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

Percentage Accurate: 99.8% → 99.9%

Time: 10.7s

Alternatives: 7

Speedup: 4.4×

• 28.0% 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, 4.4× speedup

Math

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

FPCore

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

C

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (x_m * Math.pow(Math.PI, -0.5)) * (2.0 + (((0.047619047619047616 * Math.pow(x_m, 6.0)) + (0.2 * Math.pow(x_m, 4.0))) + (0.6666666666666666 * Math.pow(x_m, 2.0))));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (x_m * math.pow(math.pi, -0.5)) * (2.0 + (((0.047619047619047616 * math.pow(x_m, 6.0)) + (0.2 * math.pow(x_m, 4.0))) + (0.6666666666666666 * math.pow(x_m, 2.0))))

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (x_m * (pi ^ -0.5)) * (2.0 + (((0.047619047619047616 * (x_m ^ 6.0)) + (0.2 * (x_m ^ 4.0))) + (0.6666666666666666 * (x_m ^ 2.0))));
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

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 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. inv-pow 99.9%

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

   3. sqrt-pow1 99.9%

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

   4. metadata-eval 99.9%

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

7. Applied egg-rr 99.9%

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

   1. *-lft-identity 99.9%

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

9. Simplified 99.9%

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

    1. expm1-log1p-u 99.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|{\pi}^{-0.5} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)\right)} \]

    2. expm1-undefine 36.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|{\pi}^{-0.5} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)} - 1} \]

11. Applied egg-rr 4.4%

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

    1. sub-neg 4.4%

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

    2. metadata-eval 4.4%

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

    3. +-commutative 4.4%

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

    4. log1p-undefine 4.4%

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

    5. rem-exp-log 4.4%

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

    6. associate-+r+ 36.0%

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

    7. metadata-eval 36.0%

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

    8. remove-double-neg 36.0%

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

    9. sub-neg 36.0%

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

    10. neg-sub0 36.0%

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

13. Simplified 36.0%

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

    1. fma-undefine 36.0%

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

    2. fma-undefine 36.0%

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

    3. associate-+r+ 36.0%

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

15. Applied egg-rr 36.0%

    \[\leadsto \left(x \cdot {\pi}^{-0.5}\right) \cdot \left(2 + \color{blue}{\left(\left(0.047619047619047616 \cdot {x}^{6} + 0.2 \cdot {x}^{4}\right) + 0.6666666666666666 \cdot {x}^{2}\right)}\right) \]
16. Add Preprocessing

Alternative 2: 99.0% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

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 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. inv-pow 99.9%

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

   3. sqrt-pow1 99.9%

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

   4. metadata-eval 99.9%

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

7. Applied egg-rr 99.9%

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

   1. *-lft-identity 99.9%

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

9. Simplified 99.9%

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

    1. expm1-log1p-u 99.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|{\pi}^{-0.5} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)\right)} \]

    2. expm1-undefine 36.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|{\pi}^{-0.5} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)} - 1} \]

11. Applied egg-rr 4.4%

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

    1. sub-neg 4.4%

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

    2. metadata-eval 4.4%

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

    3. +-commutative 4.4%

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

    4. log1p-undefine 4.4%

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

    5. rem-exp-log 4.4%

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

    6. associate-+r+ 36.0%

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

    7. metadata-eval 36.0%

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

    8. remove-double-neg 36.0%

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

    9. sub-neg 36.0%

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

    10. neg-sub0 36.0%

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

13. Simplified 36.0%

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

14. Taylor expanded in x around inf 36.0%

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

Alternative 3: 98.9% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.abs(x_m) * Math.abs(((2.0 + (0.047619047619047616 * Math.pow(x_m, 6.0))) / Math.sqrt(Math.PI)));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.fabs(x_m) * math.fabs(((2.0 + (0.047619047619047616 * math.pow(x_m, 6.0))) / math.sqrt(math.pi)))

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = abs(x_m) * abs(((2.0 + (0.047619047619047616 * (x_m ^ 6.0))) / sqrt(pi)));
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

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 inf 99.1%

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

6. Taylor expanded in x around 0 98.9%

   \[\leadsto \left|x\right| \cdot \left|\frac{0.047619047619047616 \cdot {x}^{6} + \color{blue}{2}}{\sqrt{\pi}}\right| \]

7. Final simplification 98.9%

   \[\leadsto \left|x\right| \cdot \left|\frac{2 + 0.047619047619047616 \cdot {x}^{6}}{\sqrt{\pi}}\right| \]
8. Add Preprocessing

Alternative 4: 98.8% accurate, 5.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 0.05:\\ \;\;\;\;x\_m \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({x\_m}^{6} \cdot \frac{x\_m}{\sqrt{\pi}}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.05)
   (* x_m (/ 2.0 (sqrt PI)))
   (* 0.047619047619047616 (* (pow x_m 6.0) (/ x_m (sqrt PI))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.05) {
		tmp = x_m * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = 0.047619047619047616 * (pow(x_m, 6.0) * (x_m / sqrt(((double) M_PI))));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.abs(x_m) <= 0.05) {
		tmp = x_m * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = 0.047619047619047616 * (Math.pow(x_m, 6.0) * (x_m / Math.sqrt(Math.PI)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 0.05:
		tmp = x_m * (2.0 / math.sqrt(math.pi))
	else:
		tmp = 0.047619047619047616 * (math.pow(x_m, 6.0) * (x_m / math.sqrt(math.pi)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.05)
		tmp = Float64(x_m * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(0.047619047619047616 * Float64((x_m ^ 6.0) * Float64(x_m / sqrt(pi))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (abs(x_m) <= 0.05)
		tmp = x_m * (2.0 / sqrt(pi));
	else
		tmp = 0.047619047619047616 * ((x_m ^ 6.0) * (x_m / sqrt(pi)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.05], N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[x$95$m, 6.0], $MachinePrecision] * N[(x$95$m / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.050000000000000003

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.3%

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

      1. add-cbrt-cube 39.5%

         \[\leadsto \left|2 \cdot \color{blue}{\sqrt[3]{\left(\left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)}}\right| \]

      2. pow3 39.5%

         \[\leadsto \left|2 \cdot \sqrt[3]{\color{blue}{{\left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)}^{3}}}\right| \]

      3. inv-pow 39.5%

         \[\leadsto \left|2 \cdot \sqrt[3]{{\left(\sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left|x\right|\right)}^{3}}\right| \]

      4. sqrt-pow1 39.5%

         \[\leadsto \left|2 \cdot \sqrt[3]{{\left(\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left|x\right|\right)}^{3}}\right| \]

      5. metadata-eval 39.5%

         \[\leadsto \left|2 \cdot \sqrt[3]{{\left({\pi}^{\color{blue}{-0.5}} \cdot \left|x\right|\right)}^{3}}\right| \]

   7. Applied egg-rr 39.5%

      \[\leadsto \left|2 \cdot \color{blue}{\sqrt[3]{{\left({\pi}^{-0.5} \cdot \left|x\right|\right)}^{3}}}\right| \]
   8. Step-by-step derivation

      1. rem-cbrt-cube 99.3%

         \[\leadsto \left|2 \cdot \color{blue}{\left({\pi}^{-0.5} \cdot \left|x\right|\right)}\right| \]

      2. fabs-mul 99.3%

         \[\leadsto \color{blue}{\left|2\right| \cdot \left|{\pi}^{-0.5} \cdot \left|x\right|\right|} \]

      3. metadata-eval 99.3%

         \[\leadsto \color{blue}{2} \cdot \left|{\pi}^{-0.5} \cdot \left|x\right|\right| \]

      4. *-commutative 99.3%

         \[\leadsto \color{blue}{\left|{\pi}^{-0.5} \cdot \left|x\right|\right| \cdot 2} \]

      5. fabs-mul 99.3%

         \[\leadsto \color{blue}{\left(\left|{\pi}^{-0.5}\right| \cdot \left|\left|x\right|\right|\right)} \cdot 2 \]

      6. sqr-pow 99.3%

         \[\leadsto \left(\left|\color{blue}{{\pi}^{\left(\frac{-0.5}{2}\right)} \cdot {\pi}^{\left(\frac{-0.5}{2}\right)}}\right| \cdot \left|\left|x\right|\right|\right) \cdot 2 \]

      7. fabs-sqr 99.3%

         \[\leadsto \left(\color{blue}{\left({\pi}^{\left(\frac{-0.5}{2}\right)} \cdot {\pi}^{\left(\frac{-0.5}{2}\right)}\right)} \cdot \left|\left|x\right|\right|\right) \cdot 2 \]

      8. sqr-pow 99.3%

         \[\leadsto \left(\color{blue}{{\pi}^{-0.5}} \cdot \left|\left|x\right|\right|\right) \cdot 2 \]

      9. fabs-fabs 99.3%

         \[\leadsto \left({\pi}^{-0.5} \cdot \color{blue}{\left|x\right|}\right) \cdot 2 \]

      10. add-sqr-sqrt 49.9%

          \[\leadsto \left({\pi}^{-0.5} \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right) \cdot 2 \]

      11. fabs-sqr 49.9%

          \[\leadsto \left({\pi}^{-0.5} \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) \cdot 2 \]

      12. add-sqr-sqrt 52.0%

          \[\leadsto \left({\pi}^{-0.5} \cdot \color{blue}{x}\right) \cdot 2 \]

   9. Applied egg-rr 52.0%

      \[\leadsto \color{blue}{\left({\pi}^{-0.5} \cdot x\right) \cdot 2} \]
   10. Step-by-step derivation

       1. pow1 52.0%

          \[\leadsto \color{blue}{{\left(\left({\pi}^{-0.5} \cdot x\right) \cdot 2\right)}^{1}} \]

       2. *-commutative 52.0%

          \[\leadsto {\left(\color{blue}{\left(x \cdot {\pi}^{-0.5}\right)} \cdot 2\right)}^{1} \]

       3. metadata-eval 52.0%

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

       4. sqrt-pow1 52.0%

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

       5. inv-pow 52.0%

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

       6. sqrt-div 52.0%

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

       7. metadata-eval 52.0%

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

       8. un-div-inv 51.6%

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

   11. Applied egg-rr 51.6%

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

       1. unpow1 51.6%

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

       2. associate-*l/ 51.6%

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

       3. associate-/l* 52.0%

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

   13. Simplified 52.0%

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

   if 0.050000000000000003 < (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 99.8%

      \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.1%

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

      1. add-sqr-sqrt 0.0%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\left({\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      2. unpow-prod-down 0.0%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\left(\color{blue}{\left({\left(\sqrt{x}\right)}^{6} \cdot {\left(\sqrt{x}\right)}^{6}\right)} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

   7. Applied egg-rr 0.0%

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

      1. pow-sqr 0.0%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\left(\color{blue}{{\left(\sqrt{x}\right)}^{\left(2 \cdot 6\right)}} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      2. metadata-eval 0.0%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\left({\left(\sqrt{x}\right)}^{\color{blue}{12}} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

   9. Simplified 0.0%

      \[\leadsto \left|0.047619047619047616 \cdot \left(\left(\color{blue}{{\left(\sqrt{x}\right)}^{12}} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \left|\color{blue}{\sqrt{0.047619047619047616 \cdot \left(\left({\left(\sqrt{x}\right)}^{12} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)} \cdot \sqrt{0.047619047619047616 \cdot \left(\left({\left(\sqrt{x}\right)}^{12} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)}}\right| \]

       2. fabs-sqr 0.0%

          \[\leadsto \color{blue}{\sqrt{0.047619047619047616 \cdot \left(\left({\left(\sqrt{x}\right)}^{12} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)} \cdot \sqrt{0.047619047619047616 \cdot \left(\left({\left(\sqrt{x}\right)}^{12} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)}} \]

       3. add-sqr-sqrt 0.0%

          \[\leadsto \color{blue}{0.047619047619047616 \cdot \left(\left({\left(\sqrt{x}\right)}^{12} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)} \]

       4. *-commutative 0.0%

          \[\leadsto \color{blue}{\left(\left({\left(\sqrt{x}\right)}^{12} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right) \cdot 0.047619047619047616} \]

   11. Applied egg-rr 0.1%

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

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.05:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({x}^{6} \cdot \frac{x}{\sqrt{\pi}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.8% accurate, 5.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 0.05:\\ \;\;\;\;x\_m \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.05)
   (* x_m (/ 2.0 (sqrt PI)))
   (* (pow x_m 7.0) (/ 0.047619047619047616 (sqrt PI)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.05) {
		tmp = x_m * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = pow(x_m, 7.0) * (0.047619047619047616 / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.abs(x_m) <= 0.05) {
		tmp = x_m * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.pow(x_m, 7.0) * (0.047619047619047616 / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 0.05:
		tmp = x_m * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.pow(x_m, 7.0) * (0.047619047619047616 / math.sqrt(math.pi))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.05)
		tmp = Float64(x_m * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64((x_m ^ 7.0) * Float64(0.047619047619047616 / sqrt(pi)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (abs(x_m) <= 0.05)
		tmp = x_m * (2.0 / sqrt(pi));
	else
		tmp = (x_m ^ 7.0) * (0.047619047619047616 / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.05], N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x$95$m, 7.0], $MachinePrecision] * N[(0.047619047619047616 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.050000000000000003

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.3%

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

      1. add-cbrt-cube 39.5%

         \[\leadsto \left|2 \cdot \color{blue}{\sqrt[3]{\left(\left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)}}\right| \]

      2. pow3 39.5%

         \[\leadsto \left|2 \cdot \sqrt[3]{\color{blue}{{\left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)}^{3}}}\right| \]

      3. inv-pow 39.5%

         \[\leadsto \left|2 \cdot \sqrt[3]{{\left(\sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left|x\right|\right)}^{3}}\right| \]

      4. sqrt-pow1 39.5%

         \[\leadsto \left|2 \cdot \sqrt[3]{{\left(\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left|x\right|\right)}^{3}}\right| \]

      5. metadata-eval 39.5%

         \[\leadsto \left|2 \cdot \sqrt[3]{{\left({\pi}^{\color{blue}{-0.5}} \cdot \left|x\right|\right)}^{3}}\right| \]

   7. Applied egg-rr 39.5%

      \[\leadsto \left|2 \cdot \color{blue}{\sqrt[3]{{\left({\pi}^{-0.5} \cdot \left|x\right|\right)}^{3}}}\right| \]
   8. Step-by-step derivation

      1. rem-cbrt-cube 99.3%

         \[\leadsto \left|2 \cdot \color{blue}{\left({\pi}^{-0.5} \cdot \left|x\right|\right)}\right| \]

      2. fabs-mul 99.3%

         \[\leadsto \color{blue}{\left|2\right| \cdot \left|{\pi}^{-0.5} \cdot \left|x\right|\right|} \]

      3. metadata-eval 99.3%

         \[\leadsto \color{blue}{2} \cdot \left|{\pi}^{-0.5} \cdot \left|x\right|\right| \]

      4. *-commutative 99.3%

         \[\leadsto \color{blue}{\left|{\pi}^{-0.5} \cdot \left|x\right|\right| \cdot 2} \]

      5. fabs-mul 99.3%

         \[\leadsto \color{blue}{\left(\left|{\pi}^{-0.5}\right| \cdot \left|\left|x\right|\right|\right)} \cdot 2 \]

      6. sqr-pow 99.3%

         \[\leadsto \left(\left|\color{blue}{{\pi}^{\left(\frac{-0.5}{2}\right)} \cdot {\pi}^{\left(\frac{-0.5}{2}\right)}}\right| \cdot \left|\left|x\right|\right|\right) \cdot 2 \]

      7. fabs-sqr 99.3%

         \[\leadsto \left(\color{blue}{\left({\pi}^{\left(\frac{-0.5}{2}\right)} \cdot {\pi}^{\left(\frac{-0.5}{2}\right)}\right)} \cdot \left|\left|x\right|\right|\right) \cdot 2 \]

      8. sqr-pow 99.3%

         \[\leadsto \left(\color{blue}{{\pi}^{-0.5}} \cdot \left|\left|x\right|\right|\right) \cdot 2 \]

      9. fabs-fabs 99.3%

         \[\leadsto \left({\pi}^{-0.5} \cdot \color{blue}{\left|x\right|}\right) \cdot 2 \]

      10. add-sqr-sqrt 49.9%

          \[\leadsto \left({\pi}^{-0.5} \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right) \cdot 2 \]

      11. fabs-sqr 49.9%

          \[\leadsto \left({\pi}^{-0.5} \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) \cdot 2 \]

      12. add-sqr-sqrt 52.0%

          \[\leadsto \left({\pi}^{-0.5} \cdot \color{blue}{x}\right) \cdot 2 \]

   9. Applied egg-rr 52.0%

      \[\leadsto \color{blue}{\left({\pi}^{-0.5} \cdot x\right) \cdot 2} \]
   10. Step-by-step derivation

       1. pow1 52.0%

          \[\leadsto \color{blue}{{\left(\left({\pi}^{-0.5} \cdot x\right) \cdot 2\right)}^{1}} \]

       2. *-commutative 52.0%

          \[\leadsto {\left(\color{blue}{\left(x \cdot {\pi}^{-0.5}\right)} \cdot 2\right)}^{1} \]

       3. metadata-eval 52.0%

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

       4. sqrt-pow1 52.0%

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

       5. inv-pow 52.0%

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

       6. sqrt-div 52.0%

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

       7. metadata-eval 52.0%

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

       8. un-div-inv 51.6%

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

   11. Applied egg-rr 51.6%

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

       1. unpow1 51.6%

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

       2. associate-*l/ 51.6%

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

       3. associate-/l* 52.0%

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

   13. Simplified 52.0%

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

   if 0.050000000000000003 < (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 99.8%

      \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.1%

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

      1. add-sqr-sqrt 0.0%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\left({\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      2. unpow-prod-down 0.0%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\left(\color{blue}{\left({\left(\sqrt{x}\right)}^{6} \cdot {\left(\sqrt{x}\right)}^{6}\right)} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

   7. Applied egg-rr 0.0%

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

      1. pow-sqr 0.0%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\left(\color{blue}{{\left(\sqrt{x}\right)}^{\left(2 \cdot 6\right)}} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      2. metadata-eval 0.0%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\left({\left(\sqrt{x}\right)}^{\color{blue}{12}} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

   9. Simplified 0.0%

      \[\leadsto \left|0.047619047619047616 \cdot \left(\left(\color{blue}{{\left(\sqrt{x}\right)}^{12}} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \left|\color{blue}{\sqrt{0.047619047619047616 \cdot \left(\left({\left(\sqrt{x}\right)}^{12} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)} \cdot \sqrt{0.047619047619047616 \cdot \left(\left({\left(\sqrt{x}\right)}^{12} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)}}\right| \]

       2. fabs-sqr 0.0%

          \[\leadsto \color{blue}{\sqrt{0.047619047619047616 \cdot \left(\left({\left(\sqrt{x}\right)}^{12} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)} \cdot \sqrt{0.047619047619047616 \cdot \left(\left({\left(\sqrt{x}\right)}^{12} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)}} \]

       3. add-sqr-sqrt 0.0%

          \[\leadsto \color{blue}{0.047619047619047616 \cdot \left(\left({\left(\sqrt{x}\right)}^{12} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)} \]

       4. expm1-log1p-u 0.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \left(\left({\left(\sqrt{x}\right)}^{12} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)} \]

       5. expm1-undefine 0.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left(\left({\left(\sqrt{x}\right)}^{12} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)} - 1} \]

   11. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{6}\right) \cdot \frac{x}{\sqrt{\pi}}\right)} - 1} \]
   12. Step-by-step derivation

       1. sub-neg 0.0%

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

       2. metadata-eval 0.0%

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

       3. +-commutative 0.0%

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

       4. log1p-undefine 0.0%

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

       5. rem-exp-log 0.1%

          \[\leadsto -1 + \color{blue}{\left(1 + \left(0.047619047619047616 \cdot {x}^{6}\right) \cdot \frac{x}{\sqrt{\pi}}\right)} \]

       6. associate-+r+ 0.1%

          \[\leadsto \color{blue}{\left(-1 + 1\right) + \left(0.047619047619047616 \cdot {x}^{6}\right) \cdot \frac{x}{\sqrt{\pi}}} \]

       7. metadata-eval 0.1%

          \[\leadsto \color{blue}{0} + \left(0.047619047619047616 \cdot {x}^{6}\right) \cdot \frac{x}{\sqrt{\pi}} \]

       8. +-lft-identity 0.1%

          \[\leadsto \color{blue}{\left(0.047619047619047616 \cdot {x}^{6}\right) \cdot \frac{x}{\sqrt{\pi}}} \]

       9. associate-*l* 0.1%

          \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{6} \cdot \frac{x}{\sqrt{\pi}}\right)} \]

       10. associate-*r/ 0.1%

           \[\leadsto 0.047619047619047616 \cdot \color{blue}{\frac{{x}^{6} \cdot x}{\sqrt{\pi}}} \]

       11. pow-plus 0.1%

           \[\leadsto 0.047619047619047616 \cdot \frac{\color{blue}{{x}^{\left(6 + 1\right)}}}{\sqrt{\pi}} \]

       12. metadata-eval 0.1%

           \[\leadsto 0.047619047619047616 \cdot \frac{{x}^{\color{blue}{7}}}{\sqrt{\pi}} \]

       13. associate-*r/ 0.1%

           \[\leadsto \color{blue}{\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}} \]

       14. *-commutative 0.1%

           \[\leadsto \frac{\color{blue}{{x}^{7} \cdot 0.047619047619047616}}{\sqrt{\pi}} \]

       15. associate-/l* 0.1%

           \[\leadsto \color{blue}{{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}} \]

   13. Simplified 0.1%

       \[\leadsto \color{blue}{{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.9% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (x_m * Math.pow(Math.PI, -0.5)) * (2.0 + (0.047619047619047616 * Math.pow(x_m, 6.0)));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (x_m * math.pow(math.pi, -0.5)) * (2.0 + (0.047619047619047616 * math.pow(x_m, 6.0)))

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (x_m * (pi ^ -0.5)) * (2.0 + (0.047619047619047616 * (x_m ^ 6.0)));
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

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 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. inv-pow 99.9%

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

   3. sqrt-pow1 99.9%

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

   4. metadata-eval 99.9%

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

7. Applied egg-rr 99.9%

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

   1. *-lft-identity 99.9%

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

9. Simplified 99.9%

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

    1. expm1-log1p-u 99.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|{\pi}^{-0.5} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)\right)} \]

    2. expm1-undefine 36.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|{\pi}^{-0.5} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)} - 1} \]

11. Applied egg-rr 4.4%

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

    1. sub-neg 4.4%

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

    2. metadata-eval 4.4%

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

    3. +-commutative 4.4%

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

    4. log1p-undefine 4.4%

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

    5. rem-exp-log 4.4%

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

    6. associate-+r+ 36.0%

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

    7. metadata-eval 36.0%

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

    8. remove-double-neg 36.0%

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

    9. sub-neg 36.0%

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

    10. neg-sub0 36.0%

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

13. Simplified 36.0%

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

14. Taylor expanded in x around inf 36.0%

    \[\leadsto \left(x \cdot {\pi}^{-0.5}\right) \cdot \left(2 + \color{blue}{0.047619047619047616 \cdot {x}^{6}}\right) \]
15. Add Preprocessing

Alternative 7: 68.0% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 70.6%

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

   1. add-cbrt-cube 48.5%

      \[\leadsto \left|2 \cdot \color{blue}{\sqrt[3]{\left(\left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)}}\right| \]

   2. pow3 48.5%

      \[\leadsto \left|2 \cdot \sqrt[3]{\color{blue}{{\left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)}^{3}}}\right| \]

   3. inv-pow 48.5%

      \[\leadsto \left|2 \cdot \sqrt[3]{{\left(\sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left|x\right|\right)}^{3}}\right| \]

   4. sqrt-pow1 48.5%

      \[\leadsto \left|2 \cdot \sqrt[3]{{\left(\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left|x\right|\right)}^{3}}\right| \]

   5. metadata-eval 48.5%

      \[\leadsto \left|2 \cdot \sqrt[3]{{\left({\pi}^{\color{blue}{-0.5}} \cdot \left|x\right|\right)}^{3}}\right| \]

7. Applied egg-rr 48.5%

   \[\leadsto \left|2 \cdot \color{blue}{\sqrt[3]{{\left({\pi}^{-0.5} \cdot \left|x\right|\right)}^{3}}}\right| \]
8. Step-by-step derivation

   1. rem-cbrt-cube 70.6%

      \[\leadsto \left|2 \cdot \color{blue}{\left({\pi}^{-0.5} \cdot \left|x\right|\right)}\right| \]

   2. fabs-mul 70.6%

      \[\leadsto \color{blue}{\left|2\right| \cdot \left|{\pi}^{-0.5} \cdot \left|x\right|\right|} \]

   3. metadata-eval 70.6%

      \[\leadsto \color{blue}{2} \cdot \left|{\pi}^{-0.5} \cdot \left|x\right|\right| \]

   4. *-commutative 70.6%

      \[\leadsto \color{blue}{\left|{\pi}^{-0.5} \cdot \left|x\right|\right| \cdot 2} \]

   5. fabs-mul 70.6%

      \[\leadsto \color{blue}{\left(\left|{\pi}^{-0.5}\right| \cdot \left|\left|x\right|\right|\right)} \cdot 2 \]

   6. sqr-pow 70.6%

      \[\leadsto \left(\left|\color{blue}{{\pi}^{\left(\frac{-0.5}{2}\right)} \cdot {\pi}^{\left(\frac{-0.5}{2}\right)}}\right| \cdot \left|\left|x\right|\right|\right) \cdot 2 \]

   7. fabs-sqr 70.6%

      \[\leadsto \left(\color{blue}{\left({\pi}^{\left(\frac{-0.5}{2}\right)} \cdot {\pi}^{\left(\frac{-0.5}{2}\right)}\right)} \cdot \left|\left|x\right|\right|\right) \cdot 2 \]

   8. sqr-pow 70.6%

      \[\leadsto \left(\color{blue}{{\pi}^{-0.5}} \cdot \left|\left|x\right|\right|\right) \cdot 2 \]

   9. fabs-fabs 70.6%

      \[\leadsto \left({\pi}^{-0.5} \cdot \color{blue}{\left|x\right|}\right) \cdot 2 \]

   10. add-sqr-sqrt 34.5%

       \[\leadsto \left({\pi}^{-0.5} \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right) \cdot 2 \]

   11. fabs-sqr 34.5%

       \[\leadsto \left({\pi}^{-0.5} \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) \cdot 2 \]

   12. add-sqr-sqrt 36.0%

       \[\leadsto \left({\pi}^{-0.5} \cdot \color{blue}{x}\right) \cdot 2 \]

9. Applied egg-rr 36.0%

   \[\leadsto \color{blue}{\left({\pi}^{-0.5} \cdot x\right) \cdot 2} \]
10. Step-by-step derivation

    1. pow1 36.0%

       \[\leadsto \color{blue}{{\left(\left({\pi}^{-0.5} \cdot x\right) \cdot 2\right)}^{1}} \]

    2. *-commutative 36.0%

       \[\leadsto {\left(\color{blue}{\left(x \cdot {\pi}^{-0.5}\right)} \cdot 2\right)}^{1} \]

    3. metadata-eval 36.0%

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

    4. sqrt-pow1 36.0%

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

    5. inv-pow 36.0%

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

    6. sqrt-div 36.0%

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

    7. metadata-eval 36.0%

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

    8. un-div-inv 35.8%

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

11. Applied egg-rr 35.8%

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

    1. unpow1 35.8%

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

    2. associate-*l/ 35.8%

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

    3. associate-/l* 36.0%

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

13. Simplified 36.0%

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

Reproduce

herbie shell --seed 2024182 
(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)))))))