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

Percentage Accurate: 99.8% → 99.9%

Time: 11.9s

Alternatives: 9

Speedup: 3.6×

• 28.6% 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, 2.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fabs x)
  (fabs
   (/
    (+
     (fma 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(((fma(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(fma(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[(0.2 * N[Power[x, 4.0], $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. Add Preprocessing

Alternative 2: 99.9% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 99.9%

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

   1. *-commutative 99.9%

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

7. Simplified 99.9%

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

   1. pow2 99.9%

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

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

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

Alternative 3: 99.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fabs x)
  (fabs
   (*
    (sqrt (/ 1.0 PI))
    (+
     2.0
     (+
      (* 0.047619047619047616 (pow x 6.0))
      (* 0.6666666666666666 (pow x 2.0))))))))

C

double code(double x) {
	return fabs(x) * fabs((sqrt((1.0 / ((double) M_PI))) * (2.0 + ((0.047619047619047616 * pow(x, 6.0)) + (0.6666666666666666 * pow(x, 2.0))))));
}

Java

public static double code(double x) {
	return Math.abs(x) * Math.abs((Math.sqrt((1.0 / Math.PI)) * (2.0 + ((0.047619047619047616 * Math.pow(x, 6.0)) + (0.6666666666666666 * Math.pow(x, 2.0))))));
}

Python

def code(x):
	return math.fabs(x) * math.fabs((math.sqrt((1.0 / math.pi)) * (2.0 + ((0.047619047619047616 * math.pow(x, 6.0)) + (0.6666666666666666 * math.pow(x, 2.0))))))

Julia

function code(x)
	return Float64(abs(x) * abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(2.0 + Float64(Float64(0.047619047619047616 * (x ^ 6.0)) + Float64(0.6666666666666666 * (x ^ 2.0)))))))
end

MATLAB

function tmp = code(x)
	tmp = abs(x) * abs((sqrt((1.0 / pi)) * (2.0 + ((0.047619047619047616 * (x ^ 6.0)) + (0.6666666666666666 * (x ^ 2.0))))));
end

Wolfram

code[x_] := N[(N[Abs[x], $MachinePrecision] * N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(2.0 + N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 99.1%

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

6. Taylor expanded in x around 0 99.1%

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

Alternative 4: 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 99.1%

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

Alternative 5: 95.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{+22}:\\ \;\;\;\;\left|\left(x \cdot 2\right) \cdot {\pi}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\left|x \cdot \left(2 \cdot {\pi}^{-0.5}\right)\right|}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 5e+22)
   (fabs (* (* x 2.0) (pow PI -0.5)))
   (log (exp (fabs (* x (* 2.0 (pow PI -0.5))))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 5e+22) {
		tmp = fabs(((x * 2.0) * pow(((double) M_PI), -0.5)));
	} else {
		tmp = log(exp(fabs((x * (2.0 * pow(((double) M_PI), -0.5))))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 5e+22) {
		tmp = Math.abs(((x * 2.0) * Math.pow(Math.PI, -0.5)));
	} else {
		tmp = Math.log(Math.exp(Math.abs((x * (2.0 * Math.pow(Math.PI, -0.5))))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 5e+22:
		tmp = math.fabs(((x * 2.0) * math.pow(math.pi, -0.5)))
	else:
		tmp = math.log(math.exp(math.fabs((x * (2.0 * math.pow(math.pi, -0.5))))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 5e+22)
		tmp = abs(Float64(Float64(x * 2.0) * (pi ^ -0.5)));
	else
		tmp = log(exp(abs(Float64(x * Float64(2.0 * (pi ^ -0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 5e+22)
		tmp = abs(((x * 2.0) * (pi ^ -0.5)));
	else
		tmp = log(exp(abs((x * (2.0 * (pi ^ -0.5))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 5e+22], N[Abs[N[(N[(x * 2.0), $MachinePrecision] * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Log[N[Exp[N[Abs[N[(x * N[(2.0 * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 4.9999999999999996e22

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 94.3%

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

   6. Taylor expanded in x around 0 94.3%

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

   7. Taylor expanded in x around 0 93.0%

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

      1. rem-exp-log 93.0%

         \[\leadsto \left|x\right| \cdot \left|2 \cdot \sqrt{\frac{1}{\color{blue}{e^{\log \pi}}}}\right| \]

      2. exp-neg 93.0%

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

      3. unpow1/2 93.0%

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

      4. exp-prod 93.0%

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

      5. distribute-lft-neg-out 93.0%

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

      6. distribute-rgt-neg-in 93.0%

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

      7. metadata-eval 93.0%

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

      8. exp-to-pow 93.0%

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

   9. Simplified 93.0%

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

       1. pow1 93.0%

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

       2. mul-fabs 93.0%

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

   11. Applied egg-rr 93.0%

       \[\leadsto \color{blue}{{\left(\left|x \cdot \left(2 \cdot {\pi}^{-0.5}\right)\right|\right)}^{1}} \]
   12. Step-by-step derivation

       1. unpow1 93.0%

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

       2. associate-*r* 93.0%

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

   13. Simplified 93.0%

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

   if 4.9999999999999996e22 < (fabs.f64 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 100.0%

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

   5. Taylor expanded in x around 0 89.0%

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

   6. Taylor expanded in x around 0 89.0%

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

   7. Taylor expanded in x around 0 5.9%

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

      1. rem-exp-log 5.9%

         \[\leadsto \left|x\right| \cdot \left|2 \cdot \sqrt{\frac{1}{\color{blue}{e^{\log \pi}}}}\right| \]

      2. exp-neg 5.9%

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

      3. unpow1/2 5.9%

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

      4. exp-prod 5.9%

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

      5. distribute-lft-neg-out 5.9%

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

      6. distribute-rgt-neg-in 5.9%

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

      7. metadata-eval 5.9%

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

      8. exp-to-pow 5.9%

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

   9. Simplified 5.9%

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

       1. add-log-exp 95.9%

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

       2. mul-fabs 95.9%

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

   11. Applied egg-rr 95.9%

       \[\leadsto \color{blue}{\log \left(e^{\left|x \cdot \left(2 \cdot {\pi}^{-0.5}\right)\right|}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 93.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;\left|\left(x \cdot 2\right) \cdot {\pi}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;0.2 \cdot \frac{\left|x\right| \cdot {x}^{4}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.5)
   (fabs (* (* x 2.0) (pow PI -0.5)))
   (* 0.2 (/ (* (fabs x) (pow x 4.0)) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.5) {
		tmp = fabs(((x * 2.0) * pow(((double) M_PI), -0.5)));
	} else {
		tmp = 0.2 * ((fabs(x) * pow(x, 4.0)) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.5) {
		tmp = Math.abs(((x * 2.0) * Math.pow(Math.PI, -0.5)));
	} else {
		tmp = 0.2 * ((Math.abs(x) * Math.pow(x, 4.0)) / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.5:
		tmp = math.fabs(((x * 2.0) * math.pow(math.pi, -0.5)))
	else:
		tmp = 0.2 * ((math.fabs(x) * math.pow(x, 4.0)) / math.sqrt(math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.5)
		tmp = abs(Float64(Float64(x * 2.0) * (pi ^ -0.5)));
	else
		tmp = Float64(0.2 * Float64(Float64(abs(x) * (x ^ 4.0)) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.5)
		tmp = abs(((x * 2.0) * (pi ^ -0.5)));
	else
		tmp = 0.2 * ((abs(x) * (x ^ 4.0)) / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.5], N[Abs[N[(N[(x * 2.0), $MachinePrecision] * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(0.2 * N[(N[(N[Abs[x], $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.5

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.5%

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

   6. Taylor expanded in x around 0 99.6%

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

   7. Taylor expanded in x around 0 98.4%

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

      1. rem-exp-log 98.4%

         \[\leadsto \left|x\right| \cdot \left|2 \cdot \sqrt{\frac{1}{\color{blue}{e^{\log \pi}}}}\right| \]

      2. exp-neg 98.4%

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

      3. unpow1/2 98.4%

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

      4. exp-prod 98.4%

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

      5. distribute-lft-neg-out 98.4%

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

      6. distribute-rgt-neg-in 98.4%

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

      7. metadata-eval 98.4%

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

      8. exp-to-pow 98.4%

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

   9. Simplified 98.4%

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

       1. pow1 98.4%

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

       2. mul-fabs 98.4%

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

   11. Applied egg-rr 98.4%

       \[\leadsto \color{blue}{{\left(\left|x \cdot \left(2 \cdot {\pi}^{-0.5}\right)\right|\right)}^{1}} \]
   12. Step-by-step derivation

       1. unpow1 98.4%

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

       2. associate-*r* 98.4%

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

   13. Simplified 98.4%

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

   if 0.5 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{3} \cdot \left({\left(\left|x\right|\right)}^{3} \cdot \left|x\right|\right), \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{3} \cdot \left(x \cdot x\right), \mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)\right)\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 77.8%

      \[\leadsto \left|\frac{\color{blue}{0.2 \cdot \left({x}^{2} \cdot {\left(\left|x\right|\right)}^{3}\right)}}{\sqrt{\pi}}\right| \]
   6. Step-by-step derivation

      1. *-commutative 77.8%

         \[\leadsto \left|\frac{\color{blue}{\left({x}^{2} \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot 0.2}}{\sqrt{\pi}}\right| \]

      2. *-commutative 77.8%

         \[\leadsto \left|\frac{\color{blue}{\left({\left(\left|x\right|\right)}^{3} \cdot {x}^{2}\right)} \cdot 0.2}{\sqrt{\pi}}\right| \]

      3. associate-*l* 77.8%

         \[\leadsto \left|\frac{\color{blue}{{\left(\left|x\right|\right)}^{3} \cdot \left({x}^{2} \cdot 0.2\right)}}{\sqrt{\pi}}\right| \]

   7. Simplified 77.8%

      \[\leadsto \left|\frac{\color{blue}{{\left(\left|x\right|\right)}^{3} \cdot \left({x}^{2} \cdot 0.2\right)}}{\sqrt{\pi}}\right| \]
   8. Step-by-step derivation

      1. *-un-lft-identity 77.8%

         \[\leadsto \color{blue}{1 \cdot \left|\frac{{\left(\left|x\right|\right)}^{3} \cdot \left({x}^{2} \cdot 0.2\right)}{\sqrt{\pi}}\right|} \]

      2. associate-/l* 77.8%

         \[\leadsto 1 \cdot \left|\color{blue}{{\left(\left|x\right|\right)}^{3} \cdot \frac{{x}^{2} \cdot 0.2}{\sqrt{\pi}}}\right| \]

   9. Applied egg-rr 77.8%

      \[\leadsto \color{blue}{1 \cdot \left|{\left(\left|x\right|\right)}^{3} \cdot \frac{{x}^{2} \cdot 0.2}{\sqrt{\pi}}\right|} \]
   10. Step-by-step derivation

       1. *-lft-identity 77.8%

          \[\leadsto \color{blue}{\left|{\left(\left|x\right|\right)}^{3} \cdot \frac{{x}^{2} \cdot 0.2}{\sqrt{\pi}}\right|} \]

       2. rem-square-sqrt 77.8%

          \[\leadsto \left|\color{blue}{\sqrt{{\left(\left|x\right|\right)}^{3} \cdot \frac{{x}^{2} \cdot 0.2}{\sqrt{\pi}}} \cdot \sqrt{{\left(\left|x\right|\right)}^{3} \cdot \frac{{x}^{2} \cdot 0.2}{\sqrt{\pi}}}}\right| \]

       3. fabs-sqr 77.8%

          \[\leadsto \color{blue}{\sqrt{{\left(\left|x\right|\right)}^{3} \cdot \frac{{x}^{2} \cdot 0.2}{\sqrt{\pi}}} \cdot \sqrt{{\left(\left|x\right|\right)}^{3} \cdot \frac{{x}^{2} \cdot 0.2}{\sqrt{\pi}}}} \]

       4. rem-square-sqrt 77.8%

          \[\leadsto \color{blue}{{\left(\left|x\right|\right)}^{3} \cdot \frac{{x}^{2} \cdot 0.2}{\sqrt{\pi}}} \]

       5. associate-*r/ 77.8%

          \[\leadsto \color{blue}{\frac{{\left(\left|x\right|\right)}^{3} \cdot \left({x}^{2} \cdot 0.2\right)}{\sqrt{\pi}}} \]

       6. associate-*r* 77.8%

          \[\leadsto \frac{\color{blue}{\left({\left(\left|x\right|\right)}^{3} \cdot {x}^{2}\right) \cdot 0.2}}{\sqrt{\pi}} \]

       7. *-commutative 77.8%

          \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot {\left(\left|x\right|\right)}^{3}\right)} \cdot 0.2}{\sqrt{\pi}} \]

       8. *-commutative 77.8%

          \[\leadsto \frac{\color{blue}{0.2 \cdot \left({x}^{2} \cdot {\left(\left|x\right|\right)}^{3}\right)}}{\sqrt{\pi}} \]

       9. associate-/l* 77.8%

          \[\leadsto \color{blue}{0.2 \cdot \frac{{x}^{2} \cdot {\left(\left|x\right|\right)}^{3}}{\sqrt{\pi}}} \]

   11. Simplified 77.8%

       \[\leadsto \color{blue}{0.2 \cdot \frac{\left|x\right| \cdot {x}^{4}}{\sqrt{\pi}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 89.5% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;\left|\left(x \cdot 2\right) \cdot {\pi}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;0.6666666666666666 \cdot \left({\pi}^{-0.5} \cdot {\left(\left|x\right|\right)}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.5)
   (fabs (* (* x 2.0) (pow PI -0.5)))
   (* 0.6666666666666666 (* (pow PI -0.5) (pow (fabs x) 3.0)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.5) {
		tmp = fabs(((x * 2.0) * pow(((double) M_PI), -0.5)));
	} else {
		tmp = 0.6666666666666666 * (pow(((double) M_PI), -0.5) * pow(fabs(x), 3.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.5) {
		tmp = Math.abs(((x * 2.0) * Math.pow(Math.PI, -0.5)));
	} else {
		tmp = 0.6666666666666666 * (Math.pow(Math.PI, -0.5) * Math.pow(Math.abs(x), 3.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.5:
		tmp = math.fabs(((x * 2.0) * math.pow(math.pi, -0.5)))
	else:
		tmp = 0.6666666666666666 * (math.pow(math.pi, -0.5) * math.pow(math.fabs(x), 3.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.5)
		tmp = abs(Float64(Float64(x * 2.0) * (pi ^ -0.5)));
	else
		tmp = Float64(0.6666666666666666 * Float64((pi ^ -0.5) * (abs(x) ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.5)
		tmp = abs(((x * 2.0) * (pi ^ -0.5)));
	else
		tmp = 0.6666666666666666 * ((pi ^ -0.5) * (abs(x) ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.5], N[Abs[N[(N[(x * 2.0), $MachinePrecision] * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(0.6666666666666666 * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Power[N[Abs[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.5

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.5%

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

   6. Taylor expanded in x around 0 99.6%

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

   7. Taylor expanded in x around 0 98.4%

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

      1. rem-exp-log 98.4%

         \[\leadsto \left|x\right| \cdot \left|2 \cdot \sqrt{\frac{1}{\color{blue}{e^{\log \pi}}}}\right| \]

      2. exp-neg 98.4%

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

      3. unpow1/2 98.4%

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

      4. exp-prod 98.4%

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

      5. distribute-lft-neg-out 98.4%

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

      6. distribute-rgt-neg-in 98.4%

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

      7. metadata-eval 98.4%

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

      8. exp-to-pow 98.4%

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

   9. Simplified 98.4%

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

       1. pow1 98.4%

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

       2. mul-fabs 98.4%

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

   11. Applied egg-rr 98.4%

       \[\leadsto \color{blue}{{\left(\left|x \cdot \left(2 \cdot {\pi}^{-0.5}\right)\right|\right)}^{1}} \]
   12. Step-by-step derivation

       1. unpow1 98.4%

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

       2. associate-*r* 98.4%

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

   13. Simplified 98.4%

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

   if 0.5 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 68.9%

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

      1. associate-*r* 68.9%

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

      2. *-commutative 68.9%

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

      3. *-commutative 68.9%

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

      4. unpow2 68.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot \left(\left|x\right| \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right| \]

      5. sqr-abs 68.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot \left(\left|x\right| \cdot \color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)}\right)\right)\right| \]

      6. cube-mult 68.9%

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

   7. Simplified 68.9%

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

   8. Taylor expanded in x around 0 68.9%

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

      1. associate-*r* 68.9%

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

      2. fabs-mul 68.9%

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

      3. unpow-1 68.9%

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

      4. metadata-eval 68.9%

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

      5. pow-sqr 68.9%

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

      6. rem-sqrt-square 68.9%

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

      7. rem-square-sqrt 68.9%

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

      8. fabs-sqr 68.9%

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

      9. rem-square-sqrt 68.9%

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

      10. *-commutative 68.9%

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

   10. Simplified 68.9%

       \[\leadsto \color{blue}{0.6666666666666666 \cdot \left({\left(\left|x\right|\right)}^{3} \cdot {\pi}^{-0.5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;\left|\left(x \cdot 2\right) \cdot {\pi}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;0.6666666666666666 \cdot \left({\pi}^{-0.5} \cdot {\left(\left|x\right|\right)}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.0% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = abs(x) * abs((((0.047619047619047616 * (x ^ 6.0)) + 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] + 2.0), $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 99.1%

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

6. Taylor expanded in x around 0 98.4%

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

Alternative 9: 68.2% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \left|\left(x \cdot 2\right) \cdot {\pi}^{-0.5}\right| \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fabs (* (* x 2.0) (pow PI -0.5))))

C

double code(double x) {
	return fabs(((x * 2.0) * pow(((double) M_PI), -0.5)));
}

Java

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

Python

def code(x):
	return math.fabs(((x * 2.0) * math.pow(math.pi, -0.5)))

Julia

function code(x)
	return abs(Float64(Float64(x * 2.0) * (pi ^ -0.5)))
end

MATLAB

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

Wolfram

code[x_] := N[Abs[N[(N[(x * 2.0), $MachinePrecision] * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 92.9%

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

6. Taylor expanded in x around 0 92.9%

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

7. Taylor expanded in x around 0 70.2%

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

   1. rem-exp-log 70.2%

      \[\leadsto \left|x\right| \cdot \left|2 \cdot \sqrt{\frac{1}{\color{blue}{e^{\log \pi}}}}\right| \]

   2. exp-neg 70.2%

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

   3. unpow1/2 70.2%

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

   4. exp-prod 70.2%

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

   5. distribute-lft-neg-out 70.2%

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

   6. distribute-rgt-neg-in 70.2%

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

   7. metadata-eval 70.2%

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

   8. exp-to-pow 70.2%

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

9. Simplified 70.2%

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

    1. pow1 70.2%

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

    2. mul-fabs 70.2%

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

11. Applied egg-rr 70.2%

    \[\leadsto \color{blue}{{\left(\left|x \cdot \left(2 \cdot {\pi}^{-0.5}\right)\right|\right)}^{1}} \]
12. Step-by-step derivation

    1. unpow1 70.2%

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

    2. associate-*r* 70.5%

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

13. Simplified 70.5%

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

Reproduce

herbie shell --seed 2024149 
(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)))))))