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

Percentage Accurate: 99.8% → 99.9%

Time: 12.3s

Alternatives: 10

Speedup: 4.4×

• 29.2% 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, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $MachinePrecision] + N[(0.2 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * N[Power[Pi, -0.5], $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 99.4%

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

6. Applied egg-rr 36.4%

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

   1. *-commutative 36.4%

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

   2. associate-*l* 36.4%

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

   3. +-commutative 36.4%

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

8. Simplified 36.4%

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

   1. fma-udef 36.4%

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

10. Applied egg-rr 36.4%

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

11. Taylor expanded in x around 0 36.4%

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

    1. fma-def 36.4%

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

13. Simplified 36.4%

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

14. Final simplification 36.4%

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

Alternative 2: 99.9% accurate, 4.4× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (* x_m (pow PI -0.5))
  (+
   (+ 2.0 (* 0.6666666666666666 (pow x_m 2.0)))
   (+ (* 0.2 (pow x_m 4.0)) (* 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 + (0.6666666666666666 * pow(x_m, 2.0))) + ((0.2 * pow(x_m, 4.0)) + (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 + (0.6666666666666666 * Math.pow(x_m, 2.0))) + ((0.2 * Math.pow(x_m, 4.0)) + (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 + (0.6666666666666666 * math.pow(x_m, 2.0))) + ((0.2 * math.pow(x_m, 4.0)) + (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(Float64(2.0 + Float64(0.6666666666666666 * (x_m ^ 2.0))) + Float64(Float64(0.2 * (x_m ^ 4.0)) + 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 + (0.6666666666666666 * (x_m ^ 2.0))) + ((0.2 * (x_m ^ 4.0)) + (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[(N[(2.0 + N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.2 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $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 99.4%

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

6. Applied egg-rr 36.4%

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

   1. *-commutative 36.4%

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

   2. associate-*l* 36.4%

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

   3. +-commutative 36.4%

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

8. Simplified 36.4%

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

   1. fma-udef 36.4%

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

10. Applied egg-rr 36.4%

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

    1. fma-udef 36.4%

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

12. Applied egg-rr 36.4%

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

13. Final simplification 36.4%

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

Alternative 3: 99.0% accurate, 4.5× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (* x_m (pow PI -0.5))
  (+ 2.0 (fma 0.2 (pow x_m 4.0) (* 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 + fma(0.2, pow(x_m, 4.0), (0.047619047619047616 * 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 + fma(0.2, (x_m ^ 4.0), Float64(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[(0.2 * N[Power[x$95$m, 4.0], $MachinePrecision] + N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $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 99.4%

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

6. Applied egg-rr 36.4%

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

   1. *-commutative 36.4%

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

   2. associate-*l* 36.4%

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

   3. +-commutative 36.4%

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

8. Simplified 36.4%

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

9. Taylor expanded in x around 0 36.2%

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

10. Final simplification 36.2%

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

Alternative 4: 99.4% accurate, 5.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 2.15:\\ \;\;\;\;\left(x_m \cdot {\pi}^{-0.5}\right) \cdot \left(0.2 \cdot {x_m}^{4} + \left(2 + 0.6666666666666666 \cdot {x_m}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(0.2 \cdot {x_m}^{5} + 0.047619047619047616 \cdot {x_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.15)
   (*
    (* x_m (pow PI -0.5))
    (+ (* 0.2 (pow x_m 4.0)) (+ 2.0 (* 0.6666666666666666 (pow x_m 2.0)))))
   (*
    (pow PI -0.5)
    (+ (* 0.2 (pow x_m 5.0)) (* 0.047619047619047616 (pow x_m 7.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.15) {
		tmp = (x_m * pow(((double) M_PI), -0.5)) * ((0.2 * pow(x_m, 4.0)) + (2.0 + (0.6666666666666666 * pow(x_m, 2.0))));
	} else {
		tmp = pow(((double) M_PI), -0.5) * ((0.2 * pow(x_m, 5.0)) + (0.047619047619047616 * pow(x_m, 7.0)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.15) {
		tmp = (x_m * Math.pow(Math.PI, -0.5)) * ((0.2 * Math.pow(x_m, 4.0)) + (2.0 + (0.6666666666666666 * Math.pow(x_m, 2.0))));
	} else {
		tmp = Math.pow(Math.PI, -0.5) * ((0.2 * Math.pow(x_m, 5.0)) + (0.047619047619047616 * Math.pow(x_m, 7.0)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.15:
		tmp = (x_m * math.pow(math.pi, -0.5)) * ((0.2 * math.pow(x_m, 4.0)) + (2.0 + (0.6666666666666666 * math.pow(x_m, 2.0))))
	else:
		tmp = math.pow(math.pi, -0.5) * ((0.2 * math.pow(x_m, 5.0)) + (0.047619047619047616 * math.pow(x_m, 7.0)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.15)
		tmp = Float64(Float64(x_m * (pi ^ -0.5)) * Float64(Float64(0.2 * (x_m ^ 4.0)) + Float64(2.0 + Float64(0.6666666666666666 * (x_m ^ 2.0)))));
	else
		tmp = Float64((pi ^ -0.5) * Float64(Float64(0.2 * (x_m ^ 5.0)) + Float64(0.047619047619047616 * (x_m ^ 7.0))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.15)
		tmp = (x_m * (pi ^ -0.5)) * ((0.2 * (x_m ^ 4.0)) + (2.0 + (0.6666666666666666 * (x_m ^ 2.0))));
	else
		tmp = (pi ^ -0.5) * ((0.2 * (x_m ^ 5.0)) + (0.047619047619047616 * (x_m ^ 7.0)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.15], N[(N[(x$95$m * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(0.2 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(N[(0.2 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.14999999999999991

   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 99.4%

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

   6. Applied egg-rr 36.4%

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

      1. *-commutative 36.4%

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

      2. associate-*l* 36.4%

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

      3. +-commutative 36.4%

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

   8. Simplified 36.4%

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

      1. fma-udef 36.4%

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

   10. Applied egg-rr 36.4%

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

   11. Taylor expanded in x around 0 36.4%

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

   if 2.14999999999999991 < x

   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 99.4%

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

   6. Applied egg-rr 36.4%

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

   7. Taylor expanded in x around inf 3.9%

      \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.2 \cdot \left({x}^{5} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 3.9%

         \[\leadsto \color{blue}{0.2 \cdot \left({x}^{5} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)} \]

      2. associate-*r* 3.9%

         \[\leadsto \color{blue}{\left(0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}}} + 0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right) \]

      3. associate-*r* 3.9%

         \[\leadsto \left(0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}} \]

      4. distribute-rgt-out 3.9%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)} \]

      5. unpow-1 3.9%

         \[\leadsto \sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right) \]

      6. metadata-eval 3.9%

         \[\leadsto \sqrt{{\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right) \]

      7. pow-sqr 3.9%

         \[\leadsto \sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right) \]

      8. rem-sqrt-square 3.9%

         \[\leadsto \color{blue}{\left|{\pi}^{-0.5}\right|} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right) \]

      9. rem-cube-cbrt 3.9%

         \[\leadsto \left|\color{blue}{{\left(\sqrt[3]{{\pi}^{-0.5}}\right)}^{3}}\right| \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right) \]

      10. sqr-pow 3.9%

          \[\leadsto \left|\color{blue}{{\left(\sqrt[3]{{\pi}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{\pi}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)}}\right| \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right) \]

      11. fabs-sqr 3.9%

          \[\leadsto \color{blue}{\left({\left(\sqrt[3]{{\pi}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{\pi}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)}\right)} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right) \]

      12. sqr-pow 3.9%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{{\pi}^{-0.5}}\right)}^{3}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right) \]

      13. rem-cube-cbrt 3.9%

          \[\leadsto \color{blue}{{\pi}^{-0.5}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right) \]

   9. Simplified 3.9%

      \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.15:\\ \;\;\;\;\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(0.2 \cdot {x}^{4} + \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 5.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1.85:\\ \;\;\;\;x_m \cdot \left({\pi}^{-0.5} \cdot \mathsf{fma}\left(0.6666666666666666, {x_m}^{2}, 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(0.2 \cdot {x_m}^{5} + 0.047619047619047616 \cdot {x_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.85)
   (* x_m (* (pow PI -0.5) (fma 0.6666666666666666 (pow x_m 2.0) 2.0)))
   (*
    (pow PI -0.5)
    (+ (* 0.2 (pow x_m 5.0)) (* 0.047619047619047616 (pow x_m 7.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = x_m * (pow(((double) M_PI), -0.5) * fma(0.6666666666666666, pow(x_m, 2.0), 2.0));
	} else {
		tmp = pow(((double) M_PI), -0.5) * ((0.2 * pow(x_m, 5.0)) + (0.047619047619047616 * pow(x_m, 7.0)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.85)
		tmp = Float64(x_m * Float64((pi ^ -0.5) * fma(0.6666666666666666, (x_m ^ 2.0), 2.0)));
	else
		tmp = Float64((pi ^ -0.5) * Float64(Float64(0.2 * (x_m ^ 5.0)) + Float64(0.047619047619047616 * (x_m ^ 7.0))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.85], N[(x$95$m * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(N[(0.2 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   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 99.4%

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

   6. Applied egg-rr 36.4%

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

   7. Taylor expanded in x around 0 36.4%

      \[\leadsto \color{blue}{0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 36.4%

         \[\leadsto \color{blue}{\left(0.6666666666666666 \cdot {x}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}} + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right) \]

      2. associate-*r* 36.4%

         \[\leadsto \left(0.6666666666666666 \cdot {x}^{3}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} \]

      3. distribute-rgt-out 36.4%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right)} \]

      4. unpow-1 36.4%

         \[\leadsto \sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right) \]

      5. metadata-eval 36.4%

         \[\leadsto \sqrt{{\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right) \]

      6. pow-sqr 36.4%

         \[\leadsto \sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right) \]

      7. rem-sqrt-square 36.4%

         \[\leadsto \color{blue}{\left|{\pi}^{-0.5}\right|} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right) \]

      8. rem-cube-cbrt 35.8%

         \[\leadsto \left|\color{blue}{{\left(\sqrt[3]{{\pi}^{-0.5}}\right)}^{3}}\right| \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right) \]

      9. sqr-pow 35.4%

         \[\leadsto \left|\color{blue}{{\left(\sqrt[3]{{\pi}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{\pi}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)}}\right| \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right) \]

      10. fabs-sqr 35.4%

          \[\leadsto \color{blue}{\left({\left(\sqrt[3]{{\pi}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{\pi}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)}\right)} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right) \]

      11. sqr-pow 35.8%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{{\pi}^{-0.5}}\right)}^{3}} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right) \]

      12. rem-cube-cbrt 36.4%

          \[\leadsto \color{blue}{{\pi}^{-0.5}} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right) \]

      13. unpow3 36.4%

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

      14. unpow2 36.4%

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

      15. associate-*r* 36.4%

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

      16. distribute-rgt-in 36.4%

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

      17. fma-def 36.4%

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

   9. Simplified 36.4%

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

   if 1.8500000000000001 < x

   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 99.4%

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

   6. Applied egg-rr 36.4%

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

   7. Taylor expanded in x around inf 3.9%

      \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.2 \cdot \left({x}^{5} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 3.9%

         \[\leadsto \color{blue}{0.2 \cdot \left({x}^{5} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)} \]

      2. associate-*r* 3.9%

         \[\leadsto \color{blue}{\left(0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}}} + 0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right) \]

      3. associate-*r* 3.9%

         \[\leadsto \left(0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}} \]

      4. distribute-rgt-out 3.9%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)} \]

      5. unpow-1 3.9%

         \[\leadsto \sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right) \]

      6. metadata-eval 3.9%

         \[\leadsto \sqrt{{\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right) \]

      7. pow-sqr 3.9%

         \[\leadsto \sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right) \]

      8. rem-sqrt-square 3.9%

         \[\leadsto \color{blue}{\left|{\pi}^{-0.5}\right|} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right) \]

      9. rem-cube-cbrt 3.9%

         \[\leadsto \left|\color{blue}{{\left(\sqrt[3]{{\pi}^{-0.5}}\right)}^{3}}\right| \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right) \]

      10. sqr-pow 3.9%

          \[\leadsto \left|\color{blue}{{\left(\sqrt[3]{{\pi}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{\pi}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)}}\right| \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right) \]

      11. fabs-sqr 3.9%

          \[\leadsto \color{blue}{\left({\left(\sqrt[3]{{\pi}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{\pi}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)}\right)} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right) \]

      12. sqr-pow 3.9%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{{\pi}^{-0.5}}\right)}^{3}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right) \]

      13. rem-cube-cbrt 3.9%

          \[\leadsto \color{blue}{{\pi}^{-0.5}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right) \]

   9. Simplified 3.9%

      \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \left({\pi}^{-0.5} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 2.15:\\ \;\;\;\;x_m \cdot \left({\pi}^{-0.5} \cdot \mathsf{fma}\left(0.6666666666666666, {x_m}^{2}, 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({x_m}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.15)
   (* x_m (* (pow PI -0.5) (fma 0.6666666666666666 (pow x_m 2.0) 2.0)))
   (* 0.047619047619047616 (* (pow x_m 7.0) (sqrt (/ 1.0 PI))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.15) {
		tmp = x_m * (pow(((double) M_PI), -0.5) * fma(0.6666666666666666, pow(x_m, 2.0), 2.0));
	} else {
		tmp = 0.047619047619047616 * (pow(x_m, 7.0) * sqrt((1.0 / ((double) M_PI))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.15)
		tmp = Float64(x_m * Float64((pi ^ -0.5) * fma(0.6666666666666666, (x_m ^ 2.0), 2.0)));
	else
		tmp = Float64(0.047619047619047616 * Float64((x_m ^ 7.0) * sqrt(Float64(1.0 / pi))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.15], N[(x$95$m * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[x$95$m, 7.0], $MachinePrecision] * N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.14999999999999991

   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 99.4%

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

   6. Applied egg-rr 36.4%

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

   7. Taylor expanded in x around 0 36.4%

      \[\leadsto \color{blue}{0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 36.4%

         \[\leadsto \color{blue}{\left(0.6666666666666666 \cdot {x}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}} + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right) \]

      2. associate-*r* 36.4%

         \[\leadsto \left(0.6666666666666666 \cdot {x}^{3}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} \]

      3. distribute-rgt-out 36.4%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right)} \]

      4. unpow-1 36.4%

         \[\leadsto \sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right) \]

      5. metadata-eval 36.4%

         \[\leadsto \sqrt{{\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right) \]

      6. pow-sqr 36.4%

         \[\leadsto \sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right) \]

      7. rem-sqrt-square 36.4%

         \[\leadsto \color{blue}{\left|{\pi}^{-0.5}\right|} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right) \]

      8. rem-cube-cbrt 35.8%

         \[\leadsto \left|\color{blue}{{\left(\sqrt[3]{{\pi}^{-0.5}}\right)}^{3}}\right| \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right) \]

      9. sqr-pow 35.4%

         \[\leadsto \left|\color{blue}{{\left(\sqrt[3]{{\pi}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{\pi}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)}}\right| \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right) \]

      10. fabs-sqr 35.4%

          \[\leadsto \color{blue}{\left({\left(\sqrt[3]{{\pi}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{\pi}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)}\right)} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right) \]

      11. sqr-pow 35.8%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{{\pi}^{-0.5}}\right)}^{3}} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right) \]

      12. rem-cube-cbrt 36.4%

          \[\leadsto \color{blue}{{\pi}^{-0.5}} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right) \]

      13. unpow3 36.4%

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

      14. unpow2 36.4%

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

      15. associate-*r* 36.4%

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

      16. distribute-rgt-in 36.4%

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

      17. fma-def 36.4%

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

   9. Simplified 36.4%

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

   if 2.14999999999999991 < x

   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 99.4%

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

   6. Applied egg-rr 36.4%

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

   7. Taylor expanded in x around inf 3.8%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.15:\\ \;\;\;\;x \cdot \left({\pi}^{-0.5} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1.85:\\ \;\;\;\;x_m \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({x_m}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.85)
   (* x_m (/ 2.0 (sqrt PI)))
   (* 0.047619047619047616 (* (pow x_m 7.0) (sqrt (/ 1.0 PI))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = x_m * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = 0.047619047619047616 * (pow(x_m, 7.0) * sqrt((1.0 / ((double) M_PI))));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = x_m * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = 0.047619047619047616 * (Math.pow(x_m, 7.0) * Math.sqrt((1.0 / Math.PI)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.85:
		tmp = x_m * (2.0 / math.sqrt(math.pi))
	else:
		tmp = 0.047619047619047616 * (math.pow(x_m, 7.0) * math.sqrt((1.0 / math.pi)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.85)
		tmp = Float64(x_m * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(0.047619047619047616 * Float64((x_m ^ 7.0) * sqrt(Float64(1.0 / pi))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.85)
		tmp = x_m * (2.0 / sqrt(pi));
	else
		tmp = 0.047619047619047616 * ((x_m ^ 7.0) * sqrt((1.0 / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.85], N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[x$95$m, 7.0], $MachinePrecision] * N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   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 99.4%

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

   6. Applied egg-rr 36.4%

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

   7. Taylor expanded in x around 0 36.3%

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

      1. *-commutative 36.3%

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

      2. associate-*r* 36.3%

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

      3. unpow-1 36.3%

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

      4. metadata-eval 36.3%

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

      5. pow-sqr 36.3%

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

      6. rem-sqrt-square 36.3%

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

      7. rem-cube-cbrt 35.8%

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

      8. sqr-pow 35.4%

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

      9. fabs-sqr 35.4%

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

      10. sqr-pow 35.8%

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

      11. rem-cube-cbrt 36.3%

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

   9. Simplified 36.3%

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

       1. expm1-log1p-u 36.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot {\pi}^{-0.5}\right) \cdot x\right)\right)} \]

       2. expm1-udef 4.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(2 \cdot {\pi}^{-0.5}\right) \cdot x\right)} - 1} \]

   11. Applied egg-rr 4.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(2 \cdot \frac{x}{\sqrt{\pi}}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 36.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \frac{x}{\sqrt{\pi}}\right)\right)} \]

       2. expm1-log1p 36.1%

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

       3. associate-*r/ 36.1%

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

       4. associate-*l/ 36.3%

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

   13. Simplified 36.3%

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

   if 1.8500000000000001 < x

   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 99.4%

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

   6. Applied egg-rr 36.4%

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

   7. Taylor expanded in x around inf 3.8%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.8% accurate, 8.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1.85:\\ \;\;\;\;x_m \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;{x_m}^{7} \cdot \sqrt{\frac{0.0022675736961451248}{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.85)
   (* x_m (/ 2.0 (sqrt PI)))
   (* (pow x_m 7.0) (sqrt (/ 0.0022675736961451248 PI)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = x_m * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = pow(x_m, 7.0) * sqrt((0.0022675736961451248 / ((double) M_PI)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = x_m * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.pow(x_m, 7.0) * Math.sqrt((0.0022675736961451248 / Math.PI));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.85:
		tmp = x_m * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.pow(x_m, 7.0) * math.sqrt((0.0022675736961451248 / math.pi))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.85)
		tmp = Float64(x_m * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64((x_m ^ 7.0) * sqrt(Float64(0.0022675736961451248 / pi)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.85)
		tmp = x_m * (2.0 / sqrt(pi));
	else
		tmp = (x_m ^ 7.0) * sqrt((0.0022675736961451248 / pi));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.85], N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x$95$m, 7.0], $MachinePrecision] * N[Sqrt[N[(0.0022675736961451248 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   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 99.4%

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

   6. Applied egg-rr 36.4%

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

   7. Taylor expanded in x around 0 36.3%

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

      1. *-commutative 36.3%

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

      2. associate-*r* 36.3%

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

      3. unpow-1 36.3%

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

      4. metadata-eval 36.3%

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

      5. pow-sqr 36.3%

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

      6. rem-sqrt-square 36.3%

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

      7. rem-cube-cbrt 35.8%

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

      8. sqr-pow 35.4%

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

      9. fabs-sqr 35.4%

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

      10. sqr-pow 35.8%

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

      11. rem-cube-cbrt 36.3%

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

   9. Simplified 36.3%

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

       1. expm1-log1p-u 36.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot {\pi}^{-0.5}\right) \cdot x\right)\right)} \]

       2. expm1-udef 4.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(2 \cdot {\pi}^{-0.5}\right) \cdot x\right)} - 1} \]

   11. Applied egg-rr 4.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(2 \cdot \frac{x}{\sqrt{\pi}}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 36.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \frac{x}{\sqrt{\pi}}\right)\right)} \]

       2. expm1-log1p 36.1%

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

       3. associate-*r/ 36.1%

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

       4. associate-*l/ 36.3%

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

   13. Simplified 36.3%

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

   if 1.8500000000000001 < x

   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 99.4%

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

   6. Applied egg-rr 36.4%

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

   7. Taylor expanded in x around inf 3.8%

      \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 3.8%

         \[\leadsto \color{blue}{\left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right) \cdot 0.047619047619047616} \]

      2. associate-*l* 3.8%

         \[\leadsto \color{blue}{{x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.047619047619047616\right)} \]

      3. reciprocal-define 3.8%

         \[\leadsto {x}^{7} \cdot \left(\sqrt{\color{blue}{\mathsf{reciprocal}\left(\pi\right)}} \cdot 0.047619047619047616\right) \]

   9. Simplified 3.8%

      \[\leadsto \color{blue}{{x}^{7} \cdot \left(\sqrt{\mathsf{reciprocal}\left(\pi\right)} \cdot 0.047619047619047616\right)} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 3.8%

          \[\leadsto {x}^{7} \cdot \color{blue}{\left(\sqrt{\sqrt{\mathsf{reciprocal}\left(\pi\right)} \cdot 0.047619047619047616} \cdot \sqrt{\sqrt{\mathsf{reciprocal}\left(\pi\right)} \cdot 0.047619047619047616}\right)} \]

       2. sqrt-unprod 3.8%

          \[\leadsto {x}^{7} \cdot \color{blue}{\sqrt{\left(\sqrt{\mathsf{reciprocal}\left(\pi\right)} \cdot 0.047619047619047616\right) \cdot \left(\sqrt{\mathsf{reciprocal}\left(\pi\right)} \cdot 0.047619047619047616\right)}} \]

       3. *-commutative 3.8%

          \[\leadsto {x}^{7} \cdot \sqrt{\left(\sqrt{\mathsf{reciprocal}\left(\pi\right)} \cdot 0.047619047619047616\right) \cdot \color{blue}{\left(0.047619047619047616 \cdot \sqrt{\mathsf{reciprocal}\left(\pi\right)}\right)}} \]

       4. reciprocal-undefine 3.8%

          \[\leadsto {x}^{7} \cdot \sqrt{\left(\sqrt{\mathsf{reciprocal}\left(\pi\right)} \cdot 0.047619047619047616\right) \cdot \left(0.047619047619047616 \cdot \sqrt{\color{blue}{\frac{1}{\pi}}}\right)} \]

       5. sqrt-div 3.8%

          \[\leadsto {x}^{7} \cdot \sqrt{\left(\sqrt{\mathsf{reciprocal}\left(\pi\right)} \cdot 0.047619047619047616\right) \cdot \left(0.047619047619047616 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right)} \]

       6. metadata-eval 3.8%

          \[\leadsto {x}^{7} \cdot \sqrt{\left(\sqrt{\mathsf{reciprocal}\left(\pi\right)} \cdot 0.047619047619047616\right) \cdot \left(0.047619047619047616 \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right)} \]

       7. pow1/2 3.8%

          \[\leadsto {x}^{7} \cdot \sqrt{\left(\sqrt{\mathsf{reciprocal}\left(\pi\right)} \cdot 0.047619047619047616\right) \cdot \left(0.047619047619047616 \cdot \frac{1}{\color{blue}{{\pi}^{0.5}}}\right)} \]

       8. pow-flip 3.8%

          \[\leadsto {x}^{7} \cdot \sqrt{\left(\sqrt{\mathsf{reciprocal}\left(\pi\right)} \cdot 0.047619047619047616\right) \cdot \left(0.047619047619047616 \cdot \color{blue}{{\pi}^{\left(-0.5\right)}}\right)} \]

       9. metadata-eval 3.8%

          \[\leadsto {x}^{7} \cdot \sqrt{\left(\sqrt{\mathsf{reciprocal}\left(\pi\right)} \cdot 0.047619047619047616\right) \cdot \left(0.047619047619047616 \cdot {\pi}^{\color{blue}{-0.5}}\right)} \]

       10. reciprocal-undefine 3.8%

           \[\leadsto {x}^{7} \cdot \sqrt{\left(\sqrt{\color{blue}{\frac{1}{\pi}}} \cdot 0.047619047619047616\right) \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)} \]

       11. sqrt-div 3.8%

           \[\leadsto {x}^{7} \cdot \sqrt{\left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}} \cdot 0.047619047619047616\right) \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)} \]

       12. metadata-eval 3.8%

           \[\leadsto {x}^{7} \cdot \sqrt{\left(\frac{\color{blue}{1}}{\sqrt{\pi}} \cdot 0.047619047619047616\right) \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)} \]

       13. pow1/2 3.8%

           \[\leadsto {x}^{7} \cdot \sqrt{\left(\frac{1}{\color{blue}{{\pi}^{0.5}}} \cdot 0.047619047619047616\right) \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)} \]

       14. pow-flip 3.8%

           \[\leadsto {x}^{7} \cdot \sqrt{\left(\color{blue}{{\pi}^{\left(-0.5\right)}} \cdot 0.047619047619047616\right) \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)} \]

       15. metadata-eval 3.8%

           \[\leadsto {x}^{7} \cdot \sqrt{\left({\pi}^{\color{blue}{-0.5}} \cdot 0.047619047619047616\right) \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)} \]

       16. *-commutative 3.8%

           \[\leadsto {x}^{7} \cdot \sqrt{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right) \cdot \color{blue}{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right)}} \]

   11. Applied egg-rr 3.8%

       \[\leadsto {x}^{7} \cdot \color{blue}{\sqrt{\mathsf{reciprocal}\left(\pi\right) \cdot 0.0022675736961451248}} \]
   12. Step-by-step derivation

       1. reciprocal-define 3.8%

          \[\leadsto {x}^{7} \cdot \sqrt{\color{blue}{\frac{1}{\pi}} \cdot 0.0022675736961451248} \]

       2. associate-*l/ 3.8%

          \[\leadsto {x}^{7} \cdot \sqrt{\color{blue}{\frac{1 \cdot 0.0022675736961451248}{\pi}}} \]

       3. metadata-eval 3.8%

          \[\leadsto {x}^{7} \cdot \sqrt{\frac{\color{blue}{0.0022675736961451248}}{\pi}} \]

   13. Simplified 3.8%

       \[\leadsto {x}^{7} \cdot \color{blue}{\sqrt{\frac{0.0022675736961451248}{\pi}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot \sqrt{\frac{0.0022675736961451248}{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.1% 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 99.4%

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

6. Applied egg-rr 36.4%

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

7. Taylor expanded in x around 0 36.3%

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

   1. *-commutative 36.3%

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

   2. associate-*r* 36.3%

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

   3. unpow-1 36.3%

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

   4. metadata-eval 36.3%

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

   5. pow-sqr 36.3%

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

   6. rem-sqrt-square 36.3%

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

   7. rem-cube-cbrt 35.8%

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

   8. sqr-pow 35.4%

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

   9. fabs-sqr 35.4%

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

   10. sqr-pow 35.8%

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

   11. rem-cube-cbrt 36.3%

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

9. Simplified 36.3%

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

    1. expm1-log1p-u 36.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot {\pi}^{-0.5}\right) \cdot x\right)\right)} \]

    2. expm1-udef 4.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(2 \cdot {\pi}^{-0.5}\right) \cdot x\right)} - 1} \]

11. Applied egg-rr 4.4%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(2 \cdot \frac{x}{\sqrt{\pi}}\right)} - 1} \]
12. Step-by-step derivation

    1. expm1-def 36.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \frac{x}{\sqrt{\pi}}\right)\right)} \]

    2. expm1-log1p 36.1%

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

    3. associate-*r/ 36.1%

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

    4. associate-*l/ 36.3%

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

13. Simplified 36.3%

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

14. Final simplification 36.3%

    \[\leadsto x \cdot \frac{2}{\sqrt{\pi}} \]
15. Add Preprocessing

Alternative 10: 4.1% accurate, 1849.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.0;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 0.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.0

Julia

x_m = abs(x)
function code(x_m)
	return 0.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 0.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.0

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 99.4%

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

6. Applied egg-rr 36.4%

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

7. Taylor expanded in x around 0 36.3%

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

   1. *-commutative 36.3%

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

   2. associate-*r* 36.3%

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

   3. unpow-1 36.3%

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

   4. metadata-eval 36.3%

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

   5. pow-sqr 36.3%

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

   6. rem-sqrt-square 36.3%

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

   7. rem-cube-cbrt 35.8%

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

   8. sqr-pow 35.4%

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

   9. fabs-sqr 35.4%

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

   10. sqr-pow 35.8%

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

   11. rem-cube-cbrt 36.3%

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

9. Simplified 36.3%

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

    1. expm1-log1p-u 36.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot {\pi}^{-0.5}\right) \cdot x\right)\right)} \]

    2. expm1-udef 4.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(2 \cdot {\pi}^{-0.5}\right) \cdot x\right)} - 1} \]

11. Applied egg-rr 4.4%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(2 \cdot \frac{x}{\sqrt{\pi}}\right)} - 1} \]

12. Taylor expanded in x around 0 4.3%

    \[\leadsto \color{blue}{1} - 1 \]

13. Final simplification 4.3%

    \[\leadsto 0 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(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)))))))