *-commutative N/A

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

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

           \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{\mathsf{neg}\left(\frac{250000000000000}{238732414637843}\right)}\right)}\right) \]

       20. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{\frac{-250000000000000}{238732414637843}}\right)\right) \]

       21. metadata-eval 6.4%

           \[\leadsto \mathsf{*.f64}\left(x, \frac{-238732414637843}{250000000000000}\right) \]

   13. Applied egg-rr 6.4%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7:\\ \;\;\;\;0.954929658551372 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.954929658551372\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 5.0% accurate, 3.7× 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 99.8%

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

   1. associate-*r* N/A

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

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

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

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{238732414637843}{250000000000000} - \frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{238732414637843}{250000000000000} + \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(\mathsf{neg}\left(\left(\frac{6450306886639899}{50000000000000000} \cdot x\right) \cdot x\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(\mathsf{neg}\left(x \cdot \left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot x\right)\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{6450306886639899}{50000000000000000}\right)\right)}\right)\right)\right)\right) \]

   13. metadata-eval 99.8%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{238732414637843}{250000000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-6450306886639899}{50000000000000000}\right)\right)\right)\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{x \cdot \left(0.954929658551372 + x \cdot \left(x \cdot -0.12900613773279798\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

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

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

   5. distribute-rgt-out-- N/A

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

   6. associate-*r* N/A

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

   7. flip-- N/A

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

   8. clear-num N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{238732414637843}{250000000000000} \cdot x + \frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)}{\left(\frac{238732414637843}{250000000000000} \cdot x\right) \cdot \left(\frac{238732414637843}{250000000000000} \cdot x\right) - \left(\frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)}\right)}\right) \]

   10. clear-num N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{\left(\frac{238732414637843}{250000000000000} \cdot x\right) \cdot \left(\frac{238732414637843}{250000000000000} \cdot x\right) - \left(\frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)}{\frac{238732414637843}{250000000000000} \cdot x + \frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)}}}\right)\right) \]

   11. flip-- N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\frac{238732414637843}{250000000000000} \cdot x - \color{blue}{\frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)}}\right)\right) \]

   12. fmm-def N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\mathsf{fma}\left(\frac{238732414637843}{250000000000000}, \color{blue}{x}, \mathsf{neg}\left(\frac{6450306886639899}{50000000000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)}\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\mathsf{fma}\left(\frac{238732414637843}{250000000000000}, x, \mathsf{neg}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{6450306886639899}{50000000000000000}\right)\right)}\right)\right) \]

6. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{x \cdot \left(0.954929658551372 + -0.12900613773279798 \cdot \left(x \cdot x\right)\right)}}} \]

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{250000000000000}{238732414637843}}{x}\right)}\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f64 46.6%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\frac{250000000000000}{238732414637843}, \color{blue}{x}\right)\right) \]

9. Simplified 46.6%

   \[\leadsto \frac{1}{\color{blue}{\frac{1.0471975511965979}{x}}} \]
10. Step-by-step derivation

    1. clear-num N/A

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

    2. /-lowering-/.f64 46.6%

       \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\frac{250000000000000}{238732414637843}}\right) \]

11. Applied egg-rr 46.6%

    \[\leadsto \color{blue}{\frac{x}{1.0471975511965979}} \]
12. Step-by-step derivation

    1. clear-num N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{\frac{250000000000000}{238732414637843}}{x}}} \]

    2. div-inv N/A

       \[\leadsto \frac{1}{\frac{250000000000000}{238732414637843} \cdot \color{blue}{\frac{1}{x}}} \]

    3. associate-/r* N/A

       \[\leadsto \frac{\frac{1}{\frac{250000000000000}{238732414637843}}}{\color{blue}{\frac{1}{x}}} \]

    4. frac-2neg N/A

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

    5. metadata-eval N/A

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

    6. metadata-eval N/A

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

    7. metadata-eval N/A

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

    8. metadata-eval N/A

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

    9. div-inv N/A

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

    10. inv-pow N/A

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

    11. sqr-pow N/A

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

    12. unpow-prod-down N/A

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

    13. sqr-neg N/A

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

    14. unpow-prod-down N/A

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

    15. sqr-pow N/A

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

    16. inv-pow N/A

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

    17. remove-double-div N/A

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

    18. *-commutative N/A

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

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

        \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{\mathsf{neg}\left(\frac{250000000000000}{238732414637843}\right)}\right)}\right) \]

    20. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{\frac{-250000000000000}{238732414637843}}\right)\right) \]

    21. metadata-eval 5.4%

        \[\leadsto \mathsf{*.f64}\left(x, \frac{-238732414637843}{250000000000000}\right) \]

13. Applied egg-rr 5.4%

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

Reproduce

herbie shell --seed 2024150 
(FPCore (x)
  :name "Rosa's Benchmark"
  :precision binary64
  (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))