Jmat.Real.erf

Percentage Accurate: 79.0% → 99.7%

Time: 15.7s

Alternatives: 12

Speedup: 53.4×

• Could not converge on a ground truth

  1. reg0 = (bf #e5293.846798494105314603075385093688964844)

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\\ 1 - \left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + t\_0 \cdot \left(1.421413741 + t\_0 \cdot \left(-1.453152027 + t\_0 \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))))
   (-
    1.0
    (*
     (*
      t_0
      (+
       0.254829592
       (*
        t_0
        (+
         -0.284496736
         (*
          t_0
          (+ 1.421413741 (* t_0 (+ -1.453152027 (* t_0 1.061405429)))))))))
     (exp (- (* (fabs x) (fabs x))))))))

C

double code(double x) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * fabs(x)));
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * exp(-(fabs(x) * fabs(x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = 1.0d0 / (1.0d0 + (0.3275911d0 * abs(x)))
    code = 1.0d0 - ((t_0 * (0.254829592d0 + (t_0 * ((-0.284496736d0) + (t_0 * (1.421413741d0 + (t_0 * ((-1.453152027d0) + (t_0 * 1.061405429d0))))))))) * exp(-(abs(x) * abs(x))))
end function

Java

public static double code(double x) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * Math.abs(x)));
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * Math.exp(-(Math.abs(x) * Math.abs(x))));
}

Python

def code(x):
	t_0 = 1.0 / (1.0 + (0.3275911 * math.fabs(x)))
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * math.exp(-(math.fabs(x) * math.fabs(x))))

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x))))
	return Float64(1.0 - Float64(Float64(t_0 * Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + Float64(t_0 * Float64(1.421413741 + Float64(t_0 * Float64(-1.453152027 + Float64(t_0 * 1.061405429))))))))) * exp(Float64(-Float64(abs(x) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / (1.0 + (0.3275911 * abs(x)));
	tmp = 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * exp(-(abs(x) * abs(x))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[(N[(t$95$0 * N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(t$95$0 * N[(1.421413741 + N[(t$95$0 * N[(-1.453152027 + N[(t$95$0 * 1.061405429), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\\ 1 - \left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + t\_0 \cdot \left(1.421413741 + t\_0 \cdot \left(-1.453152027 + t\_0 \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))))
   (-
    1.0
    (*
     (*
      t_0
      (+
       0.254829592
       (*
        t_0
        (+
         -0.284496736
         (*
          t_0
          (+ 1.421413741 (* t_0 (+ -1.453152027 (* t_0 1.061405429)))))))))
     (exp (- (* (fabs x) (fabs x))))))))

C

double code(double x) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * fabs(x)));
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * exp(-(fabs(x) * fabs(x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = 1.0d0 / (1.0d0 + (0.3275911d0 * abs(x)))
    code = 1.0d0 - ((t_0 * (0.254829592d0 + (t_0 * ((-0.284496736d0) + (t_0 * (1.421413741d0 + (t_0 * ((-1.453152027d0) + (t_0 * 1.061405429d0))))))))) * exp(-(abs(x) * abs(x))))
end function

Java

public static double code(double x) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * Math.abs(x)));
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * Math.exp(-(Math.abs(x) * Math.abs(x))));
}

Python

def code(x):
	t_0 = 1.0 / (1.0 + (0.3275911 * math.fabs(x)))
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * math.exp(-(math.fabs(x) * math.fabs(x))))

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x))))
	return Float64(1.0 - Float64(Float64(t_0 * Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + Float64(t_0 * Float64(1.421413741 + Float64(t_0 * Float64(-1.453152027 + Float64(t_0 * 1.061405429))))))))) * exp(Float64(-Float64(abs(x) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / (1.0 + (0.3275911 * abs(x)));
	tmp = 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * exp(-(abs(x) * abs(x))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[(N[(t$95$0 * N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(t$95$0 * N[(1.421413741 + N[(t$95$0 * N[(-1.453152027 + N[(t$95$0 * 1.061405429), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left({x\_m}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x\_m \cdot 1.128386358070218\right) \cdot \left(x\_m \cdot 1.128386358070218 + -1 \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.284496736}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} + \left(\left(\frac{1.453152027}{{\left(\mathsf{fma}\left(x\_m, 0.3275911, 1\right)\right)}^{3}} - \left(0.254829592 + \frac{1.061405429}{{\left(\mathsf{fma}\left(x\_m, 0.3275911, 1\right)\right)}^{4}}\right)\right) - \frac{1.421413741}{{\left(\mathsf{fma}\left(x\_m, 0.3275911, 1\right)\right)}^{2}}\right), \frac{{\left(e^{x\_m}\right)}^{\left(-x\_m\right)}}{\mathsf{fma}\left(0.3275911, \left|x\_m\right|, 1\right)}, 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 2e-7)
   (/
    (fma (pow x_m 3.0) 1.436724444676459 1e-27)
    (+
     1e-18
     (* (* x_m 1.128386358070218) (+ (* x_m 1.128386358070218) -1e-9))))
   (fma
    (+
     (/ 0.284496736 (fma x_m 0.3275911 1.0))
     (-
      (-
       (/ 1.453152027 (pow (fma x_m 0.3275911 1.0) 3.0))
       (+ 0.254829592 (/ 1.061405429 (pow (fma x_m 0.3275911 1.0) 4.0))))
      (/ 1.421413741 (pow (fma x_m 0.3275911 1.0) 2.0))))
    (/ (pow (exp x_m) (- x_m)) (fma 0.3275911 (fabs x_m) 1.0))
    1.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 2e-7) {
		tmp = fma(pow(x_m, 3.0), 1.436724444676459, 1e-27) / (1e-18 + ((x_m * 1.128386358070218) * ((x_m * 1.128386358070218) + -1e-9)));
	} else {
		tmp = fma(((0.284496736 / fma(x_m, 0.3275911, 1.0)) + (((1.453152027 / pow(fma(x_m, 0.3275911, 1.0), 3.0)) - (0.254829592 + (1.061405429 / pow(fma(x_m, 0.3275911, 1.0), 4.0)))) - (1.421413741 / pow(fma(x_m, 0.3275911, 1.0), 2.0)))), (pow(exp(x_m), -x_m) / fma(0.3275911, fabs(x_m), 1.0)), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 2e-7)
		tmp = Float64(fma((x_m ^ 3.0), 1.436724444676459, 1e-27) / Float64(1e-18 + Float64(Float64(x_m * 1.128386358070218) * Float64(Float64(x_m * 1.128386358070218) + -1e-9))));
	else
		tmp = fma(Float64(Float64(0.284496736 / fma(x_m, 0.3275911, 1.0)) + Float64(Float64(Float64(1.453152027 / (fma(x_m, 0.3275911, 1.0) ^ 3.0)) - Float64(0.254829592 + Float64(1.061405429 / (fma(x_m, 0.3275911, 1.0) ^ 4.0)))) - Float64(1.421413741 / (fma(x_m, 0.3275911, 1.0) ^ 2.0)))), Float64((exp(x_m) ^ Float64(-x_m)) / fma(0.3275911, abs(x_m), 1.0)), 1.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-7], N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459 + 1e-27), $MachinePrecision] / N[(1e-18 + N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] + -1e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.284496736 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.453152027 / N[Power[N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] - N[(0.254829592 + N[(1.061405429 / N[Power[N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.421413741 / N[Power[N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[x$95$m], $MachinePrecision], (-x$95$m)], $MachinePrecision] / N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.9999999999999999e-7

   1. Initial program 57.7%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 57.8%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing

   6. Applied egg-rr 56.8%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot e^{{x}^{2}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   7. Taylor expanded in x around 0 97.6%

      \[\leadsto \color{blue}{10^{-9} + 1.128386358070218 \cdot x} \]
   8. Step-by-step derivation

      1. *-commutative 97.6%

         \[\leadsto 10^{-9} + \color{blue}{x \cdot 1.128386358070218} \]

   9. Simplified 97.6%

      \[\leadsto \color{blue}{10^{-9} + x \cdot 1.128386358070218} \]
   10. Step-by-step derivation

       1. flip3-+ 97.6%

          \[\leadsto \color{blue}{\frac{{\left( 10^{-9} \right)}^{3} + {\left(x \cdot 1.128386358070218\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)}} \]

       2. metadata-eval 97.6%

          \[\leadsto \frac{\color{blue}{10^{-27}} + {\left(x \cdot 1.128386358070218\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       3. unpow-prod-down 97.6%

          \[\leadsto \frac{10^{-27} + \color{blue}{{x}^{3} \cdot {1.128386358070218}^{3}}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       4. metadata-eval 97.6%

          \[\leadsto \frac{10^{-27} + {x}^{3} \cdot \color{blue}{1.436724444676459}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       5. metadata-eval 97.6%

          \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{\color{blue}{10^{-18}} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       6. pow2 97.6%

          \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left(\color{blue}{{\left(x \cdot 1.128386358070218\right)}^{2}} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

   11. Applied egg-rr 97.6%

       \[\leadsto \color{blue}{\frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)}} \]
   12. Step-by-step derivation

       1. +-commutative 97.6%

          \[\leadsto \frac{\color{blue}{{x}^{3} \cdot 1.436724444676459 + 10^{-27}}}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       2. fma-define 97.6%

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       3. cancel-sign-sub-inv 97.6%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \color{blue}{\left({\left(x \cdot 1.128386358070218\right)}^{2} + \left(-10^{-9}\right) \cdot \left(x \cdot 1.128386358070218\right)\right)}} \]

       4. unpow2 97.6%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(\color{blue}{\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right)} + \left(-10^{-9}\right) \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       5. distribute-rgt-out 97.6%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \color{blue}{\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + \left(-10^{-9}\right)\right)}} \]

       6. metadata-eval 97.6%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + \color{blue}{-1 \cdot 10^{-9}}\right)} \]

   13. Simplified 97.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + -1 \cdot 10^{-9}\right)}} \]

   if 1.9999999999999999e-7 < (fabs.f64 x)

   1. Initial program 99.8%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.8%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right) - \left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]
   6. Step-by-step derivation

      1. associate--l+ 99.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + \left(1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} - \left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right)\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      2. associate-*r/ 99.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{0.284496736 \cdot 1}{1 + 0.3275911 \cdot \left|x\right|}} + \left(1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} - \left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right)\right), \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      3. metadata-eval 99.8%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.284496736}}{1 + 0.3275911 \cdot \left|x\right|} + \left(1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} - \left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right)\right), \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      4. +-commutative 99.8%

         \[\leadsto \mathsf{fma}\left(\frac{0.284496736}{\color{blue}{0.3275911 \cdot \left|x\right| + 1}} + \left(1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} - \left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right)\right), \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      5. *-commutative 99.8%

         \[\leadsto \mathsf{fma}\left(\frac{0.284496736}{\color{blue}{\left|x\right| \cdot 0.3275911} + 1} + \left(1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} - \left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right)\right), \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      6. fma-define 99.8%

         \[\leadsto \mathsf{fma}\left(\frac{0.284496736}{\color{blue}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}} + \left(1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} - \left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right)\right), \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      7. rem-square-sqrt 52.3%

         \[\leadsto \mathsf{fma}\left(\frac{0.284496736}{\mathsf{fma}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 0.3275911, 1\right)} + \left(1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} - \left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right)\right), \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      8. fabs-sqr 52.3%

         \[\leadsto \mathsf{fma}\left(\frac{0.284496736}{\mathsf{fma}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}, 0.3275911, 1\right)} + \left(1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} - \left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right)\right), \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      9. rem-square-sqrt 99.4%

         \[\leadsto \mathsf{fma}\left(\frac{0.284496736}{\mathsf{fma}\left(\color{blue}{x}, 0.3275911, 1\right)} + \left(1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} - \left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right)\right), \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      10. associate-+r+ 99.4%

          \[\leadsto \mathsf{fma}\left(\frac{0.284496736}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + \left(1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} - \color{blue}{\left(\left(0.254829592 + 1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}}\right) + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)}\right), \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

   7. Simplified 99.2%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{0.284496736}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + \left(\left(\frac{1.453152027}{{\left(\mathsf{fma}\left(x, 0.3275911, 1\right)\right)}^{3}} - \left(0.254829592 + \frac{1.061405429}{{\left(\mathsf{fma}\left(x, 0.3275911, 1\right)\right)}^{4}}\right)\right) - \frac{1.421413741}{{\left(\mathsf{fma}\left(x, 0.3275911, 1\right)\right)}^{2}}\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left({x\_m}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x\_m \cdot 1.128386358070218\right) \cdot \left(x\_m \cdot 1.128386358070218 + -1 \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.254829592 + \frac{\frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{x\_m \cdot \left(\frac{-1}{x\_m} - 0.3275911\right)} - -0.284496736}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}, \frac{{\left(e^{x\_m}\right)}^{\left(-x\_m\right)}}{\mathsf{fma}\left(0.3275911, \left|x\_m\right|, 1\right)}, 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 2e-7)
   (/
    (fma (pow x_m 3.0) 1.436724444676459 1e-27)
    (+
     1e-18
     (* (* x_m 1.128386358070218) (+ (* x_m 1.128386358070218) -1e-9))))
   (fma
    (+
     -0.254829592
     (/
      (-
       (/
        (+
         1.421413741
         (/
          (+ -1.453152027 (/ 1.061405429 (fma x_m 0.3275911 1.0)))
          (fma x_m 0.3275911 1.0)))
        (* x_m (- (/ -1.0 x_m) 0.3275911)))
       -0.284496736)
      (fma x_m 0.3275911 1.0)))
    (/ (pow (exp x_m) (- x_m)) (fma 0.3275911 (fabs x_m) 1.0))
    1.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 2e-7) {
		tmp = fma(pow(x_m, 3.0), 1.436724444676459, 1e-27) / (1e-18 + ((x_m * 1.128386358070218) * ((x_m * 1.128386358070218) + -1e-9)));
	} else {
		tmp = fma((-0.254829592 + ((((1.421413741 + ((-1.453152027 + (1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / (x_m * ((-1.0 / x_m) - 0.3275911))) - -0.284496736) / fma(x_m, 0.3275911, 1.0))), (pow(exp(x_m), -x_m) / fma(0.3275911, fabs(x_m), 1.0)), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 2e-7)
		tmp = Float64(fma((x_m ^ 3.0), 1.436724444676459, 1e-27) / Float64(1e-18 + Float64(Float64(x_m * 1.128386358070218) * Float64(Float64(x_m * 1.128386358070218) + -1e-9))));
	else
		tmp = fma(Float64(-0.254829592 + Float64(Float64(Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / Float64(x_m * Float64(Float64(-1.0 / x_m) - 0.3275911))) - -0.284496736) / fma(x_m, 0.3275911, 1.0))), Float64((exp(x_m) ^ Float64(-x_m)) / fma(0.3275911, abs(x_m), 1.0)), 1.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-7], N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459 + 1e-27), $MachinePrecision] / N[(1e-18 + N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] + -1e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.254829592 + N[(N[(N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * N[(N[(-1.0 / x$95$m), $MachinePrecision] - 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[x$95$m], $MachinePrecision], (-x$95$m)], $MachinePrecision] / N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.9999999999999999e-7

   1. Initial program 57.7%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 57.8%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing

   6. Applied egg-rr 56.8%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot e^{{x}^{2}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   7. Taylor expanded in x around 0 97.6%

      \[\leadsto \color{blue}{10^{-9} + 1.128386358070218 \cdot x} \]
   8. Step-by-step derivation

      1. *-commutative 97.6%

         \[\leadsto 10^{-9} + \color{blue}{x \cdot 1.128386358070218} \]

   9. Simplified 97.6%

      \[\leadsto \color{blue}{10^{-9} + x \cdot 1.128386358070218} \]
   10. Step-by-step derivation

       1. flip3-+ 97.6%

          \[\leadsto \color{blue}{\frac{{\left( 10^{-9} \right)}^{3} + {\left(x \cdot 1.128386358070218\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)}} \]

       2. metadata-eval 97.6%

          \[\leadsto \frac{\color{blue}{10^{-27}} + {\left(x \cdot 1.128386358070218\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       3. unpow-prod-down 97.6%

          \[\leadsto \frac{10^{-27} + \color{blue}{{x}^{3} \cdot {1.128386358070218}^{3}}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       4. metadata-eval 97.6%

          \[\leadsto \frac{10^{-27} + {x}^{3} \cdot \color{blue}{1.436724444676459}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       5. metadata-eval 97.6%

          \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{\color{blue}{10^{-18}} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       6. pow2 97.6%

          \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left(\color{blue}{{\left(x \cdot 1.128386358070218\right)}^{2}} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

   11. Applied egg-rr 97.6%

       \[\leadsto \color{blue}{\frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)}} \]
   12. Step-by-step derivation

       1. +-commutative 97.6%

          \[\leadsto \frac{\color{blue}{{x}^{3} \cdot 1.436724444676459 + 10^{-27}}}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       2. fma-define 97.6%

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       3. cancel-sign-sub-inv 97.6%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \color{blue}{\left({\left(x \cdot 1.128386358070218\right)}^{2} + \left(-10^{-9}\right) \cdot \left(x \cdot 1.128386358070218\right)\right)}} \]

       4. unpow2 97.6%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(\color{blue}{\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right)} + \left(-10^{-9}\right) \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       5. distribute-rgt-out 97.6%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \color{blue}{\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + \left(-10^{-9}\right)\right)}} \]

       6. metadata-eval 97.6%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + \color{blue}{-1 \cdot 10^{-9}}\right)} \]

   13. Simplified 97.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + -1 \cdot 10^{-9}\right)}} \]

   if 1.9999999999999999e-7 < (fabs.f64 x)

   1. Initial program 99.8%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sub-neg 99.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 + \left(-\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

   6. Applied egg-rr 99.2%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 + \left(-\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]
   7. Step-by-step derivation

      1. sub-neg 99.2%

         \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      2. fma-undefine 99.2%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{0.3275911 \cdot x + 1}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      3. *-commutative 99.2%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{x \cdot 0.3275911} + 1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      4. fma-define 99.2%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      5. fma-undefine 99.2%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{0.3275911 \cdot x + 1}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      6. *-commutative 99.2%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{x \cdot 0.3275911} + 1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      7. fma-define 99.2%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      8. fma-undefine 99.2%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{0.3275911 \cdot x + 1}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      9. *-commutative 99.2%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{x \cdot 0.3275911} + 1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      10. fma-define 99.2%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      11. fma-undefine 99.2%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{0.3275911 \cdot x + 1}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      12. *-commutative 99.2%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{x \cdot 0.3275911} + 1}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      13. fma-define 99.2%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

   8. Simplified 99.2%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

   9. Taylor expanded in x around inf 99.2%

      \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{x \cdot \left(0.3275911 + \frac{1}{x}\right)}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + -1 \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.254829592 + \frac{\frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{x \cdot \left(\frac{-1}{x} - 0.3275911\right)} - -0.284496736}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left|x\_m\right| \cdot 0.3275911\\ \mathbf{if}\;\left|x\_m\right| \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left({x\_m}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x\_m \cdot 1.128386358070218\right) \cdot \left(x\_m \cdot 1.128386358070218 + -1 \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x\_m \cdot \left(-x\_m\right)} \cdot \left(\frac{1}{1 + x\_m \cdot 0.3275911} \cdot \left(\frac{1}{1 + t\_0} \cdot \left(\left(-1 + \left(\frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} + 2.421413741\right)\right) \cdot \frac{1}{-1 - t\_0} - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (fabs x_m) 0.3275911)))
   (if (<= (fabs x_m) 2e-7)
     (/
      (fma (pow x_m 3.0) 1.436724444676459 1e-27)
      (+
       1e-18
       (* (* x_m 1.128386358070218) (+ (* x_m 1.128386358070218) -1e-9))))
     (+
      1.0
      (*
       (exp (* x_m (- x_m)))
       (*
        (/ 1.0 (+ 1.0 (* x_m 0.3275911)))
        (-
         (*
          (/ 1.0 (+ 1.0 t_0))
          (-
           (*
            (+
             -1.0
             (+
              (/
               (+ -1.453152027 (/ 1.061405429 (fma x_m 0.3275911 1.0)))
               (fma x_m 0.3275911 1.0))
              2.421413741))
            (/ 1.0 (- -1.0 t_0)))
           -0.284496736))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fabs(x_m) * 0.3275911;
	double tmp;
	if (fabs(x_m) <= 2e-7) {
		tmp = fma(pow(x_m, 3.0), 1.436724444676459, 1e-27) / (1e-18 + ((x_m * 1.128386358070218) * ((x_m * 1.128386358070218) + -1e-9)));
	} else {
		tmp = 1.0 + (exp((x_m * -x_m)) * ((1.0 / (1.0 + (x_m * 0.3275911))) * (((1.0 / (1.0 + t_0)) * (((-1.0 + (((-1.453152027 + (1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0)) + 2.421413741)) * (1.0 / (-1.0 - t_0))) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(abs(x_m) * 0.3275911)
	tmp = 0.0
	if (abs(x_m) <= 2e-7)
		tmp = Float64(fma((x_m ^ 3.0), 1.436724444676459, 1e-27) / Float64(1e-18 + Float64(Float64(x_m * 1.128386358070218) * Float64(Float64(x_m * 1.128386358070218) + -1e-9))));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(Float64(1.0 / Float64(1.0 + Float64(x_m * 0.3275911))) * Float64(Float64(Float64(1.0 / Float64(1.0 + t_0)) * Float64(Float64(Float64(-1.0 + Float64(Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0)) + 2.421413741)) * Float64(1.0 / Float64(-1.0 - t_0))) - -0.284496736)) - 0.254829592))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-7], N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459 + 1e-27), $MachinePrecision] / N[(1e-18 + N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] + -1e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-1.0 + N[(N[(N[(-1.453152027 + N[(1.061405429 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] + 2.421413741), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.9999999999999999e-7

   1. Initial program 57.7%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 57.8%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing

   6. Applied egg-rr 56.8%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot e^{{x}^{2}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   7. Taylor expanded in x around 0 97.6%

      \[\leadsto \color{blue}{10^{-9} + 1.128386358070218 \cdot x} \]
   8. Step-by-step derivation

      1. *-commutative 97.6%

         \[\leadsto 10^{-9} + \color{blue}{x \cdot 1.128386358070218} \]

   9. Simplified 97.6%

      \[\leadsto \color{blue}{10^{-9} + x \cdot 1.128386358070218} \]
   10. Step-by-step derivation

       1. flip3-+ 97.6%

          \[\leadsto \color{blue}{\frac{{\left( 10^{-9} \right)}^{3} + {\left(x \cdot 1.128386358070218\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)}} \]

       2. metadata-eval 97.6%

          \[\leadsto \frac{\color{blue}{10^{-27}} + {\left(x \cdot 1.128386358070218\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       3. unpow-prod-down 97.6%

          \[\leadsto \frac{10^{-27} + \color{blue}{{x}^{3} \cdot {1.128386358070218}^{3}}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       4. metadata-eval 97.6%

          \[\leadsto \frac{10^{-27} + {x}^{3} \cdot \color{blue}{1.436724444676459}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       5. metadata-eval 97.6%

          \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{\color{blue}{10^{-18}} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       6. pow2 97.6%

          \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left(\color{blue}{{\left(x \cdot 1.128386358070218\right)}^{2}} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

   11. Applied egg-rr 97.6%

       \[\leadsto \color{blue}{\frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)}} \]
   12. Step-by-step derivation

       1. +-commutative 97.6%

          \[\leadsto \frac{\color{blue}{{x}^{3} \cdot 1.436724444676459 + 10^{-27}}}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       2. fma-define 97.6%

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       3. cancel-sign-sub-inv 97.6%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \color{blue}{\left({\left(x \cdot 1.128386358070218\right)}^{2} + \left(-10^{-9}\right) \cdot \left(x \cdot 1.128386358070218\right)\right)}} \]

       4. unpow2 97.6%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(\color{blue}{\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right)} + \left(-10^{-9}\right) \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       5. distribute-rgt-out 97.6%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \color{blue}{\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + \left(-10^{-9}\right)\right)}} \]

       6. metadata-eval 97.6%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + \color{blue}{-1 \cdot 10^{-9}}\right)} \]

   13. Simplified 97.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + -1 \cdot 10^{-9}\right)}} \]

   if 1.9999999999999999e-7 < (fabs.f64 x)

   1. Initial program 99.8%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. expm1-log1p-u 99.8%

         \[\leadsto 1 - \left(\frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      2. log1p-define 99.8%

         \[\leadsto 1 - \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      3. +-commutative 99.8%

         \[\leadsto 1 - \left(\frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      4. fma-undefine 99.8%

         \[\leadsto 1 - \left(\frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      5. expm1-undefine 99.8%

         \[\leadsto 1 - \left(\frac{1}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      6. add-exp-log 99.8%

         \[\leadsto 1 - \left(\frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      7. add-sqr-sqrt 52.3%

         \[\leadsto 1 - \left(\frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      8. fabs-sqr 52.3%

         \[\leadsto 1 - \left(\frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      9. add-sqr-sqrt 99.4%

         \[\leadsto 1 - \left(\frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   6. Applied egg-rr 99.4%

      \[\leadsto 1 - \left(\frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   7. Step-by-step derivation

      1. fma-undefine 99.4%

         \[\leadsto 1 - \left(\frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      2. associate--l+ 99.4%

         \[\leadsto 1 - \left(\frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      3. metadata-eval 99.4%

         \[\leadsto 1 - \left(\frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      4. +-rgt-identity 99.4%

         \[\leadsto 1 - \left(\frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      5. *-commutative 99.4%

         \[\leadsto 1 - \left(\frac{1}{1 + \color{blue}{x \cdot 0.3275911}} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   8. Simplified 99.4%

      \[\leadsto 1 - \left(\frac{1}{1 + \color{blue}{x \cdot 0.3275911}} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   9. Step-by-step derivation

      1. associate-*l/ 99.4%

         \[\leadsto 1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \color{blue}{\frac{1 \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)}{1 + 0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      2. *-un-lft-identity 99.4%

         \[\leadsto 1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{\color{blue}{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      3. +-commutative 99.4%

         \[\leadsto 1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{0.3275911 \cdot \left|x\right| + 1}}}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      4. fma-undefine 99.4%

         \[\leadsto 1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      5. +-commutative 99.4%

         \[\leadsto 1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\color{blue}{0.3275911 \cdot \left|x\right| + 1}}\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      6. fma-undefine 99.4%

         \[\leadsto 1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      7. expm1-log1p-u 99.4%

         \[\leadsto 1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)\right)}\right)\right)\right) \cdot e^{-x \cdot x} \]

      8. expm1-undefine 99.4%

         \[\leadsto 1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)} - 1\right)}\right)\right)\right) \cdot e^{-x \cdot x} \]

   10. Applied egg-rr 99.3%

       \[\leadsto 1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)} - 1\right)}\right)\right)\right) \cdot e^{-x \cdot x} \]
   11. Step-by-step derivation

       1. sub-neg 99.3%

          \[\leadsto 1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)} + \left(-1\right)\right)}\right)\right)\right) \cdot e^{-x \cdot x} \]

       2. metadata-eval 99.3%

          \[\leadsto 1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(e^{\mathsf{log1p}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)} + \color{blue}{-1}\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       3. +-commutative 99.3%

          \[\leadsto 1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}\right)}\right)\right)\right) \cdot e^{-x \cdot x} \]

       4. log1p-undefine 99.3%

          \[\leadsto 1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1 + e^{\color{blue}{\log \left(1 + \left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)}}\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       5. rem-exp-log 99.3%

          \[\leadsto 1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1 + \color{blue}{\left(1 + \left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       6. associate-+r+ 99.3%

          \[\leadsto 1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1 + \color{blue}{\left(\left(1 + 1.421413741\right) + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       7. metadata-eval 99.3%

          \[\leadsto 1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1 + \left(\color{blue}{2.421413741} + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       8. fma-undefine 99.3%

          \[\leadsto 1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1 + \left(2.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{0.3275911 \cdot x + 1}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       9. *-commutative 99.3%

          \[\leadsto 1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1 + \left(2.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{x \cdot 0.3275911} + 1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       10. fma-define 99.3%

           \[\leadsto 1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1 + \left(2.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       11. fma-undefine 99.3%

           \[\leadsto 1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1 + \left(2.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{0.3275911 \cdot x + 1}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       12. *-commutative 99.3%

           \[\leadsto 1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1 + \left(2.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{x \cdot 0.3275911} + 1}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       13. fma-define 99.3%

           \[\leadsto 1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1 + \left(2.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   12. Simplified 99.3%

       \[\leadsto 1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\left(-1 + \left(2.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)\right)}\right)\right)\right) \cdot e^{-x \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + -1 \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\left(-1 + \left(\frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + 2.421413741\right)\right) \cdot \frac{1}{-1 - \left|x\right| \cdot 0.3275911} - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 0.0001:\\ \;\;\;\;\frac{\mathsf{fma}\left({x\_m}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x\_m \cdot 1.128386358070218\right) \cdot \left(x\_m \cdot 1.128386358070218 + -1 \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.254829592 \cdot \frac{e^{-{x\_m}^{2}}}{1 + \left|x\_m\right| \cdot 0.3275911}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.0001)
   (/
    (fma (pow x_m 3.0) 1.436724444676459 1e-27)
    (+
     1e-18
     (* (* x_m 1.128386358070218) (+ (* x_m 1.128386358070218) -1e-9))))
   (+
    1.0
    (*
     -0.254829592
     (/ (exp (- (pow x_m 2.0))) (+ 1.0 (* (fabs x_m) 0.3275911)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.0001) {
		tmp = fma(pow(x_m, 3.0), 1.436724444676459, 1e-27) / (1e-18 + ((x_m * 1.128386358070218) * ((x_m * 1.128386358070218) + -1e-9)));
	} else {
		tmp = 1.0 + (-0.254829592 * (exp(-pow(x_m, 2.0)) / (1.0 + (fabs(x_m) * 0.3275911))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.0001)
		tmp = Float64(fma((x_m ^ 3.0), 1.436724444676459, 1e-27) / Float64(1e-18 + Float64(Float64(x_m * 1.128386358070218) * Float64(Float64(x_m * 1.128386358070218) + -1e-9))));
	else
		tmp = Float64(1.0 + Float64(-0.254829592 * Float64(exp(Float64(-(x_m ^ 2.0))) / Float64(1.0 + Float64(abs(x_m) * 0.3275911)))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.0001], N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459 + 1e-27), $MachinePrecision] / N[(1e-18 + N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] + -1e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.254829592 * N[(N[Exp[(-N[Power[x$95$m, 2.0], $MachinePrecision])], $MachinePrecision] / N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000005e-4

   1. Initial program 57.9%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 57.9%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing

   6. Applied egg-rr 56.3%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot e^{{x}^{2}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   7. Taylor expanded in x around 0 96.9%

      \[\leadsto \color{blue}{10^{-9} + 1.128386358070218 \cdot x} \]
   8. Step-by-step derivation

      1. *-commutative 96.9%

         \[\leadsto 10^{-9} + \color{blue}{x \cdot 1.128386358070218} \]

   9. Simplified 96.9%

      \[\leadsto \color{blue}{10^{-9} + x \cdot 1.128386358070218} \]
   10. Step-by-step derivation

       1. flip3-+ 96.9%

          \[\leadsto \color{blue}{\frac{{\left( 10^{-9} \right)}^{3} + {\left(x \cdot 1.128386358070218\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)}} \]

       2. metadata-eval 96.9%

          \[\leadsto \frac{\color{blue}{10^{-27}} + {\left(x \cdot 1.128386358070218\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       3. unpow-prod-down 96.9%

          \[\leadsto \frac{10^{-27} + \color{blue}{{x}^{3} \cdot {1.128386358070218}^{3}}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       4. metadata-eval 96.9%

          \[\leadsto \frac{10^{-27} + {x}^{3} \cdot \color{blue}{1.436724444676459}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       5. metadata-eval 96.9%

          \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{\color{blue}{10^{-18}} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       6. pow2 96.9%

          \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left(\color{blue}{{\left(x \cdot 1.128386358070218\right)}^{2}} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

   11. Applied egg-rr 96.9%

       \[\leadsto \color{blue}{\frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)}} \]
   12. Step-by-step derivation

       1. +-commutative 96.9%

          \[\leadsto \frac{\color{blue}{{x}^{3} \cdot 1.436724444676459 + 10^{-27}}}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       2. fma-define 96.9%

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       3. cancel-sign-sub-inv 96.9%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \color{blue}{\left({\left(x \cdot 1.128386358070218\right)}^{2} + \left(-10^{-9}\right) \cdot \left(x \cdot 1.128386358070218\right)\right)}} \]

       4. unpow2 96.9%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(\color{blue}{\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right)} + \left(-10^{-9}\right) \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       5. distribute-rgt-out 96.9%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \color{blue}{\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + \left(-10^{-9}\right)\right)}} \]

       6. metadata-eval 96.9%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + \color{blue}{-1 \cdot 10^{-9}}\right)} \]

   13. Simplified 96.9%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + -1 \cdot 10^{-9}\right)}} \]

   if 1.00000000000000005e-4 < (fabs.f64 x)

   1. Initial program 100.0%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 + \left(-\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 + \left(-\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]
   7. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      2. fma-undefine 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{0.3275911 \cdot x + 1}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      3. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{x \cdot 0.3275911} + 1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      4. fma-define 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      5. fma-undefine 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{0.3275911 \cdot x + 1}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      6. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{x \cdot 0.3275911} + 1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      7. fma-define 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      8. fma-undefine 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{0.3275911 \cdot x + 1}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      9. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{x \cdot 0.3275911} + 1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      10. fma-define 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      11. fma-undefine 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{0.3275911 \cdot x + 1}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      12. *-commutative 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{x \cdot 0.3275911} + 1}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      13. fma-define 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

   8. Simplified 100.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

   9. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{1 + -0.254829592 \cdot \frac{e^{-1 \cdot {x}^{2}}}{1 + 0.3275911 \cdot \left|x\right|}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + -1 \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.254829592 \cdot \frac{e^{-{x}^{2}}}{1 + \left|x\right| \cdot 0.3275911}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 0.0001:\\ \;\;\;\;\frac{\mathsf{fma}\left({x\_m}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x\_m \cdot 1.128386358070218\right) \cdot \left(x\_m \cdot 1.128386358070218 + -1 \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.999999999 \cdot \frac{1}{-1 - 0.3275911 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.0001)
   (/
    (fma (pow x_m 3.0) 1.436724444676459 1e-27)
    (+
     1e-18
     (* (* x_m 1.128386358070218) (+ (* x_m 1.128386358070218) -1e-9))))
   (+ 1.0 (* 0.999999999 (/ 1.0 (- -1.0 (* 0.3275911 (log1p (expm1 x_m)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.0001) {
		tmp = fma(pow(x_m, 3.0), 1.436724444676459, 1e-27) / (1e-18 + ((x_m * 1.128386358070218) * ((x_m * 1.128386358070218) + -1e-9)));
	} else {
		tmp = 1.0 + (0.999999999 * (1.0 / (-1.0 - (0.3275911 * log1p(expm1(x_m))))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.0001)
		tmp = Float64(fma((x_m ^ 3.0), 1.436724444676459, 1e-27) / Float64(1e-18 + Float64(Float64(x_m * 1.128386358070218) * Float64(Float64(x_m * 1.128386358070218) + -1e-9))));
	else
		tmp = Float64(1.0 + Float64(0.999999999 * Float64(1.0 / Float64(-1.0 - Float64(0.3275911 * log1p(expm1(x_m)))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.0001], N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459 + 1e-27), $MachinePrecision] / N[(1e-18 + N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] + -1e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(0.999999999 * N[(1.0 / N[(-1.0 - N[(0.3275911 * N[Log[1 + N[(Exp[x$95$m] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000005e-4

   1. Initial program 57.9%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 57.9%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing

   6. Applied egg-rr 56.3%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot e^{{x}^{2}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   7. Taylor expanded in x around 0 96.9%

      \[\leadsto \color{blue}{10^{-9} + 1.128386358070218 \cdot x} \]
   8. Step-by-step derivation

      1. *-commutative 96.9%

         \[\leadsto 10^{-9} + \color{blue}{x \cdot 1.128386358070218} \]

   9. Simplified 96.9%

      \[\leadsto \color{blue}{10^{-9} + x \cdot 1.128386358070218} \]
   10. Step-by-step derivation

       1. flip3-+ 96.9%

          \[\leadsto \color{blue}{\frac{{\left( 10^{-9} \right)}^{3} + {\left(x \cdot 1.128386358070218\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)}} \]

       2. metadata-eval 96.9%

          \[\leadsto \frac{\color{blue}{10^{-27}} + {\left(x \cdot 1.128386358070218\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       3. unpow-prod-down 96.9%

          \[\leadsto \frac{10^{-27} + \color{blue}{{x}^{3} \cdot {1.128386358070218}^{3}}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       4. metadata-eval 96.9%

          \[\leadsto \frac{10^{-27} + {x}^{3} \cdot \color{blue}{1.436724444676459}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       5. metadata-eval 96.9%

          \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{\color{blue}{10^{-18}} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       6. pow2 96.9%

          \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left(\color{blue}{{\left(x \cdot 1.128386358070218\right)}^{2}} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

   11. Applied egg-rr 96.9%

       \[\leadsto \color{blue}{\frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)}} \]
   12. Step-by-step derivation

       1. +-commutative 96.9%

          \[\leadsto \frac{\color{blue}{{x}^{3} \cdot 1.436724444676459 + 10^{-27}}}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       2. fma-define 96.9%

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       3. cancel-sign-sub-inv 96.9%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \color{blue}{\left({\left(x \cdot 1.128386358070218\right)}^{2} + \left(-10^{-9}\right) \cdot \left(x \cdot 1.128386358070218\right)\right)}} \]

       4. unpow2 96.9%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(\color{blue}{\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right)} + \left(-10^{-9}\right) \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

       5. distribute-rgt-out 96.9%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \color{blue}{\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + \left(-10^{-9}\right)\right)}} \]

       6. metadata-eval 96.9%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + \color{blue}{-1 \cdot 10^{-9}}\right)} \]

   13. Simplified 96.9%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + -1 \cdot 10^{-9}\right)}} \]

   if 1.00000000000000005e-4 < (fabs.f64 x)

   1. Initial program 100.0%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 + \left(-\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 + \left(-\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]
   7. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      2. fma-undefine 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{0.3275911 \cdot x + 1}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      3. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{x \cdot 0.3275911} + 1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      4. fma-define 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      5. fma-undefine 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{0.3275911 \cdot x + 1}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      6. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{x \cdot 0.3275911} + 1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      7. fma-define 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      8. fma-undefine 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{0.3275911 \cdot x + 1}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      9. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{x \cdot 0.3275911} + 1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      10. fma-define 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      11. fma-undefine 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{0.3275911 \cdot x + 1}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      12. *-commutative 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{x \cdot 0.3275911} + 1}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      13. fma-define 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

   8. Simplified 100.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

   9. Taylor expanded in x around 0 97.8%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 51.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|} \]

       2. fabs-sqr 51.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}} \]

       3. add-sqr-sqrt 97.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{x}} \]

       4. log1p-expm1-u 99.4%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x\right)\right)}} \]

   11. Applied egg-rr 99.4%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + -1 \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.999999999 \cdot \frac{1}{-1 - 0.3275911 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 0.0001:\\ \;\;\;\;10^{-9} + {\left(\sqrt[3]{x\_m \cdot 1.128386358070218}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.999999999 \cdot \frac{1}{-1 - 0.3275911 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.0001)
   (+ 1e-9 (pow (cbrt (* x_m 1.128386358070218)) 3.0))
   (+ 1.0 (* 0.999999999 (/ 1.0 (- -1.0 (* 0.3275911 (log1p (expm1 x_m)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.0001) {
		tmp = 1e-9 + pow(cbrt((x_m * 1.128386358070218)), 3.0);
	} else {
		tmp = 1.0 + (0.999999999 * (1.0 / (-1.0 - (0.3275911 * log1p(expm1(x_m))))));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.abs(x_m) <= 0.0001) {
		tmp = 1e-9 + Math.pow(Math.cbrt((x_m * 1.128386358070218)), 3.0);
	} else {
		tmp = 1.0 + (0.999999999 * (1.0 / (-1.0 - (0.3275911 * Math.log1p(Math.expm1(x_m))))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.0001)
		tmp = Float64(1e-9 + (cbrt(Float64(x_m * 1.128386358070218)) ^ 3.0));
	else
		tmp = Float64(1.0 + Float64(0.999999999 * Float64(1.0 / Float64(-1.0 - Float64(0.3275911 * log1p(expm1(x_m)))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.0001], N[(1e-9 + N[Power[N[Power[N[(x$95$m * 1.128386358070218), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(0.999999999 * N[(1.0 / N[(-1.0 - N[(0.3275911 * N[Log[1 + N[(Exp[x$95$m] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000005e-4

   1. Initial program 57.9%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 57.9%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing

   6. Applied egg-rr 56.3%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot e^{{x}^{2}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   7. Taylor expanded in x around 0 96.9%

      \[\leadsto \color{blue}{10^{-9} + 1.128386358070218 \cdot x} \]
   8. Step-by-step derivation

      1. *-commutative 96.9%

         \[\leadsto 10^{-9} + \color{blue}{x \cdot 1.128386358070218} \]

   9. Simplified 96.9%

      \[\leadsto \color{blue}{10^{-9} + x \cdot 1.128386358070218} \]
   10. Step-by-step derivation

       1. add-cube-cbrt 96.9%

          \[\leadsto 10^{-9} + \color{blue}{\left(\sqrt[3]{x \cdot 1.128386358070218} \cdot \sqrt[3]{x \cdot 1.128386358070218}\right) \cdot \sqrt[3]{x \cdot 1.128386358070218}} \]

       2. pow3 96.9%

          \[\leadsto 10^{-9} + \color{blue}{{\left(\sqrt[3]{x \cdot 1.128386358070218}\right)}^{3}} \]

   11. Applied egg-rr 96.9%

       \[\leadsto 10^{-9} + \color{blue}{{\left(\sqrt[3]{x \cdot 1.128386358070218}\right)}^{3}} \]

   if 1.00000000000000005e-4 < (fabs.f64 x)

   1. Initial program 100.0%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 + \left(-\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 + \left(-\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]
   7. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      2. fma-undefine 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{0.3275911 \cdot x + 1}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      3. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{x \cdot 0.3275911} + 1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      4. fma-define 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      5. fma-undefine 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{0.3275911 \cdot x + 1}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      6. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{x \cdot 0.3275911} + 1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      7. fma-define 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      8. fma-undefine 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{0.3275911 \cdot x + 1}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      9. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{x \cdot 0.3275911} + 1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      10. fma-define 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      11. fma-undefine 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{0.3275911 \cdot x + 1}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      12. *-commutative 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{x \cdot 0.3275911} + 1}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      13. fma-define 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

   8. Simplified 100.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

   9. Taylor expanded in x around 0 97.8%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 51.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|} \]

       2. fabs-sqr 51.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}} \]

       3. add-sqr-sqrt 97.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{x}} \]

       4. log1p-expm1-u 99.4%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x\right)\right)}} \]

   11. Applied egg-rr 99.4%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;10^{-9} + {\left(\sqrt[3]{x \cdot 1.128386358070218}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.999999999 \cdot \frac{1}{-1 - 0.3275911 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 0.0001:\\ \;\;\;\;10^{-9} + {\left(\sqrt[3]{x\_m \cdot 1.128386358070218}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.999999999 \cdot \frac{1}{x\_m \cdot 0.3275911 + -1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.0001)
   (+ 1e-9 (pow (cbrt (* x_m 1.128386358070218)) 3.0))
   (+ 1.0 (* 0.999999999 (/ 1.0 (+ (* x_m 0.3275911) -1.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.0001) {
		tmp = 1e-9 + pow(cbrt((x_m * 1.128386358070218)), 3.0);
	} else {
		tmp = 1.0 + (0.999999999 * (1.0 / ((x_m * 0.3275911) + -1.0)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.abs(x_m) <= 0.0001) {
		tmp = 1e-9 + Math.pow(Math.cbrt((x_m * 1.128386358070218)), 3.0);
	} else {
		tmp = 1.0 + (0.999999999 * (1.0 / ((x_m * 0.3275911) + -1.0)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.0001)
		tmp = Float64(1e-9 + (cbrt(Float64(x_m * 1.128386358070218)) ^ 3.0));
	else
		tmp = Float64(1.0 + Float64(0.999999999 * Float64(1.0 / Float64(Float64(x_m * 0.3275911) + -1.0))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.0001], N[(1e-9 + N[Power[N[Power[N[(x$95$m * 1.128386358070218), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(0.999999999 * N[(1.0 / N[(N[(x$95$m * 0.3275911), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000005e-4

   1. Initial program 57.9%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 57.9%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing

   6. Applied egg-rr 56.3%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot e^{{x}^{2}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   7. Taylor expanded in x around 0 96.9%

      \[\leadsto \color{blue}{10^{-9} + 1.128386358070218 \cdot x} \]
   8. Step-by-step derivation

      1. *-commutative 96.9%

         \[\leadsto 10^{-9} + \color{blue}{x \cdot 1.128386358070218} \]

   9. Simplified 96.9%

      \[\leadsto \color{blue}{10^{-9} + x \cdot 1.128386358070218} \]
   10. Step-by-step derivation

       1. add-cube-cbrt 96.9%

          \[\leadsto 10^{-9} + \color{blue}{\left(\sqrt[3]{x \cdot 1.128386358070218} \cdot \sqrt[3]{x \cdot 1.128386358070218}\right) \cdot \sqrt[3]{x \cdot 1.128386358070218}} \]

       2. pow3 96.9%

          \[\leadsto 10^{-9} + \color{blue}{{\left(\sqrt[3]{x \cdot 1.128386358070218}\right)}^{3}} \]

   11. Applied egg-rr 96.9%

       \[\leadsto 10^{-9} + \color{blue}{{\left(\sqrt[3]{x \cdot 1.128386358070218}\right)}^{3}} \]

   if 1.00000000000000005e-4 < (fabs.f64 x)

   1. Initial program 100.0%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 + \left(-\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 + \left(-\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]
   7. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      2. fma-undefine 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{0.3275911 \cdot x + 1}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      3. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{x \cdot 0.3275911} + 1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      4. fma-define 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      5. fma-undefine 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{0.3275911 \cdot x + 1}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      6. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{x \cdot 0.3275911} + 1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      7. fma-define 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      8. fma-undefine 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{0.3275911 \cdot x + 1}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      9. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{x \cdot 0.3275911} + 1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      10. fma-define 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      11. fma-undefine 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{0.3275911 \cdot x + 1}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      12. *-commutative 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{x \cdot 0.3275911} + 1}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      13. fma-define 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

   8. Simplified 100.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

   9. Taylor expanded in x around 0 97.8%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   10. Step-by-step derivation

       1. neg-fabs 97.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left|-x\right|}} \]

       2. add-sqr-sqrt 45.9%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right|} \]

       3. fabs-sqr 45.9%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)}} \]

       4. add-sqr-sqrt 97.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left(-x\right)}} \]

       5. neg-sub0 97.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left(0 - x\right)}} \]

   11. Applied egg-rr 97.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left(0 - x\right)}} \]
   12. Step-by-step derivation

       1. neg-sub0 97.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left(-x\right)}} \]

   13. Simplified 97.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left(-x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;10^{-9} + {\left(\sqrt[3]{x \cdot 1.128386358070218}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.999999999 \cdot \frac{1}{x \cdot 0.3275911 + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 0.0001:\\ \;\;\;\;\frac{10^{-18} - {x\_m}^{2} \cdot 1.2732557730789702}{10^{-9} - x\_m \cdot 1.128386358070218}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.999999999 \cdot \frac{1}{x\_m \cdot 0.3275911 + -1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.0001)
   (/
    (- 1e-18 (* (pow x_m 2.0) 1.2732557730789702))
    (- 1e-9 (* x_m 1.128386358070218)))
   (+ 1.0 (* 0.999999999 (/ 1.0 (+ (* x_m 0.3275911) -1.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.0001) {
		tmp = (1e-18 - (pow(x_m, 2.0) * 1.2732557730789702)) / (1e-9 - (x_m * 1.128386358070218));
	} else {
		tmp = 1.0 + (0.999999999 * (1.0 / ((x_m * 0.3275911) + -1.0)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (abs(x_m) <= 0.0001d0) then
        tmp = (1d-18 - ((x_m ** 2.0d0) * 1.2732557730789702d0)) / (1d-9 - (x_m * 1.128386358070218d0))
    else
        tmp = 1.0d0 + (0.999999999d0 * (1.0d0 / ((x_m * 0.3275911d0) + (-1.0d0))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.abs(x_m) <= 0.0001) {
		tmp = (1e-18 - (Math.pow(x_m, 2.0) * 1.2732557730789702)) / (1e-9 - (x_m * 1.128386358070218));
	} else {
		tmp = 1.0 + (0.999999999 * (1.0 / ((x_m * 0.3275911) + -1.0)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 0.0001:
		tmp = (1e-18 - (math.pow(x_m, 2.0) * 1.2732557730789702)) / (1e-9 - (x_m * 1.128386358070218))
	else:
		tmp = 1.0 + (0.999999999 * (1.0 / ((x_m * 0.3275911) + -1.0)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.0001)
		tmp = Float64(Float64(1e-18 - Float64((x_m ^ 2.0) * 1.2732557730789702)) / Float64(1e-9 - Float64(x_m * 1.128386358070218)));
	else
		tmp = Float64(1.0 + Float64(0.999999999 * Float64(1.0 / Float64(Float64(x_m * 0.3275911) + -1.0))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (abs(x_m) <= 0.0001)
		tmp = (1e-18 - ((x_m ^ 2.0) * 1.2732557730789702)) / (1e-9 - (x_m * 1.128386358070218));
	else
		tmp = 1.0 + (0.999999999 * (1.0 / ((x_m * 0.3275911) + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.0001], N[(N[(1e-18 - N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 1.2732557730789702), $MachinePrecision]), $MachinePrecision] / N[(1e-9 - N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(0.999999999 * N[(1.0 / N[(N[(x$95$m * 0.3275911), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000005e-4

   1. Initial program 57.9%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 57.9%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing

   6. Applied egg-rr 56.3%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot e^{{x}^{2}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   7. Taylor expanded in x around 0 96.9%

      \[\leadsto \color{blue}{10^{-9} + 1.128386358070218 \cdot x} \]
   8. Step-by-step derivation

      1. *-commutative 96.9%

         \[\leadsto 10^{-9} + \color{blue}{x \cdot 1.128386358070218} \]

   9. Simplified 96.9%

      \[\leadsto \color{blue}{10^{-9} + x \cdot 1.128386358070218} \]
   10. Step-by-step derivation

       1. flip-+ 96.9%

          \[\leadsto \color{blue}{\frac{10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right)}{10^{-9} - x \cdot 1.128386358070218}} \]

       2. metadata-eval 96.9%

          \[\leadsto \frac{\color{blue}{10^{-18}} - \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right)}{10^{-9} - x \cdot 1.128386358070218} \]

       3. pow2 96.9%

          \[\leadsto \frac{10^{-18} - \color{blue}{{\left(x \cdot 1.128386358070218\right)}^{2}}}{10^{-9} - x \cdot 1.128386358070218} \]

   11. Applied egg-rr 96.9%

       \[\leadsto \color{blue}{\frac{10^{-18} - {\left(x \cdot 1.128386358070218\right)}^{2}}{10^{-9} - x \cdot 1.128386358070218}} \]
   12. Step-by-step derivation

       1. unpow2 96.9%

          \[\leadsto \frac{10^{-18} - \color{blue}{\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right)}}{10^{-9} - x \cdot 1.128386358070218} \]

       2. swap-sqr 96.9%

          \[\leadsto \frac{10^{-18} - \color{blue}{\left(x \cdot x\right) \cdot \left(1.128386358070218 \cdot 1.128386358070218\right)}}{10^{-9} - x \cdot 1.128386358070218} \]

       3. unpow2 96.9%

          \[\leadsto \frac{10^{-18} - \color{blue}{{x}^{2}} \cdot \left(1.128386358070218 \cdot 1.128386358070218\right)}{10^{-9} - x \cdot 1.128386358070218} \]

       4. metadata-eval 96.9%

          \[\leadsto \frac{10^{-18} - {x}^{2} \cdot \color{blue}{1.2732557730789702}}{10^{-9} - x \cdot 1.128386358070218} \]

   13. Simplified 96.9%

       \[\leadsto \color{blue}{\frac{10^{-18} - {x}^{2} \cdot 1.2732557730789702}{10^{-9} - x \cdot 1.128386358070218}} \]

   if 1.00000000000000005e-4 < (fabs.f64 x)

   1. Initial program 100.0%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 + \left(-\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 + \left(-\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]
   7. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      2. fma-undefine 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{0.3275911 \cdot x + 1}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      3. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{x \cdot 0.3275911} + 1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      4. fma-define 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      5. fma-undefine 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{0.3275911 \cdot x + 1}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      6. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{x \cdot 0.3275911} + 1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      7. fma-define 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      8. fma-undefine 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{0.3275911 \cdot x + 1}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      9. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{x \cdot 0.3275911} + 1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      10. fma-define 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      11. fma-undefine 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{0.3275911 \cdot x + 1}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      12. *-commutative 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{x \cdot 0.3275911} + 1}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      13. fma-define 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

   8. Simplified 100.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

   9. Taylor expanded in x around 0 97.8%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   10. Step-by-step derivation

       1. neg-fabs 97.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left|-x\right|}} \]

       2. add-sqr-sqrt 45.9%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right|} \]

       3. fabs-sqr 45.9%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)}} \]

       4. add-sqr-sqrt 97.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left(-x\right)}} \]

       5. neg-sub0 97.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left(0 - x\right)}} \]

   11. Applied egg-rr 97.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left(0 - x\right)}} \]
   12. Step-by-step derivation

       1. neg-sub0 97.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left(-x\right)}} \]

   13. Simplified 97.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left(-x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;\frac{10^{-18} - {x}^{2} \cdot 1.2732557730789702}{10^{-9} - x \cdot 1.128386358070218}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.999999999 \cdot \frac{1}{x \cdot 0.3275911 + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.8% accurate, 7.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 0.0001:\\ \;\;\;\;x\_m \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.999999999 \cdot \frac{1}{x\_m \cdot 0.3275911 + -1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.0001)
   (+ (* x_m 1.128386358070218) 1e-9)
   (+ 1.0 (* 0.999999999 (/ 1.0 (+ (* x_m 0.3275911) -1.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.0001) {
		tmp = (x_m * 1.128386358070218) + 1e-9;
	} else {
		tmp = 1.0 + (0.999999999 * (1.0 / ((x_m * 0.3275911) + -1.0)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (abs(x_m) <= 0.0001d0) then
        tmp = (x_m * 1.128386358070218d0) + 1d-9
    else
        tmp = 1.0d0 + (0.999999999d0 * (1.0d0 / ((x_m * 0.3275911d0) + (-1.0d0))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.abs(x_m) <= 0.0001) {
		tmp = (x_m * 1.128386358070218) + 1e-9;
	} else {
		tmp = 1.0 + (0.999999999 * (1.0 / ((x_m * 0.3275911) + -1.0)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 0.0001:
		tmp = (x_m * 1.128386358070218) + 1e-9
	else:
		tmp = 1.0 + (0.999999999 * (1.0 / ((x_m * 0.3275911) + -1.0)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.0001)
		tmp = Float64(Float64(x_m * 1.128386358070218) + 1e-9);
	else
		tmp = Float64(1.0 + Float64(0.999999999 * Float64(1.0 / Float64(Float64(x_m * 0.3275911) + -1.0))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (abs(x_m) <= 0.0001)
		tmp = (x_m * 1.128386358070218) + 1e-9;
	else
		tmp = 1.0 + (0.999999999 * (1.0 / ((x_m * 0.3275911) + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.0001], N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] + 1e-9), $MachinePrecision], N[(1.0 + N[(0.999999999 * N[(1.0 / N[(N[(x$95$m * 0.3275911), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000005e-4

   1. Initial program 57.9%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 57.9%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing

   6. Applied egg-rr 56.3%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot e^{{x}^{2}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   7. Taylor expanded in x around 0 96.9%

      \[\leadsto \color{blue}{10^{-9} + 1.128386358070218 \cdot x} \]
   8. Step-by-step derivation

      1. *-commutative 96.9%

         \[\leadsto 10^{-9} + \color{blue}{x \cdot 1.128386358070218} \]

   9. Simplified 96.9%

      \[\leadsto \color{blue}{10^{-9} + x \cdot 1.128386358070218} \]

   if 1.00000000000000005e-4 < (fabs.f64 x)

   1. Initial program 100.0%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 + \left(-\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 + \left(-\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]
   7. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      2. fma-undefine 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{0.3275911 \cdot x + 1}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      3. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{x \cdot 0.3275911} + 1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      4. fma-define 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      5. fma-undefine 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{0.3275911 \cdot x + 1}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      6. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{x \cdot 0.3275911} + 1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      7. fma-define 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      8. fma-undefine 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{0.3275911 \cdot x + 1}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      9. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{x \cdot 0.3275911} + 1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      10. fma-define 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      11. fma-undefine 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{0.3275911 \cdot x + 1}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      12. *-commutative 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{x \cdot 0.3275911} + 1}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      13. fma-define 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

   8. Simplified 100.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

   9. Taylor expanded in x around 0 97.8%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   10. Step-by-step derivation

       1. neg-fabs 97.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left|-x\right|}} \]

       2. add-sqr-sqrt 45.9%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right|} \]

       3. fabs-sqr 45.9%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)}} \]

       4. add-sqr-sqrt 97.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left(-x\right)}} \]

       5. neg-sub0 97.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left(0 - x\right)}} \]

   11. Applied egg-rr 97.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left(0 - x\right)}} \]
   12. Step-by-step derivation

       1. neg-sub0 97.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left(-x\right)}} \]

   13. Simplified 97.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left(-x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;x \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.999999999 \cdot \frac{1}{x \cdot 0.3275911 + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.1% accurate, 53.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.3:\\ \;\;\;\;x\_m \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.999999999 \cdot \frac{1}{-1 - x\_m \cdot 0.3275911}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.3)
   (+ (* x_m 1.128386358070218) 1e-9)
   (+ 1.0 (* 0.999999999 (/ 1.0 (- -1.0 (* x_m 0.3275911)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.3) {
		tmp = (x_m * 1.128386358070218) + 1e-9;
	} else {
		tmp = 1.0 + (0.999999999 * (1.0 / (-1.0 - (x_m * 0.3275911))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 2.3d0) then
        tmp = (x_m * 1.128386358070218d0) + 1d-9
    else
        tmp = 1.0d0 + (0.999999999d0 * (1.0d0 / ((-1.0d0) - (x_m * 0.3275911d0))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.3) {
		tmp = (x_m * 1.128386358070218) + 1e-9;
	} else {
		tmp = 1.0 + (0.999999999 * (1.0 / (-1.0 - (x_m * 0.3275911))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.3:
		tmp = (x_m * 1.128386358070218) + 1e-9
	else:
		tmp = 1.0 + (0.999999999 * (1.0 / (-1.0 - (x_m * 0.3275911))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.3)
		tmp = Float64(Float64(x_m * 1.128386358070218) + 1e-9);
	else
		tmp = Float64(1.0 + Float64(0.999999999 * Float64(1.0 / Float64(-1.0 - Float64(x_m * 0.3275911)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.3)
		tmp = (x_m * 1.128386358070218) + 1e-9;
	else
		tmp = 1.0 + (0.999999999 * (1.0 / (-1.0 - (x_m * 0.3275911))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.3], N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] + 1e-9), $MachinePrecision], N[(1.0 + N[(0.999999999 * N[(1.0 / N[(-1.0 - N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2999999999999998

   1. Initial program 71.6%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 71.6%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing

   6. Applied egg-rr 39.1%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot e^{{x}^{2}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   7. Taylor expanded in x around 0 65.8%

      \[\leadsto \color{blue}{10^{-9} + 1.128386358070218 \cdot x} \]
   8. Step-by-step derivation

      1. *-commutative 65.8%

         \[\leadsto 10^{-9} + \color{blue}{x \cdot 1.128386358070218} \]

   9. Simplified 65.8%

      \[\leadsto \color{blue}{10^{-9} + x \cdot 1.128386358070218} \]

   if 2.2999999999999998 < x

   1. Initial program 100.0%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 + \left(-\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 + \left(-\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]
   7. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      2. fma-undefine 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{0.3275911 \cdot x + 1}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      3. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{x \cdot 0.3275911} + 1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      4. fma-define 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      5. fma-undefine 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{0.3275911 \cdot x + 1}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      6. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{x \cdot 0.3275911} + 1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      7. fma-define 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      8. fma-undefine 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{0.3275911 \cdot x + 1}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      9. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{x \cdot 0.3275911} + 1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      10. fma-define 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      11. fma-undefine 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{0.3275911 \cdot x + 1}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      12. *-commutative 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{x \cdot 0.3275911} + 1}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      13. fma-define 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

   8. Simplified 100.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

   9. Taylor expanded in x around 0 98.4%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       2. log1p-define 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       3. +-commutative 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       4. fma-undefine 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       5. expm1-undefine 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       6. add-exp-log 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       7. add-sqr-sqrt 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       8. fabs-sqr 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       9. add-sqr-sqrt 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   11. Applied egg-rr 98.4%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \]
   12. Step-by-step derivation

       1. fma-undefine 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       2. associate--l+ 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       3. metadata-eval 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       4. +-rgt-identity 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       5. *-commutative 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + \color{blue}{x \cdot 0.3275911}} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   13. Simplified 98.4%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{x \cdot 0.3275911}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.3:\\ \;\;\;\;x \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.999999999 \cdot \frac{1}{-1 - x \cdot 0.3275911}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.3% accurate, 85.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 920000000:\\ \;\;\;\;x\_m \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;10^{-9}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 920000000.0) (+ (* x_m 1.128386358070218) 1e-9) 1e-9))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 920000000.0) {
		tmp = (x_m * 1.128386358070218) + 1e-9;
	} else {
		tmp = 1e-9;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 920000000.0d0) then
        tmp = (x_m * 1.128386358070218d0) + 1d-9
    else
        tmp = 1d-9
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 920000000.0) {
		tmp = (x_m * 1.128386358070218) + 1e-9;
	} else {
		tmp = 1e-9;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 920000000.0:
		tmp = (x_m * 1.128386358070218) + 1e-9
	else:
		tmp = 1e-9
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 920000000.0)
		tmp = Float64(Float64(x_m * 1.128386358070218) + 1e-9);
	else
		tmp = 1e-9;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 920000000.0)
		tmp = (x_m * 1.128386358070218) + 1e-9;
	else
		tmp = 1e-9;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 920000000.0], N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] + 1e-9), $MachinePrecision], 1e-9]

Derivation

1. Split input into 2 regimes

2. if x < 9.2e8

   1. Initial program 71.7%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 71.7%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing

   6. Applied egg-rr 38.9%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot e^{{x}^{2}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   7. Taylor expanded in x around 0 65.5%

      \[\leadsto \color{blue}{10^{-9} + 1.128386358070218 \cdot x} \]
   8. Step-by-step derivation

      1. *-commutative 65.5%

         \[\leadsto 10^{-9} + \color{blue}{x \cdot 1.128386358070218} \]

   9. Simplified 65.5%

      \[\leadsto \color{blue}{10^{-9} + x \cdot 1.128386358070218} \]

   if 9.2e8 < x

   1. Initial program 100.0%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing

   6. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot e^{{x}^{2}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   7. Taylor expanded in x around 0 11.1%

      \[\leadsto \color{blue}{10^{-9}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 920000000:\\ \;\;\;\;x \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;10^{-9}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.4% accurate, 856.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 10^{-9} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 1e-9)

C

x_m = fabs(x);
double code(double x_m) {
	return 1e-9;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 1d-9
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 1e-9;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 1e-9

Julia

x_m = abs(x)
function code(x_m)
	return 1e-9
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 1e-9;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 1e-9

Derivation

1. Initial program 79.1%

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

3. Simplified 79.1%

   \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
4. Add Preprocessing

6. Applied egg-rr 29.5%

   \[\leadsto \color{blue}{{\left({\left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot e^{{x}^{2}}\right)}^{3}\right)}^{0.3333333333333333}} \]

7. Taylor expanded in x around 0 53.2%

   \[\leadsto \color{blue}{10^{-9}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024141 
(FPCore (x)
  :name "Jmat.Real.erf"
  :precision binary64
  (- 1.0 (* (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ 0.254829592 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ -0.284496736 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ 1.421413741 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ -1.453152027 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) 1.061405429))))))))) (exp (- (* (fabs x) (fabs x)))))))