Rosa's Benchmark

Percentage Accurate: 99.7% → 99.8%

Time: 7.3s

Alternatives: 7

Speedup: 1.2×

• 32.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(0.954929658551372 * x + N[(-0.12900613773279798 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. cancel-sign-sub-inv 99.9%

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

   2. fma-def 99.9%

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

   3. metadata-eval 99.9%

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

   4. unpow3 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \lor \neg \left(x \leq 2.8\right):\\ \;\;\;\;x \cdot \left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.954929658551372 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -2.7) (not (<= x 2.8)))
   (* x (* -0.12900613773279798 (* x x)))
   (* 0.954929658551372 x)))

C

double code(double x) {
	double tmp;
	if ((x <= -2.7) || !(x <= 2.8)) {
		tmp = x * (-0.12900613773279798 * (x * x));
	} else {
		tmp = 0.954929658551372 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-2.7d0)) .or. (.not. (x <= 2.8d0))) then
        tmp = x * ((-0.12900613773279798d0) * (x * x))
    else
        tmp = 0.954929658551372d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -2.7) || !(x <= 2.8)) {
		tmp = x * (-0.12900613773279798 * (x * x));
	} else {
		tmp = 0.954929658551372 * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -2.7) or not (x <= 2.8):
		tmp = x * (-0.12900613773279798 * (x * x))
	else:
		tmp = 0.954929658551372 * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -2.7) || !(x <= 2.8))
		tmp = Float64(x * Float64(-0.12900613773279798 * Float64(x * x)));
	else
		tmp = Float64(0.954929658551372 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -2.7) || ~((x <= 2.8)))
		tmp = x * (-0.12900613773279798 * (x * x));
	else
		tmp = 0.954929658551372 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -2.7], N[Not[LessEqual[x, 2.8]], $MachinePrecision]], N[(x * N[(-0.12900613773279798 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.954929658551372 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7000000000000002 or 2.7999999999999998 < x

   1. Initial program 99.9%

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

      1. sqr-neg 99.9%

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

      2. associate-*r* 99.9%

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

      3. distribute-rgt-out-- 99.9%

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

      4. sub-neg 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-*r* 99.9%

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

      7. distribute-rgt-neg-in 99.9%

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

      8. remove-double-neg 99.9%

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

      9. *-commutative 99.9%

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

      10. fma-def 99.9%

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

      11. distribute-rgt-neg-out 99.9%

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

      12. distribute-lft-neg-in 99.9%

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

      13. *-commutative 99.9%

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

      14. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 99.7%

      \[\leadsto x \cdot \color{blue}{\left(-0.12900613773279798 \cdot {x}^{2}\right)} \]
   5. Step-by-step derivation

      1. unpow2 99.7%

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

   6. Simplified 99.7%

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

   if -2.7000000000000002 < x < 2.7999999999999998

   1. Initial program 99.9%

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

      1. sqr-neg 99.9%

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

      2. associate-*r* 99.9%

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

      3. distribute-rgt-out-- 99.9%

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

      4. sub-neg 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-*r* 99.9%

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

      7. distribute-rgt-neg-in 99.9%

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

      8. remove-double-neg 99.9%

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

      9. *-commutative 99.9%

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

      10. fma-def 99.9%

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

      11. distribute-rgt-neg-out 99.9%

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

      12. distribute-lft-neg-in 99.9%

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

      13. *-commutative 99.9%

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

      14. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 98.8%

      \[\leadsto \color{blue}{0.954929658551372 \cdot x} \]
   5. Step-by-step derivation

      1. *-commutative 98.8%

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

   6. Simplified 98.8%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \lor \neg \left(x \leq 2.8\right):\\ \;\;\;\;x \cdot \left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.954929658551372 \cdot x\\ \end{array} \]

Alternative 3: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7:\\ \;\;\;\;x \cdot \left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 2.8:\\ \;\;\;\;0.954929658551372 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot -0.12900613773279798\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.7)
   (* x (* -0.12900613773279798 (* x x)))
   (if (<= x 2.8)
     (* 0.954929658551372 x)
     (* x (* x (* x -0.12900613773279798))))))

C

double code(double x) {
	double tmp;
	if (x <= -2.7) {
		tmp = x * (-0.12900613773279798 * (x * x));
	} else if (x <= 2.8) {
		tmp = 0.954929658551372 * x;
	} else {
		tmp = x * (x * (x * -0.12900613773279798));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.7d0)) then
        tmp = x * ((-0.12900613773279798d0) * (x * x))
    else if (x <= 2.8d0) then
        tmp = 0.954929658551372d0 * x
    else
        tmp = x * (x * (x * (-0.12900613773279798d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.7) {
		tmp = x * (-0.12900613773279798 * (x * x));
	} else if (x <= 2.8) {
		tmp = 0.954929658551372 * x;
	} else {
		tmp = x * (x * (x * -0.12900613773279798));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.7:
		tmp = x * (-0.12900613773279798 * (x * x))
	elif x <= 2.8:
		tmp = 0.954929658551372 * x
	else:
		tmp = x * (x * (x * -0.12900613773279798))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.7)
		tmp = Float64(x * Float64(-0.12900613773279798 * Float64(x * x)));
	elseif (x <= 2.8)
		tmp = Float64(0.954929658551372 * x);
	else
		tmp = Float64(x * Float64(x * Float64(x * -0.12900613773279798)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.7)
		tmp = x * (-0.12900613773279798 * (x * x));
	elseif (x <= 2.8)
		tmp = 0.954929658551372 * x;
	else
		tmp = x * (x * (x * -0.12900613773279798));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.7], N[(x * N[(-0.12900613773279798 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.8], N[(0.954929658551372 * x), $MachinePrecision], N[(x * N[(x * N[(x * -0.12900613773279798), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.7000000000000002

   1. Initial program 99.9%

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

      1. sqr-neg 99.9%

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

      2. associate-*r* 99.9%

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

      3. distribute-rgt-out-- 99.9%

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

      4. sub-neg 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-*r* 99.9%

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

      7. distribute-rgt-neg-in 99.9%

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

      8. remove-double-neg 99.9%

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

      9. *-commutative 99.9%

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

      10. fma-def 99.9%

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

      11. distribute-rgt-neg-out 99.9%

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

      12. distribute-lft-neg-in 99.9%

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

      13. *-commutative 99.9%

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

      14. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 99.9%

      \[\leadsto x \cdot \color{blue}{\left(-0.12900613773279798 \cdot {x}^{2}\right)} \]
   5. Step-by-step derivation

      1. unpow2 99.9%

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

   6. Simplified 99.9%

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

   if -2.7000000000000002 < x < 2.7999999999999998

   1. Initial program 99.9%

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

      1. sqr-neg 99.9%

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

      2. associate-*r* 99.9%

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

      3. distribute-rgt-out-- 99.9%

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

      4. sub-neg 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-*r* 99.9%

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

      7. distribute-rgt-neg-in 99.9%

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

      8. remove-double-neg 99.9%

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

      9. *-commutative 99.9%

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

      10. fma-def 99.9%

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

      11. distribute-rgt-neg-out 99.9%

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

      12. distribute-lft-neg-in 99.9%

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

      13. *-commutative 99.9%

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

      14. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 98.8%

      \[\leadsto \color{blue}{0.954929658551372 \cdot x} \]
   5. Step-by-step derivation

      1. *-commutative 98.8%

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

   6. Simplified 98.8%

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

   if 2.7999999999999998 < x

   1. Initial program 99.9%

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

      1. sqr-neg 99.9%

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

      2. associate-*r* 99.8%

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

      3. distribute-rgt-out-- 99.9%

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

      4. sub-neg 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-*r* 99.9%

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

      7. distribute-rgt-neg-in 99.9%

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

      8. remove-double-neg 99.9%

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

      9. *-commutative 99.9%

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

      10. fma-def 99.9%

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

      11. distribute-rgt-neg-out 99.9%

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

      12. distribute-lft-neg-in 99.9%

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

      13. *-commutative 99.9%

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

      14. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.9%

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

      2. flip-+ 43.5%

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

      3. pow1 43.5%

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

      4. pow1 43.5%

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

      5. pow-prod-up 43.5%

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

      6. metadata-eval 43.5%

         \[\leadsto x \cdot \frac{{\left(x \cdot \left(x \cdot -0.12900613773279798\right)\right)}^{\color{blue}{2}} - 0.954929658551372 \cdot 0.954929658551372}{x \cdot \left(x \cdot -0.12900613773279798\right) - 0.954929658551372} \]

      7. metadata-eval 43.5%

         \[\leadsto x \cdot \frac{{\left(x \cdot \left(x \cdot -0.12900613773279798\right)\right)}^{2} - \color{blue}{0.9118906527810399}}{x \cdot \left(x \cdot -0.12900613773279798\right) - 0.954929658551372} \]

      8. fma-neg 43.5%

         \[\leadsto x \cdot \frac{{\left(x \cdot \left(x \cdot -0.12900613773279798\right)\right)}^{2} - 0.9118906527810399}{\color{blue}{\mathsf{fma}\left(x, x \cdot -0.12900613773279798, -0.954929658551372\right)}} \]

      9. metadata-eval 43.5%

         \[\leadsto x \cdot \frac{{\left(x \cdot \left(x \cdot -0.12900613773279798\right)\right)}^{2} - 0.9118906527810399}{\mathsf{fma}\left(x, x \cdot -0.12900613773279798, \color{blue}{-0.954929658551372}\right)} \]

   5. Applied egg-rr 43.5%

      \[\leadsto x \cdot \color{blue}{\frac{{\left(x \cdot \left(x \cdot -0.12900613773279798\right)\right)}^{2} - 0.9118906527810399}{\mathsf{fma}\left(x, x \cdot -0.12900613773279798, -0.954929658551372\right)}} \]

   6. Taylor expanded in x around inf 99.5%

      \[\leadsto x \cdot \color{blue}{\left(-0.12900613773279798 \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

      1. unpow2 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-*r* 99.6%

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

   8. Simplified 99.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7:\\ \;\;\;\;x \cdot \left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 2.8:\\ \;\;\;\;0.954929658551372 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot -0.12900613773279798\right)\right)\\ \end{array} \]

Alternative 4: 99.7% 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 99.9%

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

2. Final simplification 99.9%

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

Alternative 5: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. sqr-neg 99.9%

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

   2. associate-*r* 99.9%

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

   3. distribute-rgt-out-- 99.9%

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

   4. sub-neg 99.9%

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

   5. +-commutative 99.9%

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

   6. associate-*r* 99.9%

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

   7. distribute-rgt-neg-in 99.9%

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

   8. remove-double-neg 99.9%

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

   9. *-commutative 99.9%

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

   10. fma-def 99.9%

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

   11. distribute-rgt-neg-out 99.9%

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

   12. distribute-lft-neg-in 99.9%

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

   13. *-commutative 99.9%

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

   14. metadata-eval 99.9%

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

3. Simplified 99.9%

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

   1. fma-udef 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 6: 53.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7:\\ \;\;\;\;x \cdot -0.954929658551372\\ \mathbf{elif}\;x \leq 2.8:\\ \;\;\;\;0.954929658551372 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.954929658551372\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.7)
   (* x -0.954929658551372)
   (if (<= x 2.8) (* 0.954929658551372 x) (* x -0.954929658551372))))

C

double code(double x) {
	double tmp;
	if (x <= -2.7) {
		tmp = x * -0.954929658551372;
	} else if (x <= 2.8) {
		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 = x * (-0.954929658551372d0)
    else if (x <= 2.8d0) 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 = x * -0.954929658551372;
	} else if (x <= 2.8) {
		tmp = 0.954929658551372 * x;
	} else {
		tmp = x * -0.954929658551372;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.7:
		tmp = x * -0.954929658551372
	elif x <= 2.8:
		tmp = 0.954929658551372 * x
	else:
		tmp = x * -0.954929658551372
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.7)
		tmp = Float64(x * -0.954929658551372);
	elseif (x <= 2.8)
		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 = x * -0.954929658551372;
	elseif (x <= 2.8)
		tmp = 0.954929658551372 * x;
	else
		tmp = x * -0.954929658551372;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.7], N[(x * -0.954929658551372), $MachinePrecision], If[LessEqual[x, 2.8], N[(0.954929658551372 * x), $MachinePrecision], N[(x * -0.954929658551372), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7000000000000002 or 2.7999999999999998 < x

   1. Initial program 99.9%

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

      1. sqr-neg 99.9%

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

      2. associate-*r* 99.9%

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

      3. distribute-rgt-out-- 99.9%

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

      4. sub-neg 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-*r* 99.9%

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

      7. distribute-rgt-neg-in 99.9%

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

      8. remove-double-neg 99.9%

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

      9. *-commutative 99.9%

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

      10. fma-def 99.9%

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

      11. distribute-rgt-neg-out 99.9%

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

      12. distribute-lft-neg-in 99.9%

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

      13. *-commutative 99.9%

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

      14. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 0.4%

      \[\leadsto \color{blue}{0.954929658551372 \cdot x} \]
   5. Step-by-step derivation

      1. *-commutative 0.4%

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

   6. Simplified 0.4%

      \[\leadsto \color{blue}{x \cdot 0.954929658551372} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 0.2%

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

      2. sqrt-prod 25.5%

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

      3. metadata-eval 25.5%

         \[\leadsto \sqrt{x \cdot x} \cdot \color{blue}{\sqrt{0.9118906527810399}} \]

      4. sqrt-prod 25.5%

         \[\leadsto \color{blue}{\sqrt{\left(x \cdot x\right) \cdot 0.9118906527810399}} \]

      5. associate-*r* 25.5%

         \[\leadsto \sqrt{\color{blue}{x \cdot \left(x \cdot 0.9118906527810399\right)}} \]

   8. Applied egg-rr 25.5%

      \[\leadsto \color{blue}{\sqrt{x \cdot \left(x \cdot 0.9118906527810399\right)}} \]

   9. Taylor expanded in x around -inf 6.3%

      \[\leadsto \color{blue}{-0.954929658551372 \cdot x} \]
   10. Step-by-step derivation

       1. *-commutative 6.3%

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

   11. Simplified 6.3%

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

   if -2.7000000000000002 < x < 2.7999999999999998

   1. Initial program 99.9%

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

      1. sqr-neg 99.9%

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

      2. associate-*r* 99.9%

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

      3. distribute-rgt-out-- 99.9%

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

      4. sub-neg 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-*r* 99.9%

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

      7. distribute-rgt-neg-in 99.9%

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

      8. remove-double-neg 99.9%

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

      9. *-commutative 99.9%

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

      10. fma-def 99.9%

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

      11. distribute-rgt-neg-out 99.9%

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

      12. distribute-lft-neg-in 99.9%

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

      13. *-commutative 99.9%

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

      14. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 98.8%

      \[\leadsto \color{blue}{0.954929658551372 \cdot x} \]
   5. Step-by-step derivation

      1. *-commutative 98.8%

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

   6. Simplified 98.8%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7:\\ \;\;\;\;x \cdot -0.954929658551372\\ \mathbf{elif}\;x \leq 2.8:\\ \;\;\;\;0.954929658551372 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.954929658551372\\ \end{array} \]

Alternative 7: 5.1% 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 99.9%

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

   1. sqr-neg 99.9%

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

   2. associate-*r* 99.9%

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

   3. distribute-rgt-out-- 99.9%

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

   4. sub-neg 99.9%

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

   5. +-commutative 99.9%

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

   6. associate-*r* 99.9%

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

   7. distribute-rgt-neg-in 99.9%

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

   8. remove-double-neg 99.9%

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

   9. *-commutative 99.9%

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

   10. fma-def 99.9%

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

   11. distribute-rgt-neg-out 99.9%

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

   12. distribute-lft-neg-in 99.9%

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

   13. *-commutative 99.9%

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

   14. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 45.8%

   \[\leadsto \color{blue}{0.954929658551372 \cdot x} \]
5. Step-by-step derivation

   1. *-commutative 45.8%

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

6. Simplified 45.8%

   \[\leadsto \color{blue}{x \cdot 0.954929658551372} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 24.6%

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

   2. sqrt-prod 29.2%

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

   3. metadata-eval 29.2%

      \[\leadsto \sqrt{x \cdot x} \cdot \color{blue}{\sqrt{0.9118906527810399}} \]

   4. sqrt-prod 29.2%

      \[\leadsto \color{blue}{\sqrt{\left(x \cdot x\right) \cdot 0.9118906527810399}} \]

   5. associate-*r* 29.2%

      \[\leadsto \sqrt{\color{blue}{x \cdot \left(x \cdot 0.9118906527810399\right)}} \]

8. Applied egg-rr 29.2%

   \[\leadsto \color{blue}{\sqrt{x \cdot \left(x \cdot 0.9118906527810399\right)}} \]

9. Taylor expanded in x around -inf 5.0%

   \[\leadsto \color{blue}{-0.954929658551372 \cdot x} \]
10. Step-by-step derivation

    1. *-commutative 5.0%

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

11. Simplified 5.0%

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

12. Final simplification 5.0%

    \[\leadsto x \cdot -0.954929658551372 \]

Reproduce

herbie shell --seed 2023283 
(FPCore (x)
  :name "Rosa's Benchmark"
  :precision binary64
  (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))