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

Percentage Accurate: 99.9% → 99.9%

Time: 11.3s

Alternatives: 8

Speedup: 1.5×

• 28.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 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, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = Float64(abs(x) * Float64(x * x))
	t_1 = Float64(Float64(x * x) * t_0)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(fma(2.0, abs(x), Float64(0.6666666666666666 * t_0)) + Float64(0.2 * t_1)) + Float64(0.047619047619047616 * Float64(Float64(x * 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[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision] + N[(0.6666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.2 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[(N[(x * x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 3: 99.4% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[Abs[N[(N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $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 99.4%

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

   1. expm1-log1p-u 99.2%

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

   2. expm1-udef 35.1%

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

6. Applied egg-rr 35.1%

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

   1. expm1-def 99.2%

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

   2. expm1-log1p 99.4%

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

   3. unpow1 99.4%

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

   4. sqr-pow 39.5%

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

   5. fabs-sqr 39.5%

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

   6. sqr-pow 99.4%

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

   7. unpow1 99.4%

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

8. Simplified 99.4%

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

   1. fma-udef 99.4%

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

10. Applied egg-rr 99.4%

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

11. Final simplification 99.4%

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

Alternative 4: 66.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.002:\\ \;\;\;\;\frac{{\pi}^{-0.5}}{\frac{\mathsf{fma}\left(-0.16666666666666666, {x}^{2}, 0.5\right)}{x}}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7} + 0.2 \cdot {x}^{5}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.002)
   (/ (pow PI -0.5) (/ (fma -0.16666666666666666 (pow x 2.0) 0.5) x))
   (fabs
    (*
     (sqrt (/ 1.0 PI))
     (+ (* 0.047619047619047616 (pow x 7.0)) (* 0.2 (pow x 5.0)))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.002) {
		tmp = pow(((double) M_PI), -0.5) / (fma(-0.16666666666666666, pow(x, 2.0), 0.5) / x);
	} else {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * ((0.047619047619047616 * pow(x, 7.0)) + (0.2 * pow(x, 5.0)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.002)
		tmp = Float64((pi ^ -0.5) / Float64(fma(-0.16666666666666666, (x ^ 2.0), 0.5) / x));
	else
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(0.047619047619047616 * (x ^ 7.0)) + Float64(0.2 * (x ^ 5.0)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.002], N[(N[Power[Pi, -0.5], $MachinePrecision] / N[(N[(-0.16666666666666666 * N[Power[x, 2.0], $MachinePrecision] + 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2e-3

   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 99.2%

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

   5. Taylor expanded in x around 0 99.2%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{-0.16666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\pi}\right) + 0.5 \cdot \sqrt{\pi}}\right|} \]
   6. Step-by-step derivation

      1. associate-*r* 99.2%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot \sqrt{\pi}} + 0.5 \cdot \sqrt{\pi}\right|} \]

      2. distribute-rgt-out 99.2%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot \left(-0.16666666666666666 \cdot {x}^{2} + 0.5\right)}\right|} \]

      3. *-commutative 99.2%

         \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \left(\color{blue}{{x}^{2} \cdot -0.16666666666666666} + 0.5\right)\right|} \]

   7. Simplified 99.2%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)}\right|} \]
   8. Step-by-step derivation

      1. div-inv 99.8%

         \[\leadsto \color{blue}{\left|x\right| \cdot \frac{1}{\left|\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)\right|}} \]

      2. add-sqr-sqrt 56.5%

         \[\leadsto \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \frac{1}{\left|\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)\right|} \]

      3. fabs-sqr 56.5%

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \frac{1}{\left|\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)\right|} \]

      4. add-sqr-sqrt 58.7%

         \[\leadsto \color{blue}{x} \cdot \frac{1}{\left|\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)\right|} \]

      5. add-sqr-sqrt 58.7%

         \[\leadsto x \cdot \frac{1}{\left|\color{blue}{\sqrt{\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)} \cdot \sqrt{\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)}}\right|} \]

      6. fabs-sqr 58.7%

         \[\leadsto x \cdot \frac{1}{\color{blue}{\sqrt{\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)} \cdot \sqrt{\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)}}} \]

      7. add-sqr-sqrt 58.7%

         \[\leadsto x \cdot \frac{1}{\color{blue}{\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)}} \]

      8. associate-/r* 58.7%

         \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{\sqrt{\pi}}}{{x}^{2} \cdot -0.16666666666666666 + 0.5}} \]

      9. pow1/2 58.7%

         \[\leadsto x \cdot \frac{\frac{1}{\color{blue}{{\pi}^{0.5}}}}{{x}^{2} \cdot -0.16666666666666666 + 0.5} \]

      10. pow-flip 58.7%

          \[\leadsto x \cdot \frac{\color{blue}{{\pi}^{\left(-0.5\right)}}}{{x}^{2} \cdot -0.16666666666666666 + 0.5} \]

      11. metadata-eval 58.7%

          \[\leadsto x \cdot \frac{{\pi}^{\color{blue}{-0.5}}}{{x}^{2} \cdot -0.16666666666666666 + 0.5} \]

      12. fma-def 58.7%

          \[\leadsto x \cdot \frac{{\pi}^{-0.5}}{\color{blue}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]

   9. Applied egg-rr 58.7%

      \[\leadsto \color{blue}{x \cdot \frac{{\pi}^{-0.5}}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]
   10. Step-by-step derivation

       1. *-commutative 58.7%

          \[\leadsto \color{blue}{\frac{{\pi}^{-0.5}}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)} \cdot x} \]

       2. associate-*l/ 58.7%

          \[\leadsto \color{blue}{\frac{{\pi}^{-0.5} \cdot x}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]

       3. associate-/l* 58.5%

          \[\leadsto \color{blue}{\frac{{\pi}^{-0.5}}{\frac{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}{x}}} \]

       4. fma-udef 58.5%

          \[\leadsto \frac{{\pi}^{-0.5}}{\frac{\color{blue}{{x}^{2} \cdot -0.16666666666666666 + 0.5}}{x}} \]

       5. *-commutative 58.5%

          \[\leadsto \frac{{\pi}^{-0.5}}{\frac{\color{blue}{-0.16666666666666666 \cdot {x}^{2}} + 0.5}{x}} \]

       6. fma-def 58.5%

          \[\leadsto \frac{{\pi}^{-0.5}}{\frac{\color{blue}{\mathsf{fma}\left(-0.16666666666666666, {x}^{2}, 0.5\right)}}{x}} \]

   11. Simplified 58.5%

       \[\leadsto \color{blue}{\frac{{\pi}^{-0.5}}{\frac{\mathsf{fma}\left(-0.16666666666666666, {x}^{2}, 0.5\right)}{x}}} \]

   if 2e-3 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 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 99.0%

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

   7. Simplified 99.0%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7} + 0.2 \cdot {x}^{5}\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.002:\\ \;\;\;\;\frac{{\pi}^{-0.5}}{\frac{\mathsf{fma}\left(-0.16666666666666666, {x}^{2}, 0.5\right)}{x}}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7} + 0.2 \cdot {x}^{5}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.6% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return Math.abs(((x / Math.sqrt(Math.PI)) * (2.0 + ((0.2 * Math.pow(x, 4.0)) + (0.047619047619047616 * Math.pow(x, 6.0))))));
}

Python

def code(x):
	return math.fabs(((x / math.sqrt(math.pi)) * (2.0 + ((0.2 * math.pow(x, 4.0)) + (0.047619047619047616 * math.pow(x, 6.0))))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = abs(((x / sqrt(pi)) * (2.0 + ((0.2 * (x ^ 4.0)) + (0.047619047619047616 * (x ^ 6.0))))));
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

   1. expm1-log1p-u 99.2%

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

   2. expm1-udef 35.1%

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

6. Applied egg-rr 35.1%

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

   1. expm1-def 99.2%

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

   2. expm1-log1p 99.4%

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

   3. unpow1 99.4%

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

   4. sqr-pow 39.5%

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

   5. fabs-sqr 39.5%

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

   6. sqr-pow 99.4%

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

   7. unpow1 99.4%

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

8. Simplified 99.4%

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

   1. fma-udef 99.4%

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

10. Applied egg-rr 99.4%

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

11. Taylor expanded in x around 0 98.9%

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

12. Final simplification 98.9%

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

Alternative 6: 33.4% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.002:\\ \;\;\;\;\frac{{\pi}^{-0.5}}{\frac{\mathsf{fma}\left(-0.16666666666666666, {x}^{2}, 0.5\right)}{x}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.002)
   (/ (pow PI -0.5) (/ (fma -0.16666666666666666 (pow x 2.0) 0.5) x))
   (* 0.047619047619047616 (/ (pow x 7.0) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.002) {
		tmp = pow(((double) M_PI), -0.5) / (fma(-0.16666666666666666, pow(x, 2.0), 0.5) / x);
	} else {
		tmp = 0.047619047619047616 * (pow(x, 7.0) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.002)
		tmp = Float64((pi ^ -0.5) / Float64(fma(-0.16666666666666666, (x ^ 2.0), 0.5) / x));
	else
		tmp = Float64(0.047619047619047616 * Float64((x ^ 7.0) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.002], N[(N[Power[Pi, -0.5], $MachinePrecision] / N[(N[(-0.16666666666666666 * N[Power[x, 2.0], $MachinePrecision] + 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[x, 7.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2e-3

   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 99.2%

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

   5. Taylor expanded in x around 0 99.2%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{-0.16666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\pi}\right) + 0.5 \cdot \sqrt{\pi}}\right|} \]
   6. Step-by-step derivation

      1. associate-*r* 99.2%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot \sqrt{\pi}} + 0.5 \cdot \sqrt{\pi}\right|} \]

      2. distribute-rgt-out 99.2%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot \left(-0.16666666666666666 \cdot {x}^{2} + 0.5\right)}\right|} \]

      3. *-commutative 99.2%

         \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \left(\color{blue}{{x}^{2} \cdot -0.16666666666666666} + 0.5\right)\right|} \]

   7. Simplified 99.2%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)}\right|} \]
   8. Step-by-step derivation

      1. div-inv 99.8%

         \[\leadsto \color{blue}{\left|x\right| \cdot \frac{1}{\left|\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)\right|}} \]

      2. add-sqr-sqrt 56.5%

         \[\leadsto \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \frac{1}{\left|\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)\right|} \]

      3. fabs-sqr 56.5%

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \frac{1}{\left|\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)\right|} \]

      4. add-sqr-sqrt 58.7%

         \[\leadsto \color{blue}{x} \cdot \frac{1}{\left|\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)\right|} \]

      5. add-sqr-sqrt 58.7%

         \[\leadsto x \cdot \frac{1}{\left|\color{blue}{\sqrt{\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)} \cdot \sqrt{\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)}}\right|} \]

      6. fabs-sqr 58.7%

         \[\leadsto x \cdot \frac{1}{\color{blue}{\sqrt{\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)} \cdot \sqrt{\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)}}} \]

      7. add-sqr-sqrt 58.7%

         \[\leadsto x \cdot \frac{1}{\color{blue}{\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)}} \]

      8. associate-/r* 58.7%

         \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{\sqrt{\pi}}}{{x}^{2} \cdot -0.16666666666666666 + 0.5}} \]

      9. pow1/2 58.7%

         \[\leadsto x \cdot \frac{\frac{1}{\color{blue}{{\pi}^{0.5}}}}{{x}^{2} \cdot -0.16666666666666666 + 0.5} \]

      10. pow-flip 58.7%

          \[\leadsto x \cdot \frac{\color{blue}{{\pi}^{\left(-0.5\right)}}}{{x}^{2} \cdot -0.16666666666666666 + 0.5} \]

      11. metadata-eval 58.7%

          \[\leadsto x \cdot \frac{{\pi}^{\color{blue}{-0.5}}}{{x}^{2} \cdot -0.16666666666666666 + 0.5} \]

      12. fma-def 58.7%

          \[\leadsto x \cdot \frac{{\pi}^{-0.5}}{\color{blue}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]

   9. Applied egg-rr 58.7%

      \[\leadsto \color{blue}{x \cdot \frac{{\pi}^{-0.5}}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]
   10. Step-by-step derivation

       1. *-commutative 58.7%

          \[\leadsto \color{blue}{\frac{{\pi}^{-0.5}}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)} \cdot x} \]

       2. associate-*l/ 58.7%

          \[\leadsto \color{blue}{\frac{{\pi}^{-0.5} \cdot x}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]

       3. associate-/l* 58.5%

          \[\leadsto \color{blue}{\frac{{\pi}^{-0.5}}{\frac{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}{x}}} \]

       4. fma-udef 58.5%

          \[\leadsto \frac{{\pi}^{-0.5}}{\frac{\color{blue}{{x}^{2} \cdot -0.16666666666666666 + 0.5}}{x}} \]

       5. *-commutative 58.5%

          \[\leadsto \frac{{\pi}^{-0.5}}{\frac{\color{blue}{-0.16666666666666666 \cdot {x}^{2}} + 0.5}{x}} \]

       6. fma-def 58.5%

          \[\leadsto \frac{{\pi}^{-0.5}}{\frac{\color{blue}{\mathsf{fma}\left(-0.16666666666666666, {x}^{2}, 0.5\right)}}{x}} \]

   11. Simplified 58.5%

       \[\leadsto \color{blue}{\frac{{\pi}^{-0.5}}{\frac{\mathsf{fma}\left(-0.16666666666666666, {x}^{2}, 0.5\right)}{x}}} \]

   if 2e-3 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 98.8%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}\right|} \]
   6. Step-by-step derivation

      1. *-commutative 98.8%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right) \cdot 21}\right|} \]

      2. associate-*l/ 98.8%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\frac{1 \cdot \sqrt{\pi}}{{x}^{6}}} \cdot 21\right|} \]

      3. *-lft-identity 98.8%

         \[\leadsto \frac{\left|x\right|}{\left|\frac{\color{blue}{\sqrt{\pi}}}{{x}^{6}} \cdot 21\right|} \]

      4. associate-*l/ 99.0%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\frac{\sqrt{\pi} \cdot 21}{{x}^{6}}}\right|} \]

   7. Simplified 99.0%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\frac{\sqrt{\pi} \cdot 21}{{x}^{6}}}\right|} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto \frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\left|\frac{\sqrt{\pi} \cdot 21}{{x}^{6}}\right|} \]

      2. fabs-sqr 0.0%

         \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\left|\frac{\sqrt{\pi} \cdot 21}{{x}^{6}}\right|} \]

      3. add-sqr-sqrt 0.1%

         \[\leadsto \frac{\color{blue}{x}}{\left|\frac{\sqrt{\pi} \cdot 21}{{x}^{6}}\right|} \]

      4. add-sqr-sqrt 0.1%

         \[\leadsto \frac{x}{\left|\color{blue}{\sqrt{\frac{\sqrt{\pi} \cdot 21}{{x}^{6}}} \cdot \sqrt{\frac{\sqrt{\pi} \cdot 21}{{x}^{6}}}}\right|} \]

      5. fabs-sqr 0.1%

         \[\leadsto \frac{x}{\color{blue}{\sqrt{\frac{\sqrt{\pi} \cdot 21}{{x}^{6}}} \cdot \sqrt{\frac{\sqrt{\pi} \cdot 21}{{x}^{6}}}}} \]

      6. add-sqr-sqrt 0.1%

         \[\leadsto \frac{x}{\color{blue}{\frac{\sqrt{\pi} \cdot 21}{{x}^{6}}}} \]

      7. associate-/l* 0.1%

         \[\leadsto \frac{x}{\color{blue}{\frac{\sqrt{\pi}}{\frac{{x}^{6}}{21}}}} \]

      8. associate-/r/ 0.1%

         \[\leadsto \color{blue}{\frac{x}{\sqrt{\pi}} \cdot \frac{{x}^{6}}{21}} \]

      9. div-inv 0.1%

         \[\leadsto \frac{x}{\sqrt{\pi}} \cdot \color{blue}{\left({x}^{6} \cdot \frac{1}{21}\right)} \]

      10. metadata-eval 0.1%

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

   9. Applied egg-rr 0.1%

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

       1. expm1-log1p-u 0.0%

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

       2. expm1-udef 0.0%

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

   11. Applied egg-rr 0.0%

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

       1. expm1-def 0.0%

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

       2. expm1-log1p 0.1%

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

       3. associate-*r* 0.1%

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

       4. *-commutative 0.1%

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

       5. associate-*l/ 0.1%

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

       6. *-commutative 0.1%

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

       7. pow-plus 0.1%

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

       8. metadata-eval 0.1%

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

   13. Simplified 0.1%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.002:\\ \;\;\;\;\frac{{\pi}^{-0.5}}{\frac{\mathsf{fma}\left(-0.16666666666666666, {x}^{2}, 0.5\right)}{x}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 33.4% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.002:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.002)
   (* x (/ 2.0 (sqrt PI)))
   (* 0.047619047619047616 (/ (pow x 7.0) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.002) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = 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) <= 0.002) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = 0.047619047619047616 * (Math.pow(x, 7.0) / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.002:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = 0.047619047619047616 * (math.pow(x, 7.0) / math.sqrt(math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.002)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(0.047619047619047616 * Float64((x ^ 7.0) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.002)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = 0.047619047619047616 * ((x ^ 7.0) / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.002], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[x, 7.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2e-3

   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 99.2%

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

   5. Taylor expanded in x around 0 99.2%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{-0.16666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\pi}\right) + 0.5 \cdot \sqrt{\pi}}\right|} \]
   6. Step-by-step derivation

      1. associate-*r* 99.2%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot \sqrt{\pi}} + 0.5 \cdot \sqrt{\pi}\right|} \]

      2. distribute-rgt-out 99.2%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot \left(-0.16666666666666666 \cdot {x}^{2} + 0.5\right)}\right|} \]

      3. *-commutative 99.2%

         \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \left(\color{blue}{{x}^{2} \cdot -0.16666666666666666} + 0.5\right)\right|} \]

   7. Simplified 99.2%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)}\right|} \]

   8. Taylor expanded in x around 0 99.2%

      \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \left(0.5 + -0.16666666666666666 \cdot {x}^{2}\right)\right|}} \]
   9. Step-by-step derivation

      1. fabs-mul 99.2%

         \[\leadsto \frac{\left|x\right|}{\color{blue}{\left|\sqrt{\pi}\right| \cdot \left|0.5 + -0.16666666666666666 \cdot {x}^{2}\right|}} \]

      2. +-commutative 99.2%

         \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi}\right| \cdot \left|\color{blue}{-0.16666666666666666 \cdot {x}^{2} + 0.5}\right|} \]

      3. *-commutative 99.2%

         \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi}\right| \cdot \left|\color{blue}{{x}^{2} \cdot -0.16666666666666666} + 0.5\right|} \]

      4. fma-udef 99.2%

         \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi}\right| \cdot \left|\color{blue}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}\right|} \]

      5. fabs-mul 99.2%

         \[\leadsto \frac{\left|x\right|}{\color{blue}{\left|\sqrt{\pi} \cdot \mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)\right|}} \]

      6. fabs-div 99.2%

         \[\leadsto \color{blue}{\left|\frac{x}{\sqrt{\pi} \cdot \mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}\right|} \]

      7. rem-square-sqrt 56.4%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{x}{\sqrt{\pi} \cdot \mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \cdot \sqrt{\frac{x}{\sqrt{\pi} \cdot \mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}}}\right| \]

      8. fabs-sqr 56.4%

         \[\leadsto \color{blue}{\sqrt{\frac{x}{\sqrt{\pi} \cdot \mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \cdot \sqrt{\frac{x}{\sqrt{\pi} \cdot \mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}}} \]

      9. rem-square-sqrt 58.3%

         \[\leadsto \color{blue}{\frac{x}{\sqrt{\pi} \cdot \mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]

      10. associate-/r* 58.3%

          \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{\pi}}}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]

      11. fma-udef 58.3%

          \[\leadsto \frac{\frac{x}{\sqrt{\pi}}}{\color{blue}{{x}^{2} \cdot -0.16666666666666666 + 0.5}} \]

      12. *-commutative 58.3%

          \[\leadsto \frac{\frac{x}{\sqrt{\pi}}}{\color{blue}{-0.16666666666666666 \cdot {x}^{2}} + 0.5} \]

      13. fma-def 58.3%

          \[\leadsto \frac{\frac{x}{\sqrt{\pi}}}{\color{blue}{\mathsf{fma}\left(-0.16666666666666666, {x}^{2}, 0.5\right)}} \]

   10. Simplified 58.3%

       \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{\pi}}}{\mathsf{fma}\left(-0.16666666666666666, {x}^{2}, 0.5\right)}} \]

   11. Taylor expanded in x around 0 58.3%

       \[\leadsto \frac{\frac{x}{\sqrt{\pi}}}{\color{blue}{0.5}} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 58.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{x}{\sqrt{\pi}}}{0.5}\right)\right)} \]

       2. expm1-udef 6.5%

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

       3. div-inv 6.5%

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

       4. metadata-eval 6.5%

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

   13. Applied egg-rr 6.5%

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

       1. expm1-def 58.3%

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

       2. expm1-log1p 58.3%

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

       3. metadata-eval 58.3%

          \[\leadsto \frac{x}{\sqrt{\pi}} \cdot \color{blue}{\frac{1}{0.5}} \]

       4. times-frac 58.3%

          \[\leadsto \color{blue}{\frac{x \cdot 1}{\sqrt{\pi} \cdot 0.5}} \]

       5. associate-*r/ 58.7%

          \[\leadsto \color{blue}{x \cdot \frac{1}{\sqrt{\pi} \cdot 0.5}} \]

       6. *-commutative 58.7%

          \[\leadsto x \cdot \frac{1}{\color{blue}{0.5 \cdot \sqrt{\pi}}} \]

       7. associate-/r* 58.7%

          \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{0.5}}{\sqrt{\pi}}} \]

       8. metadata-eval 58.7%

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

   15. Simplified 58.7%

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

   if 2e-3 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 98.8%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}\right|} \]
   6. Step-by-step derivation

      1. *-commutative 98.8%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right) \cdot 21}\right|} \]

      2. associate-*l/ 98.8%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\frac{1 \cdot \sqrt{\pi}}{{x}^{6}}} \cdot 21\right|} \]

      3. *-lft-identity 98.8%

         \[\leadsto \frac{\left|x\right|}{\left|\frac{\color{blue}{\sqrt{\pi}}}{{x}^{6}} \cdot 21\right|} \]

      4. associate-*l/ 99.0%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\frac{\sqrt{\pi} \cdot 21}{{x}^{6}}}\right|} \]

   7. Simplified 99.0%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\frac{\sqrt{\pi} \cdot 21}{{x}^{6}}}\right|} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto \frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\left|\frac{\sqrt{\pi} \cdot 21}{{x}^{6}}\right|} \]

      2. fabs-sqr 0.0%

         \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\left|\frac{\sqrt{\pi} \cdot 21}{{x}^{6}}\right|} \]

      3. add-sqr-sqrt 0.1%

         \[\leadsto \frac{\color{blue}{x}}{\left|\frac{\sqrt{\pi} \cdot 21}{{x}^{6}}\right|} \]

      4. add-sqr-sqrt 0.1%

         \[\leadsto \frac{x}{\left|\color{blue}{\sqrt{\frac{\sqrt{\pi} \cdot 21}{{x}^{6}}} \cdot \sqrt{\frac{\sqrt{\pi} \cdot 21}{{x}^{6}}}}\right|} \]

      5. fabs-sqr 0.1%

         \[\leadsto \frac{x}{\color{blue}{\sqrt{\frac{\sqrt{\pi} \cdot 21}{{x}^{6}}} \cdot \sqrt{\frac{\sqrt{\pi} \cdot 21}{{x}^{6}}}}} \]

      6. add-sqr-sqrt 0.1%

         \[\leadsto \frac{x}{\color{blue}{\frac{\sqrt{\pi} \cdot 21}{{x}^{6}}}} \]

      7. associate-/l* 0.1%

         \[\leadsto \frac{x}{\color{blue}{\frac{\sqrt{\pi}}{\frac{{x}^{6}}{21}}}} \]

      8. associate-/r/ 0.1%

         \[\leadsto \color{blue}{\frac{x}{\sqrt{\pi}} \cdot \frac{{x}^{6}}{21}} \]

      9. div-inv 0.1%

         \[\leadsto \frac{x}{\sqrt{\pi}} \cdot \color{blue}{\left({x}^{6} \cdot \frac{1}{21}\right)} \]

      10. metadata-eval 0.1%

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

   9. Applied egg-rr 0.1%

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

       1. expm1-log1p-u 0.0%

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

       2. expm1-udef 0.0%

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

   11. Applied egg-rr 0.0%

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

       1. expm1-def 0.0%

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

       2. expm1-log1p 0.1%

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

       3. associate-*r* 0.1%

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

       4. *-commutative 0.1%

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

       5. associate-*l/ 0.1%

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

       6. *-commutative 0.1%

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

       7. pow-plus 0.1%

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

       8. metadata-eval 0.1%

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

   13. Simplified 0.1%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.002:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 33.6% 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 99.4%

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

5. Taylor expanded in x around 0 70.1%

   \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{-0.16666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\pi}\right) + 0.5 \cdot \sqrt{\pi}}\right|} \]
6. Step-by-step derivation

   1. associate-*r* 70.1%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot \sqrt{\pi}} + 0.5 \cdot \sqrt{\pi}\right|} \]

   2. distribute-rgt-out 70.1%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot \left(-0.16666666666666666 \cdot {x}^{2} + 0.5\right)}\right|} \]

   3. *-commutative 70.1%

      \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \left(\color{blue}{{x}^{2} \cdot -0.16666666666666666} + 0.5\right)\right|} \]

7. Simplified 70.1%

   \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)}\right|} \]

8. Taylor expanded in x around 0 70.1%

   \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \left(0.5 + -0.16666666666666666 \cdot {x}^{2}\right)\right|}} \]
9. Step-by-step derivation

   1. fabs-mul 70.1%

      \[\leadsto \frac{\left|x\right|}{\color{blue}{\left|\sqrt{\pi}\right| \cdot \left|0.5 + -0.16666666666666666 \cdot {x}^{2}\right|}} \]

   2. +-commutative 70.1%

      \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi}\right| \cdot \left|\color{blue}{-0.16666666666666666 \cdot {x}^{2} + 0.5}\right|} \]

   3. *-commutative 70.1%

      \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi}\right| \cdot \left|\color{blue}{{x}^{2} \cdot -0.16666666666666666} + 0.5\right|} \]

   4. fma-udef 70.1%

      \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi}\right| \cdot \left|\color{blue}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}\right|} \]

   5. fabs-mul 70.1%

      \[\leadsto \frac{\left|x\right|}{\color{blue}{\left|\sqrt{\pi} \cdot \mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)\right|}} \]

   6. fabs-div 70.1%

      \[\leadsto \color{blue}{\left|\frac{x}{\sqrt{\pi} \cdot \mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}\right|} \]

   7. rem-square-sqrt 40.2%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{x}{\sqrt{\pi} \cdot \mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \cdot \sqrt{\frac{x}{\sqrt{\pi} \cdot \mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}}}\right| \]

   8. fabs-sqr 40.2%

      \[\leadsto \color{blue}{\sqrt{\frac{x}{\sqrt{\pi} \cdot \mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \cdot \sqrt{\frac{x}{\sqrt{\pi} \cdot \mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}}} \]

   9. rem-square-sqrt 41.5%

      \[\leadsto \color{blue}{\frac{x}{\sqrt{\pi} \cdot \mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]

   10. associate-/r* 41.5%

       \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{\pi}}}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]

   11. fma-udef 41.5%

       \[\leadsto \frac{\frac{x}{\sqrt{\pi}}}{\color{blue}{{x}^{2} \cdot -0.16666666666666666 + 0.5}} \]

   12. *-commutative 41.5%

       \[\leadsto \frac{\frac{x}{\sqrt{\pi}}}{\color{blue}{-0.16666666666666666 \cdot {x}^{2}} + 0.5} \]

   13. fma-def 41.5%

       \[\leadsto \frac{\frac{x}{\sqrt{\pi}}}{\color{blue}{\mathsf{fma}\left(-0.16666666666666666, {x}^{2}, 0.5\right)}} \]

10. Simplified 41.5%

    \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{\pi}}}{\mathsf{fma}\left(-0.16666666666666666, {x}^{2}, 0.5\right)}} \]

11. Taylor expanded in x around 0 40.9%

    \[\leadsto \frac{\frac{x}{\sqrt{\pi}}}{\color{blue}{0.5}} \]
12. Step-by-step derivation

    1. expm1-log1p-u 40.8%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{x}{\sqrt{\pi}}}{0.5}\right)\right)} \]

    2. expm1-udef 4.6%

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

    3. div-inv 4.6%

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

    4. metadata-eval 4.6%

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

13. Applied egg-rr 4.6%

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

    1. expm1-def 40.8%

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

    2. expm1-log1p 40.9%

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

    3. metadata-eval 40.9%

       \[\leadsto \frac{x}{\sqrt{\pi}} \cdot \color{blue}{\frac{1}{0.5}} \]

    4. times-frac 40.9%

       \[\leadsto \color{blue}{\frac{x \cdot 1}{\sqrt{\pi} \cdot 0.5}} \]

    5. associate-*r/ 41.1%

       \[\leadsto \color{blue}{x \cdot \frac{1}{\sqrt{\pi} \cdot 0.5}} \]

    6. *-commutative 41.1%

       \[\leadsto x \cdot \frac{1}{\color{blue}{0.5 \cdot \sqrt{\pi}}} \]

    7. associate-/r* 41.1%

       \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{0.5}}{\sqrt{\pi}}} \]

    8. metadata-eval 41.1%

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

15. Simplified 41.1%

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

16. Final simplification 41.1%

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

Reproduce

herbie shell --seed 2024011 
(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)))))))