Jmat.Real.erfi, branch x less than or equal to 0.5

Percentage Accurate: 99.8% → 99.9%

Time: 11.0s

Alternatives: 9

Speedup: 3.6×

• 29.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
      (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x))))));
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x))))));
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x))))))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
      (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x))))));
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x))))));
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x))))))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Alternative 1: 99.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ x\_m \cdot \left(t\_0 \cdot \left(2 + 0.6666666666666666 \cdot {x\_m}^{2}\right)\right) + \left(t\_0 \cdot \left({x\_m}^{2} \cdot 0.047619047619047616 + 0.2\right)\right) \cdot {x\_m}^{5} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (+
    (* x_m (* t_0 (+ 2.0 (* 0.6666666666666666 (pow x_m 2.0)))))
    (* (* t_0 (+ (* (pow x_m 2.0) 0.047619047619047616) 0.2)) (pow x_m 5.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	return (x_m * (t_0 * (2.0 + (0.6666666666666666 * pow(x_m, 2.0))))) + ((t_0 * ((pow(x_m, 2.0) * 0.047619047619047616) + 0.2)) * pow(x_m, 5.0));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	return (x_m * (t_0 * (2.0 + (0.6666666666666666 * Math.pow(x_m, 2.0))))) + ((t_0 * ((Math.pow(x_m, 2.0) * 0.047619047619047616) + 0.2)) * Math.pow(x_m, 5.0));
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = math.sqrt((1.0 / math.pi))
	return (x_m * (t_0 * (2.0 + (0.6666666666666666 * math.pow(x_m, 2.0))))) + ((t_0 * ((math.pow(x_m, 2.0) * 0.047619047619047616) + 0.2)) * math.pow(x_m, 5.0))

Julia

x_m = abs(x)
function code(x_m)
	t_0 = sqrt(Float64(1.0 / pi))
	return Float64(Float64(x_m * Float64(t_0 * Float64(2.0 + Float64(0.6666666666666666 * (x_m ^ 2.0))))) + Float64(Float64(t_0 * Float64(Float64((x_m ^ 2.0) * 0.047619047619047616) + 0.2)) * (x_m ^ 5.0)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	t_0 = sqrt((1.0 / pi));
	tmp = (x_m * (t_0 * (2.0 + (0.6666666666666666 * (x_m ^ 2.0))))) + ((t_0 * (((x_m ^ 2.0) * 0.047619047619047616) + 0.2)) * (x_m ^ 5.0));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, N[(N[(x$95$m * N[(t$95$0 * N[(2.0 + N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 * N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 0.047619047619047616), $MachinePrecision] + 0.2), $MachinePrecision]), $MachinePrecision] * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 99.8%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|} \]
6. Step-by-step derivation

   1. expm1-log1p-u 99.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)\right)} \]

   2. expm1-undefine 40.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)} - 1} \]

7. Applied egg-rr 3.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right)\right)} - 1} \]
8. Step-by-step derivation

   1. log1p-undefine 3.4%

      \[\leadsto e^{\color{blue}{\log \left(1 + x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right)\right)}} - 1 \]

   2. rem-exp-log 3.5%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right)\right)} - 1 \]

   3. +-commutative 3.5%

      \[\leadsto \color{blue}{\left(x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) + 1\right)} - 1 \]

   4. associate--l+ 31.0%

      \[\leadsto \color{blue}{x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) + \left(1 - 1\right)} \]

   5. metadata-eval 31.0%

      \[\leadsto x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) + \color{blue}{0} \]

   6. metadata-eval 31.0%

      \[\leadsto x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) + \color{blue}{\left(-0\right)} \]

   7. sub-neg 31.0%

      \[\leadsto \color{blue}{x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) - 0} \]

   8. --rgt-identity 31.0%

      \[\leadsto \color{blue}{x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right)} \]

   9. *-commutative 31.0%

      \[\leadsto \color{blue}{\left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) \cdot x} \]

9. Simplified 31.0%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.2, {x}^{4}, 2\right)\right) + 0.6666666666666666 \cdot {x}^{2}\right) \cdot \left(x \cdot {\pi}^{-0.5}\right)} \]
10. Step-by-step derivation

    1. fma-undefine 31.0%

       \[\leadsto \left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \color{blue}{0.2 \cdot {x}^{4} + 2}\right) + 0.6666666666666666 \cdot {x}^{2}\right) \cdot \left(x \cdot {\pi}^{-0.5}\right) \]

11. Applied egg-rr 31.0%

    \[\leadsto \left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \color{blue}{0.2 \cdot {x}^{4} + 2}\right) + 0.6666666666666666 \cdot {x}^{2}\right) \cdot \left(x \cdot {\pi}^{-0.5}\right) \]

12. Taylor expanded in x around 0 31.0%

    \[\leadsto \color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{\pi}} + {x}^{2} \cdot \left(0.6666666666666666 \cdot \sqrt{\frac{1}{\pi}} + {x}^{2} \cdot \left(0.047619047619047616 \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.2 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)} \]
13. Step-by-step derivation

    1. distribute-lft-in 31.0%

       \[\leadsto x \cdot \left(2 \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left({x}^{2} \cdot \left(0.6666666666666666 \cdot \sqrt{\frac{1}{\pi}}\right) + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.047619047619047616 \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.2 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)}\right) \]

    2. associate-+r+ 31.0%

       \[\leadsto x \cdot \color{blue}{\left(\left(2 \cdot \sqrt{\frac{1}{\pi}} + {x}^{2} \cdot \left(0.6666666666666666 \cdot \sqrt{\frac{1}{\pi}}\right)\right) + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.047619047619047616 \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.2 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)} \]

    3. +-commutative 31.0%

       \[\leadsto x \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(0.6666666666666666 \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \sqrt{\frac{1}{\pi}}\right)} + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.047619047619047616 \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.2 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) \]

14. Simplified 31.0%

    \[\leadsto \color{blue}{x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{2} + 2\right)\right) + \left(\sqrt{\frac{1}{\pi}} \cdot \left({x}^{2} \cdot 0.047619047619047616 + 0.2\right)\right) \cdot {x}^{5}} \]

15. Final simplification 31.0%

    \[\leadsto x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right) + \left(\sqrt{\frac{1}{\pi}} \cdot \left({x}^{2} \cdot 0.047619047619047616 + 0.2\right)\right) \cdot {x}^{5} \]
16. Add Preprocessing

Alternative 2: 99.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \left(0.6666666666666666 \cdot {x\_m}^{2} + \mathsf{fma}\left(0.047619047619047616, {x\_m}^{6}, 2 + 0.2 \cdot {x\_m}^{4}\right)\right) \cdot \left(x\_m \cdot {\pi}^{-0.5}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (+
   (* 0.6666666666666666 (pow x_m 2.0))
   (fma 0.047619047619047616 (pow x_m 6.0) (+ 2.0 (* 0.2 (pow x_m 4.0)))))
  (* x_m (pow PI -0.5))))

C

x_m = fabs(x);
double code(double x_m) {
	return ((0.6666666666666666 * pow(x_m, 2.0)) + fma(0.047619047619047616, pow(x_m, 6.0), (2.0 + (0.2 * pow(x_m, 4.0))))) * (x_m * pow(((double) M_PI), -0.5));
}

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(Float64(0.6666666666666666 * (x_m ^ 2.0)) + fma(0.047619047619047616, (x_m ^ 6.0), Float64(2.0 + Float64(0.2 * (x_m ^ 4.0))))) * Float64(x_m * (pi ^ -0.5)))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $MachinePrecision] + N[(2.0 + N[(0.2 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 99.8%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|} \]
6. Step-by-step derivation

   1. expm1-log1p-u 99.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)\right)} \]

   2. expm1-undefine 40.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)} - 1} \]

7. Applied egg-rr 3.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right)\right)} - 1} \]
8. Step-by-step derivation

   1. log1p-undefine 3.4%

      \[\leadsto e^{\color{blue}{\log \left(1 + x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right)\right)}} - 1 \]

   2. rem-exp-log 3.5%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right)\right)} - 1 \]

   3. +-commutative 3.5%

      \[\leadsto \color{blue}{\left(x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) + 1\right)} - 1 \]

   4. associate--l+ 31.0%

      \[\leadsto \color{blue}{x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) + \left(1 - 1\right)} \]

   5. metadata-eval 31.0%

      \[\leadsto x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) + \color{blue}{0} \]

   6. metadata-eval 31.0%

      \[\leadsto x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) + \color{blue}{\left(-0\right)} \]

   7. sub-neg 31.0%

      \[\leadsto \color{blue}{x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) - 0} \]

   8. --rgt-identity 31.0%

      \[\leadsto \color{blue}{x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right)} \]

   9. *-commutative 31.0%

      \[\leadsto \color{blue}{\left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) \cdot x} \]

9. Simplified 31.0%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.2, {x}^{4}, 2\right)\right) + 0.6666666666666666 \cdot {x}^{2}\right) \cdot \left(x \cdot {\pi}^{-0.5}\right)} \]
10. Step-by-step derivation

    1. fma-undefine 31.0%

       \[\leadsto \left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \color{blue}{0.2 \cdot {x}^{4} + 2}\right) + 0.6666666666666666 \cdot {x}^{2}\right) \cdot \left(x \cdot {\pi}^{-0.5}\right) \]

11. Applied egg-rr 31.0%

    \[\leadsto \left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \color{blue}{0.2 \cdot {x}^{4} + 2}\right) + 0.6666666666666666 \cdot {x}^{2}\right) \cdot \left(x \cdot {\pi}^{-0.5}\right) \]

12. Final simplification 31.0%

    \[\leadsto \left(0.6666666666666666 \cdot {x}^{2} + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2 + 0.2 \cdot {x}^{4}\right)\right) \cdot \left(x \cdot {\pi}^{-0.5}\right) \]
13. Add Preprocessing

Alternative 3: 99.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \left|x\_m \cdot \frac{2 + \mathsf{fma}\left(0.2, {x\_m}^{4}, 0.047619047619047616 \cdot {x\_m}^{6}\right)}{\sqrt{\pi}}\right| \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (fabs
  (*
   x_m
   (/
    (+ 2.0 (fma 0.2 (pow x_m 4.0) (* 0.047619047619047616 (pow x_m 6.0))))
    (sqrt PI)))))

C

x_m = fabs(x);
double code(double x_m) {
	return fabs((x_m * ((2.0 + fma(0.2, pow(x_m, 4.0), (0.047619047619047616 * pow(x_m, 6.0)))) / sqrt(((double) M_PI)))));
}

Julia

x_m = abs(x)
function code(x_m)
	return abs(Float64(x_m * Float64(Float64(2.0 + fma(0.2, (x_m ^ 4.0), Float64(0.047619047619047616 * (x_m ^ 6.0)))) / sqrt(pi))))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Abs[N[(x$95$m * N[(N[(2.0 + N[(0.2 * N[Power[x$95$m, 4.0], $MachinePrecision] + N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 99.2%

   \[\leadsto \left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \color{blue}{2}}{\sqrt{\pi}}\right| \]
6. Step-by-step derivation

   1. mul-fabs 99.2%

      \[\leadsto \color{blue}{\left|x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2}{\sqrt{\pi}}\right|} \]

   2. +-commutative 99.2%

      \[\leadsto \left|x \cdot \frac{\color{blue}{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}}{\sqrt{\pi}}\right| \]

7. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\left|x \cdot \frac{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|} \]
8. Add Preprocessing

Alternative 4: 99.1% accurate, 4.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x\_m \cdot \frac{\mathsf{fma}\left(0.047619047619047616, {x\_m}^{6}, 2 + 0.2 \cdot {x\_m}^{4}\right)}{\sqrt{\pi}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  x_m
  (/
   (fma 0.047619047619047616 (pow x_m 6.0) (+ 2.0 (* 0.2 (pow x_m 4.0))))
   (sqrt PI))))

C

x_m = fabs(x);
double code(double x_m) {
	return x_m * (fma(0.047619047619047616, pow(x_m, 6.0), (2.0 + (0.2 * pow(x_m, 4.0)))) / sqrt(((double) M_PI)));
}

Julia

x_m = abs(x)
function code(x_m)
	return Float64(x_m * Float64(fma(0.047619047619047616, (x_m ^ 6.0), Float64(2.0 + Float64(0.2 * (x_m ^ 4.0)))) / sqrt(pi)))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * N[(N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $MachinePrecision] + N[(2.0 + N[(0.2 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 99.2%

   \[\leadsto \left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \color{blue}{2}}{\sqrt{\pi}}\right| \]
6. Step-by-step derivation

   1. expm1-log1p-u 98.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2}{\sqrt{\pi}}\right|\right)\right)} \]

   2. expm1-undefine 39.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2}{\sqrt{\pi}}\right|\right)} - 1} \]

7. Applied egg-rr 3.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right)} - 1} \]
8. Step-by-step derivation

   1. log1p-undefine 3.4%

      \[\leadsto e^{\color{blue}{\log \left(1 + x \cdot \frac{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right)}} - 1 \]

   2. rem-exp-log 3.5%

      \[\leadsto \color{blue}{\left(1 + x \cdot \frac{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right)} - 1 \]

   3. +-commutative 3.5%

      \[\leadsto \color{blue}{\left(x \cdot \frac{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}} + 1\right)} - 1 \]

   4. associate--l+ 31.0%

      \[\leadsto \color{blue}{x \cdot \frac{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}} + \left(1 - 1\right)} \]

   5. metadata-eval 31.0%

      \[\leadsto x \cdot \frac{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}} + \color{blue}{0} \]

   6. metadata-eval 31.0%

      \[\leadsto x \cdot \frac{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}} + \color{blue}{\left(-0\right)} \]

   7. sub-neg 31.0%

      \[\leadsto \color{blue}{x \cdot \frac{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}} - 0} \]

   8. --rgt-identity 31.0%

      \[\leadsto \color{blue}{x \cdot \frac{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}} \]

9. Simplified 31.0%

   \[\leadsto \color{blue}{x \cdot \frac{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.2, {x}^{4}, 2\right)\right)}{\sqrt{\pi}}} \]
10. Step-by-step derivation

    1. fma-undefine 31.0%

       \[\leadsto \left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \color{blue}{0.2 \cdot {x}^{4} + 2}\right) + 0.6666666666666666 \cdot {x}^{2}\right) \cdot \left(x \cdot {\pi}^{-0.5}\right) \]

11. Applied egg-rr 31.0%

    \[\leadsto x \cdot \frac{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \color{blue}{0.2 \cdot {x}^{4} + 2}\right)}{\sqrt{\pi}} \]

12. Final simplification 31.0%

    \[\leadsto x \cdot \frac{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2 + 0.2 \cdot {x}^{4}\right)}{\sqrt{\pi}} \]
13. Add Preprocessing

Alternative 5: 97.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \mathbf{if}\;\left|x\_m\right| \leq 2.5 \cdot 10^{-13}:\\ \;\;\;\;x\_m \cdot \left(t\_0 \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(0.047619047619047616 \cdot \left(t\_0 \cdot {x\_m}^{6}\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (if (<= (fabs x_m) 2.5e-13)
     (* x_m (* t_0 2.0))
     (* x_m (* 0.047619047619047616 (* t_0 (pow x_m 6.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	double tmp;
	if (fabs(x_m) <= 2.5e-13) {
		tmp = x_m * (t_0 * 2.0);
	} else {
		tmp = x_m * (0.047619047619047616 * (t_0 * pow(x_m, 6.0)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	double tmp;
	if (Math.abs(x_m) <= 2.5e-13) {
		tmp = x_m * (t_0 * 2.0);
	} else {
		tmp = x_m * (0.047619047619047616 * (t_0 * Math.pow(x_m, 6.0)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = math.sqrt((1.0 / math.pi))
	tmp = 0
	if math.fabs(x_m) <= 2.5e-13:
		tmp = x_m * (t_0 * 2.0)
	else:
		tmp = x_m * (0.047619047619047616 * (t_0 * math.pow(x_m, 6.0)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = sqrt(Float64(1.0 / pi))
	tmp = 0.0
	if (abs(x_m) <= 2.5e-13)
		tmp = Float64(x_m * Float64(t_0 * 2.0));
	else
		tmp = Float64(x_m * Float64(0.047619047619047616 * Float64(t_0 * (x_m ^ 6.0))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = sqrt((1.0 / pi));
	tmp = 0.0;
	if (abs(x_m) <= 2.5e-13)
		tmp = x_m * (t_0 * 2.0);
	else
		tmp = x_m * (0.047619047619047616 * (t_0 * (x_m ^ 6.0)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2.5e-13], N[(x$95$m * N[(t$95$0 * 2.0), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(0.047619047619047616 * N[(t$95$0 * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2.49999999999999995e-13

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.8%

      \[\leadsto \color{blue}{\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 99.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)\right)} \]

      2. expm1-undefine 5.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)} - 1} \]

   7. Applied egg-rr 5.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right)\right)} - 1} \]
   8. Step-by-step derivation

      1. log1p-undefine 5.4%

         \[\leadsto e^{\color{blue}{\log \left(1 + x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right)\right)}} - 1 \]

      2. rem-exp-log 5.4%

         \[\leadsto \color{blue}{\left(1 + x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right)\right)} - 1 \]

      3. +-commutative 5.4%

         \[\leadsto \color{blue}{\left(x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) + 1\right)} - 1 \]

      4. associate--l+ 49.5%

         \[\leadsto \color{blue}{x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) + \left(1 - 1\right)} \]

      5. metadata-eval 49.5%

         \[\leadsto x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) + \color{blue}{0} \]

      6. metadata-eval 49.5%

         \[\leadsto x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) + \color{blue}{\left(-0\right)} \]

      7. sub-neg 49.5%

         \[\leadsto \color{blue}{x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) - 0} \]

      8. --rgt-identity 49.5%

         \[\leadsto \color{blue}{x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right)} \]

      9. *-commutative 49.5%

         \[\leadsto \color{blue}{\left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) \cdot x} \]

   9. Simplified 49.5%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.2, {x}^{4}, 2\right)\right) + 0.6666666666666666 \cdot {x}^{2}\right) \cdot \left(x \cdot {\pi}^{-0.5}\right)} \]

   10. Taylor expanded in x around 0 49.5%

       \[\leadsto \color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 49.5%

          \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} \]

       2. *-commutative 49.5%

          \[\leadsto \color{blue}{\left(x \cdot 2\right)} \cdot \sqrt{\frac{1}{\pi}} \]

       3. associate-*l* 49.5%

          \[\leadsto \color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{\pi}}\right)} \]

       4. *-commutative 49.5%

          \[\leadsto x \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot 2\right)} \]

   12. Simplified 49.5%

       \[\leadsto \color{blue}{x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 2\right)} \]

   if 2.49999999999999995e-13 < (fabs.f64 x)

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 98.1%

      \[\leadsto \left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \color{blue}{2}}{\sqrt{\pi}}\right| \]
   6. Step-by-step derivation

      1. expm1-log1p-u 96.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2}{\sqrt{\pi}}\right|\right)\right)} \]

      2. expm1-undefine 96.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2}{\sqrt{\pi}}\right|\right)} - 1} \]

   7. Applied egg-rr 0.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right)} - 1} \]
   8. Step-by-step derivation

      1. log1p-undefine 0.1%

         \[\leadsto e^{\color{blue}{\log \left(1 + x \cdot \frac{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right)}} - 1 \]

      2. rem-exp-log 0.2%

         \[\leadsto \color{blue}{\left(1 + x \cdot \frac{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right)} - 1 \]

      3. +-commutative 0.2%

         \[\leadsto \color{blue}{\left(x \cdot \frac{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}} + 1\right)} - 1 \]

      4. associate--l+ 0.2%

         \[\leadsto \color{blue}{x \cdot \frac{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}} + \left(1 - 1\right)} \]

      5. metadata-eval 0.2%

         \[\leadsto x \cdot \frac{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}} + \color{blue}{0} \]

      6. metadata-eval 0.2%

         \[\leadsto x \cdot \frac{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}} + \color{blue}{\left(-0\right)} \]

      7. sub-neg 0.2%

         \[\leadsto \color{blue}{x \cdot \frac{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}} - 0} \]

      8. --rgt-identity 0.2%

         \[\leadsto \color{blue}{x \cdot \frac{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}} \]

   9. Simplified 0.2%

      \[\leadsto \color{blue}{x \cdot \frac{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.2, {x}^{4}, 2\right)\right)}{\sqrt{\pi}}} \]

   10. Taylor expanded in x around inf 0.2%

       \[\leadsto x \cdot \color{blue}{\left(0.047619047619047616 \cdot \left({x}^{6} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2.5 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{6}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% accurate, 6.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x\_m \cdot \frac{\mathsf{fma}\left(0.047619047619047616, {x\_m}^{6}, 2\right)}{\sqrt{\pi}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (* x_m (/ (fma 0.047619047619047616 (pow x_m 6.0) 2.0) (sqrt PI))))

C

x_m = fabs(x);
double code(double x_m) {
	return x_m * (fma(0.047619047619047616, pow(x_m, 6.0), 2.0) / sqrt(((double) M_PI)));
}

Julia

x_m = abs(x)
function code(x_m)
	return Float64(x_m * Float64(fma(0.047619047619047616, (x_m ^ 6.0), 2.0) / sqrt(pi)))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * N[(N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 99.2%

   \[\leadsto \left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \color{blue}{2}}{\sqrt{\pi}}\right| \]
6. Step-by-step derivation

   1. expm1-log1p-u 98.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2}{\sqrt{\pi}}\right|\right)\right)} \]

   2. expm1-undefine 39.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2}{\sqrt{\pi}}\right|\right)} - 1} \]

7. Applied egg-rr 3.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right)} - 1} \]
8. Step-by-step derivation

   1. log1p-undefine 3.4%

      \[\leadsto e^{\color{blue}{\log \left(1 + x \cdot \frac{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right)}} - 1 \]

   2. rem-exp-log 3.5%

      \[\leadsto \color{blue}{\left(1 + x \cdot \frac{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right)} - 1 \]

   3. +-commutative 3.5%

      \[\leadsto \color{blue}{\left(x \cdot \frac{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}} + 1\right)} - 1 \]

   4. associate--l+ 31.0%

      \[\leadsto \color{blue}{x \cdot \frac{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}} + \left(1 - 1\right)} \]

   5. metadata-eval 31.0%

      \[\leadsto x \cdot \frac{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}} + \color{blue}{0} \]

   6. metadata-eval 31.0%

      \[\leadsto x \cdot \frac{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}} + \color{blue}{\left(-0\right)} \]

   7. sub-neg 31.0%

      \[\leadsto \color{blue}{x \cdot \frac{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}} - 0} \]

   8. --rgt-identity 31.0%

      \[\leadsto \color{blue}{x \cdot \frac{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}} \]

9. Simplified 31.0%

   \[\leadsto \color{blue}{x \cdot \frac{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.2, {x}^{4}, 2\right)\right)}{\sqrt{\pi}}} \]

10. Taylor expanded in x around 0 31.0%

    \[\leadsto x \cdot \frac{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \color{blue}{2}\right)}{\sqrt{\pi}} \]
11. Add Preprocessing

Alternative 7: 98.7% accurate, 8.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \mathbf{if}\;x\_m \leq 0.001:\\ \;\;\;\;x\_m \cdot \left(t\_0 \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left(t\_0 \cdot {x\_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (if (<= x_m 0.001)
     (* x_m (* t_0 2.0))
     (* 0.047619047619047616 (* t_0 (pow x_m 7.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	double tmp;
	if (x_m <= 0.001) {
		tmp = x_m * (t_0 * 2.0);
	} else {
		tmp = 0.047619047619047616 * (t_0 * pow(x_m, 7.0));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	double tmp;
	if (x_m <= 0.001) {
		tmp = x_m * (t_0 * 2.0);
	} else {
		tmp = 0.047619047619047616 * (t_0 * Math.pow(x_m, 7.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = math.sqrt((1.0 / math.pi))
	tmp = 0
	if x_m <= 0.001:
		tmp = x_m * (t_0 * 2.0)
	else:
		tmp = 0.047619047619047616 * (t_0 * math.pow(x_m, 7.0))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = sqrt(Float64(1.0 / pi))
	tmp = 0.0
	if (x_m <= 0.001)
		tmp = Float64(x_m * Float64(t_0 * 2.0));
	else
		tmp = Float64(0.047619047619047616 * Float64(t_0 * (x_m ^ 7.0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = sqrt((1.0 / pi));
	tmp = 0.0;
	if (x_m <= 0.001)
		tmp = x_m * (t_0 * 2.0);
	else
		tmp = 0.047619047619047616 * (t_0 * (x_m ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$95$m, 0.001], N[(x$95$m * N[(t$95$0 * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(t$95$0 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1e-3

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.8%

      \[\leadsto \color{blue}{\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 99.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)\right)} \]

      2. expm1-undefine 40.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)} - 1} \]

   7. Applied egg-rr 3.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right)\right)} - 1} \]
   8. Step-by-step derivation

      1. log1p-undefine 3.4%

         \[\leadsto e^{\color{blue}{\log \left(1 + x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right)\right)}} - 1 \]

      2. rem-exp-log 3.5%

         \[\leadsto \color{blue}{\left(1 + x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right)\right)} - 1 \]

      3. +-commutative 3.5%

         \[\leadsto \color{blue}{\left(x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) + 1\right)} - 1 \]

      4. associate--l+ 31.0%

         \[\leadsto \color{blue}{x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) + \left(1 - 1\right)} \]

      5. metadata-eval 31.0%

         \[\leadsto x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) + \color{blue}{0} \]

      6. metadata-eval 31.0%

         \[\leadsto x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) + \color{blue}{\left(-0\right)} \]

      7. sub-neg 31.0%

         \[\leadsto \color{blue}{x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) - 0} \]

      8. --rgt-identity 31.0%

         \[\leadsto \color{blue}{x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right)} \]

      9. *-commutative 31.0%

         \[\leadsto \color{blue}{\left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) \cdot x} \]

   9. Simplified 31.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.2, {x}^{4}, 2\right)\right) + 0.6666666666666666 \cdot {x}^{2}\right) \cdot \left(x \cdot {\pi}^{-0.5}\right)} \]

   10. Taylor expanded in x around 0 31.1%

       \[\leadsto \color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 31.1%

          \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} \]

       2. *-commutative 31.1%

          \[\leadsto \color{blue}{\left(x \cdot 2\right)} \cdot \sqrt{\frac{1}{\pi}} \]

       3. associate-*l* 31.1%

          \[\leadsto \color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{\pi}}\right)} \]

       4. *-commutative 31.1%

          \[\leadsto x \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot 2\right)} \]

   12. Simplified 31.1%

       \[\leadsto \color{blue}{x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 2\right)} \]

   if 1e-3 < x

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.8%

      \[\leadsto \color{blue}{\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 99.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)\right)} \]

      2. expm1-undefine 40.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)} - 1} \]

   7. Applied egg-rr 3.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right)\right)} - 1} \]
   8. Step-by-step derivation

      1. log1p-undefine 3.4%

         \[\leadsto e^{\color{blue}{\log \left(1 + x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right)\right)}} - 1 \]

      2. rem-exp-log 3.5%

         \[\leadsto \color{blue}{\left(1 + x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right)\right)} - 1 \]

      3. +-commutative 3.5%

         \[\leadsto \color{blue}{\left(x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) + 1\right)} - 1 \]

      4. associate--l+ 31.0%

         \[\leadsto \color{blue}{x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) + \left(1 - 1\right)} \]

      5. metadata-eval 31.0%

         \[\leadsto x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) + \color{blue}{0} \]

      6. metadata-eval 31.0%

         \[\leadsto x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) + \color{blue}{\left(-0\right)} \]

      7. sub-neg 31.0%

         \[\leadsto \color{blue}{x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) - 0} \]

      8. --rgt-identity 31.0%

         \[\leadsto \color{blue}{x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right)} \]

      9. *-commutative 31.0%

         \[\leadsto \color{blue}{\left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) \cdot x} \]

   9. Simplified 31.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.2, {x}^{4}, 2\right)\right) + 0.6666666666666666 \cdot {x}^{2}\right) \cdot \left(x \cdot {\pi}^{-0.5}\right)} \]

   10. Taylor expanded in x around inf 3.5%

       \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.001:\\ \;\;\;\;x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.2% accurate, 17.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x\_m \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 2\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* x_m (* (sqrt (/ 1.0 PI)) 2.0)))

C

x_m = fabs(x);
double code(double x_m) {
	return x_m * (sqrt((1.0 / ((double) M_PI))) * 2.0);
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return x_m * (Math.sqrt((1.0 / Math.PI)) * 2.0);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return x_m * (math.sqrt((1.0 / math.pi)) * 2.0)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(x_m * Float64(sqrt(Float64(1.0 / pi)) * 2.0))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = x_m * (sqrt((1.0 / pi)) * 2.0);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 99.8%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|} \]
6. Step-by-step derivation

   1. expm1-log1p-u 99.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)\right)} \]

   2. expm1-undefine 40.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)} - 1} \]

7. Applied egg-rr 3.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right)\right)} - 1} \]
8. Step-by-step derivation

   1. log1p-undefine 3.4%

      \[\leadsto e^{\color{blue}{\log \left(1 + x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right)\right)}} - 1 \]

   2. rem-exp-log 3.5%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right)\right)} - 1 \]

   3. +-commutative 3.5%

      \[\leadsto \color{blue}{\left(x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) + 1\right)} - 1 \]

   4. associate--l+ 31.0%

      \[\leadsto \color{blue}{x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) + \left(1 - 1\right)} \]

   5. metadata-eval 31.0%

      \[\leadsto x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) + \color{blue}{0} \]

   6. metadata-eval 31.0%

      \[\leadsto x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) + \color{blue}{\left(-0\right)} \]

   7. sub-neg 31.0%

      \[\leadsto \color{blue}{x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) - 0} \]

   8. --rgt-identity 31.0%

      \[\leadsto \color{blue}{x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right)} \]

   9. *-commutative 31.0%

      \[\leadsto \color{blue}{\left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) \cdot x} \]

9. Simplified 31.0%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.2, {x}^{4}, 2\right)\right) + 0.6666666666666666 \cdot {x}^{2}\right) \cdot \left(x \cdot {\pi}^{-0.5}\right)} \]

10. Taylor expanded in x around 0 31.1%

    \[\leadsto \color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)} \]
11. Step-by-step derivation

    1. associate-*r* 31.1%

       \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} \]

    2. *-commutative 31.1%

       \[\leadsto \color{blue}{\left(x \cdot 2\right)} \cdot \sqrt{\frac{1}{\pi}} \]

    3. associate-*l* 31.1%

       \[\leadsto \color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{\pi}}\right)} \]

    4. *-commutative 31.1%

       \[\leadsto x \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot 2\right)} \]

12. Simplified 31.1%

    \[\leadsto \color{blue}{x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 2\right)} \]
13. Add Preprocessing

Alternative 9: 66.8% accurate, 17.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 2 \cdot \frac{x\_m}{\sqrt{\pi}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* 2.0 (/ x_m (sqrt PI))))

C

x_m = fabs(x);
double code(double x_m) {
	return 2.0 * (x_m / sqrt(((double) M_PI)));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 2.0 * (x_m / Math.sqrt(Math.PI));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 2.0 * (x_m / math.sqrt(math.pi))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(2.0 * Float64(x_m / sqrt(pi)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 2.0 * (x_m / sqrt(pi));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(2.0 * N[(x$95$m / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 99.8%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|} \]
6. Step-by-step derivation

   1. expm1-log1p-u 99.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)\right)} \]

   2. expm1-undefine 40.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)} - 1} \]

7. Applied egg-rr 3.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right)\right)} - 1} \]
8. Step-by-step derivation

   1. log1p-undefine 3.4%

      \[\leadsto e^{\color{blue}{\log \left(1 + x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right)\right)}} - 1 \]

   2. rem-exp-log 3.5%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right)\right)} - 1 \]

   3. +-commutative 3.5%

      \[\leadsto \color{blue}{\left(x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) + 1\right)} - 1 \]

   4. associate--l+ 31.0%

      \[\leadsto \color{blue}{x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) + \left(1 - 1\right)} \]

   5. metadata-eval 31.0%

      \[\leadsto x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) + \color{blue}{0} \]

   6. metadata-eval 31.0%

      \[\leadsto x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) + \color{blue}{\left(-0\right)} \]

   7. sub-neg 31.0%

      \[\leadsto \color{blue}{x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) - 0} \]

   8. --rgt-identity 31.0%

      \[\leadsto \color{blue}{x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right)} \]

   9. *-commutative 31.0%

      \[\leadsto \color{blue}{\left({\pi}^{-0.5} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)\right)\right) \cdot x} \]

9. Simplified 31.0%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.2, {x}^{4}, 2\right)\right) + 0.6666666666666666 \cdot {x}^{2}\right) \cdot \left(x \cdot {\pi}^{-0.5}\right)} \]

10. Taylor expanded in x around 0 31.1%

    \[\leadsto \color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)} \]
11. Step-by-step derivation

    1. associate-*r* 31.1%

       \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} \]

    2. *-commutative 31.1%

       \[\leadsto \color{blue}{\left(x \cdot 2\right)} \cdot \sqrt{\frac{1}{\pi}} \]

    3. associate-*l* 31.1%

       \[\leadsto \color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{\pi}}\right)} \]

    4. *-commutative 31.1%

       \[\leadsto x \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot 2\right)} \]

12. Simplified 31.1%

    \[\leadsto \color{blue}{x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 2\right)} \]
13. Step-by-step derivation

    1. add-sqr-sqrt 29.4%

       \[\leadsto \color{blue}{\sqrt{x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 2\right)} \cdot \sqrt{x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 2\right)}} \]

    2. pow2 29.4%

       \[\leadsto \color{blue}{{\left(\sqrt{x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 2\right)}\right)}^{2}} \]

14. Applied egg-rr 29.4%

    \[\leadsto \color{blue}{{\left(\sqrt{x \cdot \frac{2}{\sqrt{\pi}}}\right)}^{2}} \]
15. Step-by-step derivation

    1. unpow2 29.4%

       \[\leadsto \color{blue}{\sqrt{x \cdot \frac{2}{\sqrt{\pi}}} \cdot \sqrt{x \cdot \frac{2}{\sqrt{\pi}}}} \]

    2. rem-square-sqrt 31.1%

       \[\leadsto \color{blue}{x \cdot \frac{2}{\sqrt{\pi}}} \]

    3. associate-*r/ 30.9%

       \[\leadsto \color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}} \]

    4. associate-*l/ 30.9%

       \[\leadsto \color{blue}{\frac{x}{\sqrt{\pi}} \cdot 2} \]

    5. *-commutative 30.9%

       \[\leadsto \color{blue}{2 \cdot \frac{x}{\sqrt{\pi}}} \]

16. Simplified 30.9%

    \[\leadsto \color{blue}{2 \cdot \frac{x}{\sqrt{\pi}}} \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024103 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  :precision binary64
  :pre (<= x 0.5)
  (fabs (* (/ 1.0 (sqrt PI)) (+ (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1.0 5.0) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1.0 21.0) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))))