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

Percentage Accurate: 99.8% → 99.8%

Time: 11.6s

Alternatives: 14

Speedup: 3.6×

• 28.7% 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: 99.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| \cdot \left(x \cdot x\right)\\ t_1 := \left|x\right| \cdot \left(\left|x\right| \cdot t\_0\right)\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + 0.6666666666666666 \cdot t\_0\right) + 0.2 \cdot t\_1\right) + 0.047619047619047616 \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot t\_1\right)\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (* x x))) (t_1 (* (fabs x) (* (fabs x) t_0))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* 0.6666666666666666 t_0)) (* 0.2 t_1))
      (* 0.047619047619047616 (* (fabs x) (* (fabs x) t_1))))))))

C

double code(double x) {
	double t_0 = fabs(x) * (x * x);
	double t_1 = fabs(x) * (fabs(x) * t_0);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (fabs(x) * (fabs(x) * t_1))))));
}

Java

public static double code(double x) {
	double t_0 = Math.abs(x) * (x * x);
	double t_1 = Math.abs(x) * (Math.abs(x) * t_0);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (Math.abs(x) * (Math.abs(x) * t_1))))));
}

Python

def code(x):
	t_0 = math.fabs(x) * (x * x)
	t_1 = math.fabs(x) * (math.fabs(x) * t_0)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (math.fabs(x) * (math.fabs(x) * t_1))))))

Julia

function code(x)
	t_0 = Float64(abs(x) * Float64(x * x))
	t_1 = Float64(abs(x) * Float64(abs(x) * t_0))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(0.6666666666666666 * t_0)) + Float64(0.2 * t_1)) + Float64(0.047619047619047616 * Float64(abs(x) * Float64(abs(x) * t_1))))))
end

MATLAB

function tmp = code(x)
	t_0 = abs(x) * (x * x);
	t_1 = abs(x) * (abs(x) * t_0);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (abs(x) * (abs(x) * t_1))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$0), $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[(0.6666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.2 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(abs(x) * abs(Float64(Float64(Float64((x ^ 4.0) * Float64(0.2 + Float64(0.047619047619047616 * Float64(x * x)))) + 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[Power[x, 4.0], $MachinePrecision] * N[(0.2 + N[(0.047619047619047616 * N[(x * x), $MachinePrecision]), $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 99.8%

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

5. Taylor expanded in x around 0 99.8%

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

   1. *-commutative 99.8%

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

7. Simplified 99.8%

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

   1. pow2 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 3: 99.2% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(abs(x) * abs(Float64(Float64(fma(0.6666666666666666, Float64(x * x), 2.0) + Float64(0.047619047619047616 * (x ^ 6.0))) / 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[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around inf 98.8%

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

6. Final simplification 98.8%

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

Alternative 4: 99.0% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return Math.abs(x) * Math.abs(((2.0 + (Math.pow(x, 4.0) * (0.2 + (0.047619047619047616 * (x * x))))) / Math.sqrt(Math.PI)));
}

Python

def code(x):
	return math.fabs(x) * math.fabs(((2.0 + (math.pow(x, 4.0) * (0.2 + (0.047619047619047616 * (x * x))))) / math.sqrt(math.pi)))

Julia

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

MATLAB

function tmp = code(x)
	tmp = abs(x) * abs(((2.0 + ((x ^ 4.0) * (0.2 + (0.047619047619047616 * (x * x))))) / sqrt(pi)));
end

Wolfram

code[x_] := N[(N[Abs[x], $MachinePrecision] * N[Abs[N[(N[(2.0 + 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 99.8%

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

5. Taylor expanded in x around 0 99.8%

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

   1. *-commutative 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in x around 0 98.6%

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

   1. pow2 99.8%

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

10. Applied egg-rr 98.6%

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

11. Final simplification 98.6%

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

Alternative 5: 67.9% accurate, 4.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 2.0)
   (* (/ x (sqrt PI)) (fma 0.6666666666666666 (pow x 2.0) 2.0))
   (fabs (/ (* 0.047619047619047616 (pow x 7.0)) (sqrt PI)))))

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 2.0], N[(N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. associate-*r* 99.3%

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

      2. unpow2 99.3%

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

      3. sqr-abs 99.3%

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

      4. unpow3 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-*r* 99.3%

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

      7. distribute-rgt-in 99.3%

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

      8. fma-define 99.3%

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

      9. cube-mult 99.3%

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

      10. sqr-abs 99.3%

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

      11. unpow2 99.3%

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

      12. fma-define 99.3%

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

   7. Simplified 99.3%

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

      1. add-sqr-sqrt 98.7%

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

      2. fabs-sqr 98.7%

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

      3. add-sqr-sqrt 99.3%

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

      4. associate-*r* 99.3%

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

      5. *-commutative 99.3%

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

      6. add-sqr-sqrt 52.7%

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

      7. fabs-sqr 52.7%

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

      8. add-sqr-sqrt 54.8%

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

      9. sqrt-div 54.8%

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

      10. metadata-eval 54.8%

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

      11. un-div-inv 54.5%

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

   9. Applied egg-rr 54.5%

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

   if 2 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 97.6%

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

      1. *-commutative 97.6%

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

      2. *-commutative 97.6%

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

      3. associate-*r* 97.7%

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

      4. associate-*l* 97.7%

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

      5. *-commutative 97.7%

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

      6. associate-*l* 97.6%

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

      7. *-commutative 97.6%

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

   7. Simplified 97.6%

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

      1. add-sqr-sqrt 97.5%

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

      2. fabs-sqr 97.5%

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

      3. add-sqr-sqrt 97.6%

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

      4. expm1-log1p-u 96.6%

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

      5. expm1-undefine 96.6%

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

   9. Applied egg-rr 0.0%

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

       1. expm1-define 0.0%

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

       2. *-commutative 0.0%

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

       3. associate-*l* 0.0%

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

       4. *-commutative 0.0%

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

       5. associate-*r/ 0.0%

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

       6. pow-plus 0.0%

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

       7. metadata-eval 0.0%

          \[\leadsto \left|\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \frac{{x}^{\color{blue}{7}}}{\sqrt{\pi}}\right)\right)\right| \]

   11. Simplified 0.0%

       \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\right)\right)}\right| \]
   12. Step-by-step derivation

       1. expm1-log1p-u 97.7%

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

       2. associate-*r/ 97.7%

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

   13. Applied egg-rr 97.7%

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

Alternative 6: 68.0% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 2.0], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 98.7%

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

      1. add-sqr-sqrt 98.1%

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

      2. fabs-sqr 98.1%

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

      3. add-sqr-sqrt 98.7%

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

      4. associate-*r* 98.7%

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

      5. add-sqr-sqrt 52.3%

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

      6. fabs-sqr 52.3%

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

      7. add-sqr-sqrt 54.4%

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

      8. sqrt-div 54.4%

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

      9. metadata-eval 54.4%

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

      10. un-div-inv 54.4%

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

   7. Applied egg-rr 54.4%

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

   if 2 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 97.6%

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

      1. *-commutative 97.6%

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

      2. *-commutative 97.6%

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

      3. associate-*r* 97.7%

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

      4. associate-*l* 97.7%

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

      5. *-commutative 97.7%

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

      6. associate-*l* 97.6%

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

      7. *-commutative 97.6%

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

   7. Simplified 97.6%

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

      1. add-sqr-sqrt 97.5%

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

      2. fabs-sqr 97.5%

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

      3. add-sqr-sqrt 97.6%

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

      4. expm1-log1p-u 96.6%

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

      5. expm1-undefine 96.6%

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

   9. Applied egg-rr 0.0%

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

       1. expm1-define 0.0%

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

       2. *-commutative 0.0%

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

       3. associate-*l* 0.0%

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

       4. *-commutative 0.0%

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

       5. associate-*r/ 0.0%

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

       6. pow-plus 0.0%

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

       7. metadata-eval 0.0%

          \[\leadsto \left|\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \frac{{x}^{\color{blue}{7}}}{\sqrt{\pi}}\right)\right)\right| \]

   11. Simplified 0.0%

       \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\right)\right)}\right| \]
   12. Step-by-step derivation

       1. expm1-log1p-u 97.7%

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

       2. associate-*r/ 97.7%

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

   13. Applied egg-rr 97.7%

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

4. Final simplification 68.1%

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

Alternative 7: 98.9% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around inf 98.8%

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

6. Taylor expanded in x around 0 98.3%

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

7. Final simplification 98.3%

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

Alternative 8: 35.0% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = abs(Float64(0.047619047619047616 / Float64(sqrt(pi) / (x ^ 7.0))));
	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 = abs((0.047619047619047616 / (sqrt(pi) / (x ^ 7.0))));
	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[Abs[N[(0.047619047619047616 / N[(N[Sqrt[Pi], $MachinePrecision] / N[Power[x, 7.0], $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 99.8%

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

   5. Taylor expanded in x around 0 69.4%

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

      1. add-sqr-sqrt 69.0%

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

      2. fabs-sqr 69.0%

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

      3. add-sqr-sqrt 69.4%

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

      4. associate-*r* 69.4%

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

      5. add-sqr-sqrt 35.8%

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

      6. fabs-sqr 35.8%

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

      7. add-sqr-sqrt 37.3%

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

      8. sqrt-div 37.3%

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

      9. metadata-eval 37.3%

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

      10. un-div-inv 37.3%

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

   7. Applied egg-rr 37.3%

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

   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 99.8%

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

   5. Taylor expanded in x around inf 34.9%

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

      1. *-commutative 34.9%

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

      2. *-commutative 34.9%

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

      3. associate-*r* 34.9%

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

      4. associate-*l* 34.9%

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

      5. *-commutative 34.9%

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

      6. associate-*l* 34.9%

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

      7. *-commutative 34.9%

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

   7. Simplified 34.9%

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

      1. add-sqr-sqrt 34.9%

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

      2. fabs-sqr 34.9%

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

      3. add-sqr-sqrt 34.9%

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

      4. expm1-log1p-u 34.6%

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

      5. expm1-undefine 34.4%

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

   9. Applied egg-rr 3.9%

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

       1. expm1-define 4.0%

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

       2. *-commutative 4.0%

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

       3. associate-*l* 4.0%

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

       4. *-commutative 4.0%

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

       5. associate-*r/ 4.0%

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

       6. pow-plus 4.0%

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

       7. metadata-eval 4.0%

          \[\leadsto \left|\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \frac{{x}^{\color{blue}{7}}}{\sqrt{\pi}}\right)\right)\right| \]

   11. Simplified 4.0%

       \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\right)\right)}\right| \]
   12. Step-by-step derivation

       1. expm1-log1p-u 34.9%

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

       2. clear-num 34.9%

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

       3. un-div-inv 34.9%

          \[\leadsto \left|\color{blue}{\frac{0.047619047619047616}{\frac{\sqrt{\pi}}{{x}^{7}}}}\right| \]

   13. Applied egg-rr 34.9%

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

4. Final simplification 37.3%

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

Alternative 9: 35.0% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.fabs((0.047619047619047616 * (math.pow(x, 7.0) / math.sqrt(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 = abs(Float64(0.047619047619047616 * Float64((x ^ 7.0) / sqrt(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 = abs((0.047619047619047616 * ((x ^ 7.0) / sqrt(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[Abs[N[(0.047619047619047616 * N[(N[Power[x, 7.0], $MachinePrecision] / N[Sqrt[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 99.8%

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

   5. Taylor expanded in x around 0 69.4%

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

      1. add-sqr-sqrt 69.0%

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

      2. fabs-sqr 69.0%

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

      3. add-sqr-sqrt 69.4%

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

      4. associate-*r* 69.4%

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

      5. add-sqr-sqrt 35.8%

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

      6. fabs-sqr 35.8%

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

      7. add-sqr-sqrt 37.3%

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

      8. sqrt-div 37.3%

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

      9. metadata-eval 37.3%

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

      10. un-div-inv 37.3%

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

   7. Applied egg-rr 37.3%

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

   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 99.8%

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

   5. Taylor expanded in x around inf 34.9%

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

      1. *-commutative 34.9%

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

      2. *-commutative 34.9%

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

      3. associate-*r* 34.9%

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

      4. associate-*l* 34.9%

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

      5. *-commutative 34.9%

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

      6. associate-*l* 34.9%

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

      7. *-commutative 34.9%

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

   7. Simplified 34.9%

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

      1. pow1 34.9%

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

      2. associate-*r* 34.9%

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

      3. add-sqr-sqrt 2.2%

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

      4. fabs-sqr 2.2%

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

      5. add-sqr-sqrt 34.9%

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

      6. sqrt-div 34.9%

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

      7. metadata-eval 34.9%

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

      8. un-div-inv 34.9%

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

   9. Applied egg-rr 34.9%

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

       1. unpow1 34.9%

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

       2. associate-*r* 34.9%

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

       3. *-commutative 34.9%

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

       4. associate-*l* 35.0%

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

       5. associate-*r/ 34.9%

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

       6. pow-plus 34.9%

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

       7. metadata-eval 34.9%

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

   11. Simplified 34.9%

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

4. Final simplification 37.3%

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

Alternative 10: 35.0% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{6} \cdot \left(0.047619047619047616 \cdot \frac{x}{\sqrt{\pi}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (* x (/ 2.0 (sqrt PI)))
   (* (pow x 6.0) (* 0.047619047619047616 (/ x (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = pow(x, 6.0) * (0.047619047619047616 * (x / sqrt(((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 = Math.pow(x, 6.0) * (0.047619047619047616 * (x / Math.sqrt(Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.pow(x, 6.0) * (0.047619047619047616 * (x / math.sqrt(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((x ^ 6.0) * Float64(0.047619047619047616 * Float64(x / sqrt(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 = (x ^ 6.0) * (0.047619047619047616 * (x / sqrt(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[(N[Power[x, 6.0], $MachinePrecision] * N[(0.047619047619047616 * N[(x / N[Sqrt[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 99.8%

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

   5. Taylor expanded in x around 0 69.4%

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

      1. add-sqr-sqrt 69.0%

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

      2. fabs-sqr 69.0%

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

      3. add-sqr-sqrt 69.4%

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

      4. associate-*r* 69.4%

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

      5. add-sqr-sqrt 35.8%

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

      6. fabs-sqr 35.8%

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

      7. add-sqr-sqrt 37.3%

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

      8. sqrt-div 37.3%

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

      9. metadata-eval 37.3%

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

      10. un-div-inv 37.3%

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

   7. Applied egg-rr 37.3%

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

   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 99.8%

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

   5. Taylor expanded in x around inf 34.9%

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

      1. *-commutative 34.9%

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

      2. *-commutative 34.9%

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

      3. associate-*r* 34.9%

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

      4. associate-*l* 34.9%

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

      5. *-commutative 34.9%

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

      6. associate-*l* 34.9%

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

      7. *-commutative 34.9%

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

   7. Simplified 34.9%

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

      1. add-sqr-sqrt 34.9%

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

      2. fabs-sqr 34.9%

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

      3. add-sqr-sqrt 34.9%

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

      4. associate-*r* 34.9%

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

      5. add-sqr-sqrt 2.2%

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

      6. fabs-sqr 2.2%

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

      7. add-sqr-sqrt 3.9%

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

      8. sqrt-div 3.9%

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

      9. metadata-eval 3.9%

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

      10. un-div-inv 3.9%

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

   9. Applied egg-rr 3.9%

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

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{6} \cdot \left(0.047619047619047616 \cdot \frac{x}{\sqrt{\pi}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.0% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (* x (/ 2.0 (sqrt PI)))
   (* (pow x 7.0) (* 0.047619047619047616 (pow PI -0.5)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = pow(x, 7.0) * (0.047619047619047616 * pow(((double) M_PI), -0.5));
	}
	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 = Math.pow(x, 7.0) * (0.047619047619047616 * Math.pow(Math.PI, -0.5));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.pow(x, 7.0) * (0.047619047619047616 * math.pow(math.pi, -0.5))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64((x ^ 7.0) * Float64(0.047619047619047616 * (pi ^ -0.5)));
	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 = (x ^ 7.0) * (0.047619047619047616 * (pi ^ -0.5));
	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[(N[Power[x, 7.0], $MachinePrecision] * N[(0.047619047619047616 * N[Power[Pi, -0.5], $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 99.8%

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

   5. Taylor expanded in x around 0 69.4%

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

      1. add-sqr-sqrt 69.0%

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

      2. fabs-sqr 69.0%

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

      3. add-sqr-sqrt 69.4%

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

      4. associate-*r* 69.4%

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

      5. add-sqr-sqrt 35.8%

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

      6. fabs-sqr 35.8%

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

      7. add-sqr-sqrt 37.3%

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

      8. sqrt-div 37.3%

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

      9. metadata-eval 37.3%

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

      10. un-div-inv 37.3%

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

   7. Applied egg-rr 37.3%

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

   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 99.8%

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

   5. Taylor expanded in x around inf 34.9%

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

      1. *-commutative 34.9%

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

      2. *-commutative 34.9%

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

      3. associate-*r* 34.9%

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

      4. associate-*l* 34.9%

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

      5. *-commutative 34.9%

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

      6. associate-*l* 34.9%

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

      7. *-commutative 34.9%

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

   7. Simplified 34.9%

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

      1. add-sqr-sqrt 34.9%

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

      2. fabs-sqr 34.9%

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

      3. add-sqr-sqrt 34.9%

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

      4. associate-*l* 34.9%

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

      5. add-sqr-sqrt 2.2%

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

      6. fabs-sqr 2.2%

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

      7. add-sqr-sqrt 3.9%

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

      8. inv-pow 3.9%

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

      9. sqrt-pow1 3.9%

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

      10. metadata-eval 3.9%

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

   9. Applied egg-rr 3.9%

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

       1. *-commutative 3.9%

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

       2. associate-*r* 3.9%

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

       3. associate-*l* 3.9%

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

   11. Simplified 3.9%

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

4. Final simplification 37.3%

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

Alternative 12: 35.0% accurate, 16.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-21}:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(x \cdot x\right) \cdot \frac{4}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4e-21) (* x (/ 2.0 (sqrt PI))) (sqrt (* (* x x) (/ 4.0 PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 4e-21) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = sqrt(((x * x) * (4.0 / ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 4e-21) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.sqrt(((x * x) * (4.0 / Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4e-21:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.sqrt(((x * x) * (4.0 / math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4e-21)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = sqrt(Float64(Float64(x * x) * Float64(4.0 / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4e-21)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = sqrt(((x * x) * (4.0 / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4e-21], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x * x), $MachinePrecision] * N[(4.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.99999999999999963e-21

   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 99.8%

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

   5. Taylor expanded in x around 0 69.1%

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

      1. add-sqr-sqrt 68.7%

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

      2. fabs-sqr 68.7%

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

      3. add-sqr-sqrt 69.1%

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

      4. associate-*r* 69.1%

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

      5. add-sqr-sqrt 34.8%

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

      6. fabs-sqr 34.8%

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

      7. add-sqr-sqrt 36.4%

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

      8. sqrt-div 36.4%

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

      9. metadata-eval 36.4%

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

      10. un-div-inv 36.4%

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

   7. Applied egg-rr 36.4%

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

   if 3.99999999999999963e-21 < x

   1. Initial program 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in x around 0 82.2%

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

      1. add-sqr-sqrt 81.9%

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

      2. fabs-sqr 81.9%

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

      3. add-sqr-sqrt 82.2%

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

      4. associate-*r* 82.2%

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

      5. add-sqr-sqrt 82.2%

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

      6. fabs-sqr 82.2%

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

      7. add-sqr-sqrt 82.2%

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

      8. sqrt-div 82.2%

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

      9. metadata-eval 82.2%

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

      10. un-div-inv 82.2%

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

   7. Applied egg-rr 82.2%

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

      1. associate-*l/ 82.2%

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

      2. associate-*r/ 82.2%

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

      3. add-sqr-sqrt 81.9%

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

      4. sqrt-unprod 82.2%

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

      5. associate-*r/ 82.2%

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

      6. associate-*l/ 81.9%

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

      7. *-commutative 81.9%

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

      8. associate-*r/ 81.9%

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

      9. associate-*l/ 82.2%

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

      10. *-commutative 82.2%

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

      11. swap-sqr 81.9%

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

      12. pow2 81.9%

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

      13. frac-times 81.6%

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

      14. metadata-eval 81.6%

          \[\leadsto \sqrt{{x}^{2} \cdot \frac{\color{blue}{4}}{\sqrt{\pi} \cdot \sqrt{\pi}}} \]

      15. add-sqr-sqrt 82.5%

          \[\leadsto \sqrt{{x}^{2} \cdot \frac{4}{\color{blue}{\pi}}} \]

   9. Applied egg-rr 82.5%

      \[\leadsto \color{blue}{\sqrt{{x}^{2} \cdot \frac{4}{\pi}}} \]
   10. Step-by-step derivation

       1. pow2 99.4%

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

   11. Applied egg-rr 82.5%

       \[\leadsto \sqrt{\color{blue}{\left(x \cdot x\right)} \cdot \frac{4}{\pi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-21}:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(x \cdot x\right) \cdot \frac{4}{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 35.0% 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 99.8%

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

5. Taylor expanded in x around 0 69.4%

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

   1. add-sqr-sqrt 69.0%

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

   2. fabs-sqr 69.0%

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

   3. add-sqr-sqrt 69.4%

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

   4. associate-*r* 69.4%

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

   5. add-sqr-sqrt 35.8%

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

   6. fabs-sqr 35.8%

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

   7. add-sqr-sqrt 37.3%

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

   8. sqrt-div 37.3%

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

   9. metadata-eval 37.3%

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

   10. un-div-inv 37.3%

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

7. Applied egg-rr 37.3%

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

8. Final simplification 37.3%

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

Alternative 14: 4.1% accurate, 1849.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 69.4%

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

   1. expm1-log1p-u 69.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)\right|\right)\right)} \]

   2. expm1-undefine 6.6%

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

7. Applied egg-rr 4.2%

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

8. Taylor expanded in x around 0 4.4%

   \[\leadsto \color{blue}{1} - 1 \]
9. Step-by-step derivation

   1. metadata-eval 4.4%

      \[\leadsto \color{blue}{0} \]

10. Applied egg-rr 4.4%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024131 
(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)))))))