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

Percentage Accurate: 99.8% → 99.9%

Time: 12.1s

Alternatives: 7

Speedup: 3.0×

• 28.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 3.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fabs x)
  (fabs
   (/
    (+
     (+ (* 0.2 (pow x 4.0)) (* 0.047619047619047616 (pow x 6.0)))
     (fma 0.6666666666666666 (* x x) 2.0))
    (sqrt PI)))))

C

double code(double x) {
	return fabs(x) * fabs(((((0.2 * pow(x, 4.0)) + (0.047619047619047616 * pow(x, 6.0))) + fma(0.6666666666666666, (x * x), 2.0)) / sqrt(((double) M_PI))));
}

Julia

function code(x)
	return Float64(abs(x) * abs(Float64(Float64(Float64(Float64(0.2 * (x ^ 4.0)) + Float64(0.047619047619047616 * (x ^ 6.0))) + fma(0.6666666666666666, Float64(x * x), 2.0)) / sqrt(pi))))
end

Wolfram

code[x_] := N[(N[Abs[x], $MachinePrecision] * N[Abs[N[(N[(N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

   1. metadata-eval 99.9%

      \[\leadsto \left|x\right| \cdot \left|\frac{\mathsf{fma}\left(\color{blue}{\frac{1}{5}}, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

   2. metadata-eval 99.9%

      \[\leadsto \left|x\right| \cdot \left|\frac{\mathsf{fma}\left(\frac{1}{5}, {x}^{\color{blue}{\left(2 + 2\right)}}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

   3. pow-prod-up 99.9%

      \[\leadsto \left|x\right| \cdot \left|\frac{\mathsf{fma}\left(\frac{1}{5}, \color{blue}{{x}^{2} \cdot {x}^{2}}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

   4. pow-prod-down 99.9%

      \[\leadsto \left|x\right| \cdot \left|\frac{\mathsf{fma}\left(\frac{1}{5}, \color{blue}{{\left(x \cdot x\right)}^{2}}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

   5. sqr-abs 99.9%

      \[\leadsto \left|x\right| \cdot \left|\frac{\mathsf{fma}\left(\frac{1}{5}, {\color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)}}^{2}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

   6. pow-prod-down 99.9%

      \[\leadsto \left|x\right| \cdot \left|\frac{\mathsf{fma}\left(\frac{1}{5}, \color{blue}{{\left(\left|x\right|\right)}^{2} \cdot {\left(\left|x\right|\right)}^{2}}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

   7. pow-sqr 99.9%

      \[\leadsto \left|x\right| \cdot \left|\frac{\mathsf{fma}\left(\frac{1}{5}, \color{blue}{{\left(\left|x\right|\right)}^{\left(2 \cdot 2\right)}}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

   8. metadata-eval 99.9%

      \[\leadsto \left|x\right| \cdot \left|\frac{\mathsf{fma}\left(\frac{1}{5}, {\left(\left|x\right|\right)}^{\color{blue}{4}}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

   9. fma-define 99.9%

      \[\leadsto \left|x\right| \cdot \left|\frac{\color{blue}{\left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

   10. metadata-eval 99.9%

       \[\leadsto \left|x\right| \cdot \left|\frac{\left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{\color{blue}{\left(2 + 2\right)}} + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

   11. pow-prod-up 99.9%

       \[\leadsto \left|x\right| \cdot \left|\frac{\left(\frac{1}{5} \cdot \color{blue}{\left({\left(\left|x\right|\right)}^{2} \cdot {\left(\left|x\right|\right)}^{2}\right)} + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

   12. pow-prod-down 99.9%

       \[\leadsto \left|x\right| \cdot \left|\frac{\left(\frac{1}{5} \cdot \color{blue}{{\left(\left|x\right| \cdot \left|x\right|\right)}^{2}} + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

   13. sqr-abs 99.9%

       \[\leadsto \left|x\right| \cdot \left|\frac{\left(\frac{1}{5} \cdot {\color{blue}{\left(x \cdot x\right)}}^{2} + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

   14. pow-prod-down 99.9%

       \[\leadsto \left|x\right| \cdot \left|\frac{\left(\frac{1}{5} \cdot \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

   15. pow-prod-up 99.9%

       \[\leadsto \left|x\right| \cdot \left|\frac{\left(\frac{1}{5} \cdot \color{blue}{{x}^{\left(2 + 2\right)}} + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

   16. metadata-eval 99.9%

       \[\leadsto \left|x\right| \cdot \left|\frac{\left(\frac{1}{5} \cdot {x}^{\color{blue}{4}} + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

   17. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 99.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left|x\right| \cdot \left|\frac{0.047619047619047616 \cdot {x}^{6} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fabs x)
  (fabs
   (/
    (+
     (* 0.047619047619047616 (pow x 6.0))
     (fma 0.6666666666666666 (* x x) 2.0))
    (sqrt PI)))))

C

double code(double x) {
	return fabs(x) * fabs((((0.047619047619047616 * pow(x, 6.0)) + fma(0.6666666666666666, (x * x), 2.0)) / sqrt(((double) M_PI))));
}

Julia

function code(x)
	return Float64(abs(x) * abs(Float64(Float64(Float64(0.047619047619047616 * (x ^ 6.0)) + fma(0.6666666666666666, Float64(x * x), 2.0)) / sqrt(pi))))
end

Wolfram

code[x_] := N[(N[Abs[x], $MachinePrecision] * N[Abs[N[(N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around inf 99.0%

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

6. Final simplification 99.0%

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

Alternative 3: 34.5% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[x, 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 99.8%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 98.7%

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

   6. Taylor expanded in x around inf 98.6%

      \[\leadsto \left|x\right| \cdot \left|\frac{\color{blue}{0.047619047619047616 \cdot {x}^{6}} + 2}{\sqrt{\pi}}\right| \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 28.9%

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

      2. fabs-sqr 28.9%

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

      3. add-sqr-sqrt 30.6%

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

      4. add-sqr-sqrt 30.0%

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

      5. fabs-sqr 30.0%

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

      6. add-sqr-sqrt 30.6%

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

      7. clear-num 30.6%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\sqrt{\pi}}{0.047619047619047616 \cdot {x}^{6} + 2}}} \]

      8. un-div-inv 30.4%

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

      9. *-commutative 30.4%

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

      10. fma-define 30.4%

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

   8. Applied egg-rr 30.4%

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

      1. associate-/r/ 30.4%

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

      2. associate-*l/ 30.4%

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

      3. associate-*r/ 30.6%

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

   10. Simplified 30.6%

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

   11. Taylor expanded in x around 0 30.6%

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

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 98.7%

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

   6. Taylor expanded in x around inf 98.6%

      \[\leadsto \left|x\right| \cdot \left|\frac{\color{blue}{0.047619047619047616 \cdot {x}^{6}} + 2}{\sqrt{\pi}}\right| \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 28.9%

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

      2. fabs-sqr 28.9%

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

      3. add-sqr-sqrt 30.6%

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

      4. add-sqr-sqrt 30.0%

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

      5. fabs-sqr 30.0%

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

      6. add-sqr-sqrt 30.6%

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

      7. clear-num 30.6%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\sqrt{\pi}}{0.047619047619047616 \cdot {x}^{6} + 2}}} \]

      8. un-div-inv 30.4%

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

      9. *-commutative 30.4%

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

      10. fma-define 30.4%

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

   8. Applied egg-rr 30.4%

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

      1. associate-/r/ 30.4%

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

      2. associate-*l/ 30.4%

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

      3. associate-*r/ 30.6%

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

   10. Simplified 30.6%

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

   11. Taylor expanded in x around inf 3.5%

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

4. Final simplification 30.6%

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

Alternative 4: 34.5% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.2 \cdot {x}^{5}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.8) (* x (/ 2.0 (sqrt PI))) (/ (* 0.2 (pow x 5.0)) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (x <= 1.8) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = (0.2 * pow(x, 5.0)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.8) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = (0.2 * Math.pow(x, 5.0)) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.8:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = (0.2 * math.pow(x, 5.0)) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.8)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(Float64(0.2 * (x ^ 5.0)) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.8)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = (0.2 * (x ^ 5.0)) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.8], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.80000000000000004

   1. Initial program 99.8%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 98.7%

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

   6. Taylor expanded in x around inf 98.6%

      \[\leadsto \left|x\right| \cdot \left|\frac{\color{blue}{0.047619047619047616 \cdot {x}^{6}} + 2}{\sqrt{\pi}}\right| \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 28.9%

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

      2. fabs-sqr 28.9%

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

      3. add-sqr-sqrt 30.6%

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

      4. add-sqr-sqrt 30.0%

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

      5. fabs-sqr 30.0%

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

      6. add-sqr-sqrt 30.6%

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

      7. clear-num 30.6%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\sqrt{\pi}}{0.047619047619047616 \cdot {x}^{6} + 2}}} \]

      8. un-div-inv 30.4%

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

      9. *-commutative 30.4%

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

      10. fma-define 30.4%

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

   8. Applied egg-rr 30.4%

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

      1. associate-/r/ 30.4%

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

      2. associate-*l/ 30.4%

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

      3. associate-*r/ 30.6%

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

   10. Simplified 30.6%

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

   11. Taylor expanded in x around 0 30.6%

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

   if 1.80000000000000004 < x

   1. Initial program 99.8%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 92.0%

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

      1. expm1-log1p-u 92.0%

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

      2. expm1-undefine 32.0%

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

   7. Applied egg-rr 4.5%

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

      1. log1p-undefine 4.5%

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

      2. rem-exp-log 4.6%

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

      3. +-commutative 4.6%

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

      4. associate--l+ 30.7%

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

      5. metadata-eval 30.7%

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

      6. metadata-eval 30.7%

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

      7. sub-neg 30.7%

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

      8. --rgt-identity 30.7%

         \[\leadsto \color{blue}{x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}{\sqrt{\pi}}} \]

      9. associate-*r/ 30.5%

         \[\leadsto \color{blue}{\frac{x \cdot \mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}{\sqrt{\pi}}} \]

      10. associate-*l/ 30.5%

          \[\leadsto \color{blue}{\frac{x}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)} \]

   9. Simplified 30.5%

      \[\leadsto \color{blue}{\frac{x}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)} \]

   10. Taylor expanded in x around inf 3.6%

       \[\leadsto \frac{x}{\sqrt{\pi}} \cdot \color{blue}{\left(0.2 \cdot {x}^{4}\right)} \]
   11. Step-by-step derivation

       1. *-commutative 3.6%

          \[\leadsto \color{blue}{\left(0.2 \cdot {x}^{4}\right) \cdot \frac{x}{\sqrt{\pi}}} \]

       2. clear-num 3.6%

          \[\leadsto \left(0.2 \cdot {x}^{4}\right) \cdot \color{blue}{\frac{1}{\frac{\sqrt{\pi}}{x}}} \]

       3. un-div-inv 3.6%

          \[\leadsto \color{blue}{\frac{0.2 \cdot {x}^{4}}{\frac{\sqrt{\pi}}{x}}} \]

   12. Applied egg-rr 3.6%

       \[\leadsto \color{blue}{\frac{0.2 \cdot {x}^{4}}{\frac{\sqrt{\pi}}{x}}} \]
   13. Step-by-step derivation

       1. associate-/r/ 3.6%

          \[\leadsto \color{blue}{\frac{0.2 \cdot {x}^{4}}{\sqrt{\pi}} \cdot x} \]

       2. associate-*l/ 3.6%

          \[\leadsto \color{blue}{\frac{\left(0.2 \cdot {x}^{4}\right) \cdot x}{\sqrt{\pi}}} \]

       3. associate-*l* 3.6%

          \[\leadsto \frac{\color{blue}{0.2 \cdot \left({x}^{4} \cdot x\right)}}{\sqrt{\pi}} \]

       4. pow-plus 3.6%

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

       5. metadata-eval 3.6%

          \[\leadsto \frac{0.2 \cdot {x}^{\color{blue}{5}}}{\sqrt{\pi}} \]

   14. Simplified 3.6%

       \[\leadsto \color{blue}{\frac{0.2 \cdot {x}^{5}}{\sqrt{\pi}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.8:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.2 \cdot {x}^{5}}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 34.4% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{0.047619047619047616 \cdot {x}^{6} + 2}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (/ (+ (* 0.047619047619047616 (pow x 6.0)) 2.0) (sqrt PI))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(x * N[(N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 98.7%

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

6. Taylor expanded in x around inf 98.6%

   \[\leadsto \left|x\right| \cdot \left|\frac{\color{blue}{0.047619047619047616 \cdot {x}^{6}} + 2}{\sqrt{\pi}}\right| \]
7. Step-by-step derivation

   1. add-sqr-sqrt 28.9%

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

   2. fabs-sqr 28.9%

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

   3. add-sqr-sqrt 30.6%

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

   4. add-sqr-sqrt 30.0%

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

   5. fabs-sqr 30.0%

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

   6. add-sqr-sqrt 30.6%

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

   7. clear-num 30.6%

      \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\sqrt{\pi}}{0.047619047619047616 \cdot {x}^{6} + 2}}} \]

   8. un-div-inv 30.4%

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

   9. *-commutative 30.4%

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

   10. fma-define 30.4%

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

8. Applied egg-rr 30.4%

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

   1. associate-/r/ 30.4%

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

   2. associate-*l/ 30.4%

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

   3. associate-*r/ 30.6%

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

10. Simplified 30.6%

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

    1. fma-undefine 30.6%

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

12. Applied egg-rr 30.6%

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

13. Final simplification 30.6%

    \[\leadsto x \cdot \frac{0.047619047619047616 \cdot {x}^{6} + 2}{\sqrt{\pi}} \]
14. Add Preprocessing

Alternative 6: 34.5% accurate, 17.3× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* (sqrt (/ 1.0 PI)) (* x 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 98.7%

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

6. Taylor expanded in x around inf 98.6%

   \[\leadsto \left|x\right| \cdot \left|\frac{\color{blue}{0.047619047619047616 \cdot {x}^{6}} + 2}{\sqrt{\pi}}\right| \]
7. Step-by-step derivation

   1. add-sqr-sqrt 28.9%

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

   2. fabs-sqr 28.9%

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

   3. add-sqr-sqrt 30.6%

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

   4. add-sqr-sqrt 30.0%

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

   5. fabs-sqr 30.0%

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

   6. add-sqr-sqrt 30.6%

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

   7. clear-num 30.6%

      \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\sqrt{\pi}}{0.047619047619047616 \cdot {x}^{6} + 2}}} \]

   8. un-div-inv 30.4%

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

   9. *-commutative 30.4%

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

   10. fma-define 30.4%

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

8. Applied egg-rr 30.4%

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

   1. associate-/r/ 30.4%

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

   2. associate-*l/ 30.4%

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

   3. associate-*r/ 30.6%

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

10. Simplified 30.6%

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

11. Taylor expanded in x around 0 30.6%

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

    1. associate-*r* 30.6%

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

13. Simplified 30.6%

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

14. Final simplification 30.6%

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

Alternative 7: 34.5% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{2}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (/ 2.0 (sqrt PI))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 98.7%

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

6. Taylor expanded in x around inf 98.6%

   \[\leadsto \left|x\right| \cdot \left|\frac{\color{blue}{0.047619047619047616 \cdot {x}^{6}} + 2}{\sqrt{\pi}}\right| \]
7. Step-by-step derivation

   1. add-sqr-sqrt 28.9%

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

   2. fabs-sqr 28.9%

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

   3. add-sqr-sqrt 30.6%

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

   4. add-sqr-sqrt 30.0%

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

   5. fabs-sqr 30.0%

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

   6. add-sqr-sqrt 30.6%

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

   7. clear-num 30.6%

      \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\sqrt{\pi}}{0.047619047619047616 \cdot {x}^{6} + 2}}} \]

   8. un-div-inv 30.4%

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

   9. *-commutative 30.4%

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

   10. fma-define 30.4%

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

8. Applied egg-rr 30.4%

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

   1. associate-/r/ 30.4%

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

   2. associate-*l/ 30.4%

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

   3. associate-*r/ 30.6%

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

10. Simplified 30.6%

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

11. Taylor expanded in x around 0 30.6%

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

12. Final simplification 30.6%

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

Reproduce

herbie shell --seed 2024076 
(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)))))))