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

Percentage Accurate: 99.8% → 99.8%

Time: 6.8s

Alternatives: 6

Speedup: 2.1×

• Unsound rule application detected in e-graph. Results may not be sound.

• 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, 2.1× speedup

Math

\[\begin{array}{l} \\ \left|{\pi}^{-0.5} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (pow PI -0.5)
   (fma
    0.047619047619047616
    (pow x 7.0)
    (fma 2.0 x (fma 0.6666666666666666 (pow x 3.0) (* 0.2 (pow x 5.0))))))))

C

double code(double x) {
	return fabs((pow(((double) M_PI), -0.5) * fma(0.047619047619047616, pow(x, 7.0), fma(2.0, x, fma(0.6666666666666666, pow(x, 3.0), (0.2 * pow(x, 5.0)))))));
}

Julia

function code(x)
	return abs(Float64((pi ^ -0.5) * fma(0.047619047619047616, (x ^ 7.0), fma(2.0, x, fma(0.6666666666666666, (x ^ 3.0), Float64(0.2 * (x ^ 5.0)))))))
end

Wolfram

code[x_] := N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision] + N[(2.0 * x + N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision] + N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. distribute-lft-in 99.9%

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

3. Applied egg-rr 99.9%

   \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right)}\right| \]
4. Step-by-step derivation

   1. distribute-lft-out 99.9%

      \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + 0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right)}\right| \]

   2. +-commutative 99.9%

      \[\leadsto \left|{\pi}^{-0.5} \cdot \color{blue}{\left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right) + \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right)}\right| \]

   3. fma-udef 99.9%

      \[\leadsto \left|{\pi}^{-0.5} \cdot \color{blue}{\mathsf{fma}\left(0.047619047619047616, x \cdot {x}^{6}, \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right)}\right| \]

   4. *-commutative 99.9%

      \[\leadsto \left|{\pi}^{-0.5} \cdot \mathsf{fma}\left(0.047619047619047616, \color{blue}{{x}^{6} \cdot x}, \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right)\right| \]

   5. pow-plus 99.9%

      \[\leadsto \left|{\pi}^{-0.5} \cdot \mathsf{fma}\left(0.047619047619047616, \color{blue}{{x}^{\left(6 + 1\right)}}, \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right)\right| \]

   6. metadata-eval 99.9%

      \[\leadsto \left|{\pi}^{-0.5} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{\color{blue}{7}}, \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right)\right| \]

5. Simplified 99.9%

   \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right)}\right| \]

6. Final simplification 99.9%

   \[\leadsto \left|{\pi}^{-0.5} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right)\right| \]

Alternative 2: 99.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \left|{\pi}^{-0.5} \cdot \left(\left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right) + x \cdot 2\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (pow PI -0.5)
   (+
    (+ (* 0.2 (pow x 5.0)) (* 0.047619047619047616 (pow x 7.0)))
    (* x 2.0)))))

C

double code(double x) {
	return fabs((pow(((double) M_PI), -0.5) * (((0.2 * pow(x, 5.0)) + (0.047619047619047616 * pow(x, 7.0))) + (x * 2.0))));
}

Java

public static double code(double x) {
	return Math.abs((Math.pow(Math.PI, -0.5) * (((0.2 * Math.pow(x, 5.0)) + (0.047619047619047616 * Math.pow(x, 7.0))) + (x * 2.0))));
}

Python

def code(x):
	return math.fabs((math.pow(math.pi, -0.5) * (((0.2 * math.pow(x, 5.0)) + (0.047619047619047616 * math.pow(x, 7.0))) + (x * 2.0))))

Julia

function code(x)
	return abs(Float64((pi ^ -0.5) * Float64(Float64(Float64(0.2 * (x ^ 5.0)) + Float64(0.047619047619047616 * (x ^ 7.0))) + Float64(x * 2.0))))
end

MATLAB

function tmp = code(x)
	tmp = abs(((pi ^ -0.5) * (((0.2 * (x ^ 5.0)) + (0.047619047619047616 * (x ^ 7.0))) + (x * 2.0))));
end

Wolfram

code[x_] := N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(N[(N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. distribute-lft-in 99.9%

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

3. Applied egg-rr 99.9%

   \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right)}\right| \]
4. Step-by-step derivation

   1. +-commutative 99.9%

      \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right) + {\pi}^{-0.5} \cdot \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)}\right| \]

   2. distribute-lft-out 99.9%

      \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right) + \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right)}\right| \]

   3. fma-udef 99.9%

      \[\leadsto \left|{\pi}^{-0.5} \cdot \color{blue}{\mathsf{fma}\left(0.047619047619047616, x \cdot {x}^{6}, \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right)}\right| \]

5. Simplified 99.9%

   \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \mathsf{fma}\left(0.047619047619047616, x \cdot {x}^{6}, \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right)}\right| \]

6. Taylor expanded in x around inf 99.5%

   \[\leadsto \left|{\pi}^{-0.5} \cdot \mathsf{fma}\left(0.047619047619047616, x \cdot {x}^{6}, \mathsf{fma}\left(2, x, \color{blue}{0.2 \cdot {x}^{5}}\right)\right)\right| \]

7. Taylor expanded in x around 0 99.5%

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

8. Final simplification 99.5%

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

Alternative 3: 99.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6:\\ \;\;\;\;\left|\frac{0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.6)
   (fabs
    (/ (+ (* 0.2 (pow x 5.0)) (* 0.047619047619047616 (pow x 7.0))) (sqrt PI)))
   (fabs (* (pow PI -0.5) (* x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.6) {
		tmp = fabs((((0.2 * pow(x, 5.0)) + (0.047619047619047616 * pow(x, 7.0))) / sqrt(((double) M_PI))));
	} else {
		tmp = fabs((pow(((double) M_PI), -0.5) * (x * 2.0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.6) {
		tmp = Math.abs((((0.2 * Math.pow(x, 5.0)) + (0.047619047619047616 * Math.pow(x, 7.0))) / Math.sqrt(Math.PI)));
	} else {
		tmp = Math.abs((Math.pow(Math.PI, -0.5) * (x * 2.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.6:
		tmp = math.fabs((((0.2 * math.pow(x, 5.0)) + (0.047619047619047616 * math.pow(x, 7.0))) / math.sqrt(math.pi)))
	else:
		tmp = math.fabs((math.pow(math.pi, -0.5) * (x * 2.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.6)
		tmp = abs(Float64(Float64(Float64(0.2 * (x ^ 5.0)) + Float64(0.047619047619047616 * (x ^ 7.0))) / sqrt(pi)));
	else
		tmp = abs(Float64((pi ^ -0.5) * Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.6)
		tmp = abs((((0.2 * (x ^ 5.0)) + (0.047619047619047616 * (x ^ 7.0))) / sqrt(pi)));
	else
		tmp = abs(((pi ^ -0.5) * (x * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.6], N[Abs[N[(N[(N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6000000000000001

   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. Step-by-step derivation

      1. distribute-rgt-in 99.8%

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

   3. Applied egg-rr 99.9%

      \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)}{\sqrt{\pi}} + \frac{0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)}{\sqrt{\pi}}}\right| \]
   4. Step-by-step derivation

      1. +-commutative 99.9%

         \[\leadsto \left|\color{blue}{\frac{0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)}{\sqrt{\pi}} + \frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)}{\sqrt{\pi}}}\right| \]

      2. *-rgt-identity 99.9%

         \[\leadsto \left|\color{blue}{\frac{0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)}{\sqrt{\pi}} \cdot 1} + \frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)}{\sqrt{\pi}}\right| \]

      3. associate-*l/ 99.9%

         \[\leadsto \left|\color{blue}{\frac{\left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right) \cdot 1}{\sqrt{\pi}}} + \frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)}{\sqrt{\pi}}\right| \]

      4. *-lft-identity 99.9%

         \[\leadsto \left|\frac{\left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right) \cdot 1}{\color{blue}{1 \cdot \sqrt{\pi}}} + \frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)}{\sqrt{\pi}}\right| \]

      5. times-frac 99.9%

         \[\leadsto \left|\color{blue}{\frac{0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)}{1} \cdot \frac{1}{\sqrt{\pi}}} + \frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)}{\sqrt{\pi}}\right| \]

      6. /-rgt-identity 99.9%

         \[\leadsto \left|\color{blue}{\left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right)} \cdot \frac{1}{\sqrt{\pi}} + \frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)}{\sqrt{\pi}}\right| \]

      7. *-rgt-identity 99.9%

         \[\leadsto \left|\left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right) \cdot \frac{1}{\sqrt{\pi}} + \frac{\color{blue}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) \cdot 1}}{\sqrt{\pi}}\right| \]

      8. *-lft-identity 99.9%

         \[\leadsto \left|\left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right) \cdot \frac{1}{\sqrt{\pi}} + \frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) \cdot 1}{\color{blue}{1 \cdot \sqrt{\pi}}}\right| \]

      9. times-frac 99.9%

         \[\leadsto \left|\left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right) \cdot \frac{1}{\sqrt{\pi}} + \color{blue}{\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)}{1} \cdot \frac{1}{\sqrt{\pi}}}\right| \]

      10. /-rgt-identity 99.9%

          \[\leadsto \left|\left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right) \cdot \frac{1}{\sqrt{\pi}} + \color{blue}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)} \cdot \frac{1}{\sqrt{\pi}}\right| \]

   5. Simplified 99.9%

      \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}}\right| \]

   6. Taylor expanded in x around inf 99.1%

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

   if -1.6000000000000001 < x

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Step-by-step derivation

      1. distribute-lft-in 99.9%

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

   3. Applied egg-rr 99.9%

      \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right)}\right| \]
   4. Step-by-step derivation

      1. +-commutative 99.9%

         \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right) + {\pi}^{-0.5} \cdot \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)}\right| \]

      2. distribute-lft-out 99.9%

         \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right) + \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right)}\right| \]

      3. fma-udef 99.9%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \color{blue}{\mathsf{fma}\left(0.047619047619047616, x \cdot {x}^{6}, \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right)}\right| \]

   5. Simplified 99.9%

      \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \mathsf{fma}\left(0.047619047619047616, x \cdot {x}^{6}, \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right)}\right| \]

   6. Taylor expanded in x around 0 99.7%

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

      1. associate-*r* 99.7%

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

      2. *-commutative 99.7%

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

      3. unpow-1 99.7%

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

      4. metadata-eval 99.7%

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

      5. pow-sqr 99.7%

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

      6. rem-sqrt-square 99.7%

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

      7. metadata-eval 99.7%

         \[\leadsto \left|\left|{\pi}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \cdot \left(2 \cdot x\right)\right| \]

      8. pow-sqr 99.7%

         \[\leadsto \left|\left|\color{blue}{{\pi}^{-0.25} \cdot {\pi}^{-0.25}}\right| \cdot \left(2 \cdot x\right)\right| \]

      9. fabs-sqr 99.7%

         \[\leadsto \left|\color{blue}{\left({\pi}^{-0.25} \cdot {\pi}^{-0.25}\right)} \cdot \left(2 \cdot x\right)\right| \]

      10. pow-sqr 99.7%

          \[\leadsto \left|\color{blue}{{\pi}^{\left(2 \cdot -0.25\right)}} \cdot \left(2 \cdot x\right)\right| \]

      11. metadata-eval 99.7%

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

      12. *-commutative 99.7%

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

   8. Simplified 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6:\\ \;\;\;\;\left|\frac{0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right|\\ \end{array} \]

Alternative 4: 98.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.86:\\ \;\;\;\;\left|\frac{0.047619047619047616}{\sqrt{\pi} \cdot {x}^{-7}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.86)
   (fabs (/ 0.047619047619047616 (* (sqrt PI) (pow x -7.0))))
   (fabs (* (pow PI -0.5) (* x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.86) {
		tmp = fabs((0.047619047619047616 / (sqrt(((double) M_PI)) * pow(x, -7.0))));
	} else {
		tmp = fabs((pow(((double) M_PI), -0.5) * (x * 2.0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.86) {
		tmp = Math.abs((0.047619047619047616 / (Math.sqrt(Math.PI) * Math.pow(x, -7.0))));
	} else {
		tmp = Math.abs((Math.pow(Math.PI, -0.5) * (x * 2.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.86:
		tmp = math.fabs((0.047619047619047616 / (math.sqrt(math.pi) * math.pow(x, -7.0))))
	else:
		tmp = math.fabs((math.pow(math.pi, -0.5) * (x * 2.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.86)
		tmp = abs(Float64(0.047619047619047616 / Float64(sqrt(pi) * (x ^ -7.0))));
	else
		tmp = abs(Float64((pi ^ -0.5) * Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.86)
		tmp = abs((0.047619047619047616 / (sqrt(pi) * (x ^ -7.0))));
	else
		tmp = abs(((pi ^ -0.5) * (x * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.86], N[Abs[N[(0.047619047619047616 / N[(N[Sqrt[Pi], $MachinePrecision] * N[Power[x, -7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8600000000000001

   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. Step-by-step derivation

      1. distribute-rgt-in 99.8%

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

   3. Applied egg-rr 99.9%

      \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)}{\sqrt{\pi}} + \frac{0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)}{\sqrt{\pi}}}\right| \]
   4. Step-by-step derivation

      1. +-commutative 99.9%

         \[\leadsto \left|\color{blue}{\frac{0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)}{\sqrt{\pi}} + \frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)}{\sqrt{\pi}}}\right| \]

      2. *-rgt-identity 99.9%

         \[\leadsto \left|\color{blue}{\frac{0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)}{\sqrt{\pi}} \cdot 1} + \frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)}{\sqrt{\pi}}\right| \]

      3. associate-*l/ 99.9%

         \[\leadsto \left|\color{blue}{\frac{\left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right) \cdot 1}{\sqrt{\pi}}} + \frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)}{\sqrt{\pi}}\right| \]

      4. *-lft-identity 99.9%

         \[\leadsto \left|\frac{\left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right) \cdot 1}{\color{blue}{1 \cdot \sqrt{\pi}}} + \frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)}{\sqrt{\pi}}\right| \]

      5. times-frac 99.9%

         \[\leadsto \left|\color{blue}{\frac{0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)}{1} \cdot \frac{1}{\sqrt{\pi}}} + \frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)}{\sqrt{\pi}}\right| \]

      6. /-rgt-identity 99.9%

         \[\leadsto \left|\color{blue}{\left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right)} \cdot \frac{1}{\sqrt{\pi}} + \frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)}{\sqrt{\pi}}\right| \]

      7. *-rgt-identity 99.9%

         \[\leadsto \left|\left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right) \cdot \frac{1}{\sqrt{\pi}} + \frac{\color{blue}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) \cdot 1}}{\sqrt{\pi}}\right| \]

      8. *-lft-identity 99.9%

         \[\leadsto \left|\left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right) \cdot \frac{1}{\sqrt{\pi}} + \frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) \cdot 1}{\color{blue}{1 \cdot \sqrt{\pi}}}\right| \]

      9. times-frac 99.9%

         \[\leadsto \left|\left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right) \cdot \frac{1}{\sqrt{\pi}} + \color{blue}{\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)}{1} \cdot \frac{1}{\sqrt{\pi}}}\right| \]

      10. /-rgt-identity 99.9%

          \[\leadsto \left|\left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right) \cdot \frac{1}{\sqrt{\pi}} + \color{blue}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)} \cdot \frac{1}{\sqrt{\pi}}\right| \]

   5. Simplified 99.9%

      \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}}\right| \]

   6. Taylor expanded in x around inf 98.8%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-udef 0.0%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right)} - 1}\right| \]

   8. Applied egg-rr 0.0%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right)} - 1}\right| \]
   9. Step-by-step derivation

      1. expm1-def 0.0%

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

      2. expm1-log1p 98.8%

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

      3. associate-/l* 98.8%

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

   10. Simplified 98.8%

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

       1. div-inv 98.8%

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

       2. pow-flip 98.8%

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

       3. metadata-eval 98.8%

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

   12. Applied egg-rr 98.8%

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

   if -1.8600000000000001 < x

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Step-by-step derivation

      1. distribute-lft-in 99.9%

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

   3. Applied egg-rr 99.9%

      \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right)}\right| \]
   4. Step-by-step derivation

      1. +-commutative 99.9%

         \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right) + {\pi}^{-0.5} \cdot \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)}\right| \]

      2. distribute-lft-out 99.9%

         \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right) + \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right)}\right| \]

      3. fma-udef 99.9%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \color{blue}{\mathsf{fma}\left(0.047619047619047616, x \cdot {x}^{6}, \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right)}\right| \]

   5. Simplified 99.9%

      \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \mathsf{fma}\left(0.047619047619047616, x \cdot {x}^{6}, \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right)}\right| \]

   6. Taylor expanded in x around 0 99.7%

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

      1. associate-*r* 99.7%

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

      2. *-commutative 99.7%

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

      3. unpow-1 99.7%

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

      4. metadata-eval 99.7%

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

      5. pow-sqr 99.7%

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

      6. rem-sqrt-square 99.7%

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

      7. metadata-eval 99.7%

         \[\leadsto \left|\left|{\pi}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \cdot \left(2 \cdot x\right)\right| \]

      8. pow-sqr 99.7%

         \[\leadsto \left|\left|\color{blue}{{\pi}^{-0.25} \cdot {\pi}^{-0.25}}\right| \cdot \left(2 \cdot x\right)\right| \]

      9. fabs-sqr 99.7%

         \[\leadsto \left|\color{blue}{\left({\pi}^{-0.25} \cdot {\pi}^{-0.25}\right)} \cdot \left(2 \cdot x\right)\right| \]

      10. pow-sqr 99.7%

          \[\leadsto \left|\color{blue}{{\pi}^{\left(2 \cdot -0.25\right)}} \cdot \left(2 \cdot x\right)\right| \]

      11. metadata-eval 99.7%

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

      12. *-commutative 99.7%

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

   8. Simplified 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.86:\\ \;\;\;\;\left|\frac{0.047619047619047616}{\sqrt{\pi} \cdot {x}^{-7}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right|\\ \end{array} \]

Alternative 5: 67.2% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. distribute-lft-in 99.9%

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

3. Applied egg-rr 99.9%

   \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right)}\right| \]
4. Step-by-step derivation

   1. +-commutative 99.9%

      \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right) + {\pi}^{-0.5} \cdot \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)}\right| \]

   2. distribute-lft-out 99.9%

      \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right) + \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right)}\right| \]

   3. fma-udef 99.9%

      \[\leadsto \left|{\pi}^{-0.5} \cdot \color{blue}{\mathsf{fma}\left(0.047619047619047616, x \cdot {x}^{6}, \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right)}\right| \]

5. Simplified 99.9%

   \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \mathsf{fma}\left(0.047619047619047616, x \cdot {x}^{6}, \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right)}\right| \]

6. Taylor expanded in x around 0 68.6%

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

   1. associate-*r* 68.6%

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

   2. *-commutative 68.6%

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

   3. unpow-1 68.6%

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

   4. metadata-eval 68.6%

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

   5. pow-sqr 68.6%

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

   6. rem-sqrt-square 68.6%

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

   7. metadata-eval 68.6%

      \[\leadsto \left|\left|{\pi}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \cdot \left(2 \cdot x\right)\right| \]

   8. pow-sqr 68.6%

      \[\leadsto \left|\left|\color{blue}{{\pi}^{-0.25} \cdot {\pi}^{-0.25}}\right| \cdot \left(2 \cdot x\right)\right| \]

   9. fabs-sqr 68.6%

      \[\leadsto \left|\color{blue}{\left({\pi}^{-0.25} \cdot {\pi}^{-0.25}\right)} \cdot \left(2 \cdot x\right)\right| \]

   10. pow-sqr 68.6%

       \[\leadsto \left|\color{blue}{{\pi}^{\left(2 \cdot -0.25\right)}} \cdot \left(2 \cdot x\right)\right| \]

   11. metadata-eval 68.6%

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

   12. *-commutative 68.6%

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

8. Simplified 68.6%

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

9. Final simplification 68.6%

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

Alternative 6: 66.8% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. distribute-rgt-in 99.9%

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

3. Applied egg-rr 99.4%

   \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)}{\sqrt{\pi}} + \frac{0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)}{\sqrt{\pi}}}\right| \]
4. Step-by-step derivation

   1. +-commutative 99.4%

      \[\leadsto \left|\color{blue}{\frac{0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)}{\sqrt{\pi}} + \frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)}{\sqrt{\pi}}}\right| \]

   2. *-rgt-identity 99.4%

      \[\leadsto \left|\color{blue}{\frac{0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)}{\sqrt{\pi}} \cdot 1} + \frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)}{\sqrt{\pi}}\right| \]

   3. associate-*l/ 99.4%

      \[\leadsto \left|\color{blue}{\frac{\left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right) \cdot 1}{\sqrt{\pi}}} + \frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)}{\sqrt{\pi}}\right| \]

   4. *-lft-identity 99.4%

      \[\leadsto \left|\frac{\left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right) \cdot 1}{\color{blue}{1 \cdot \sqrt{\pi}}} + \frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)}{\sqrt{\pi}}\right| \]

   5. times-frac 99.4%

      \[\leadsto \left|\color{blue}{\frac{0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)}{1} \cdot \frac{1}{\sqrt{\pi}}} + \frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)}{\sqrt{\pi}}\right| \]

   6. /-rgt-identity 99.4%

      \[\leadsto \left|\color{blue}{\left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right)} \cdot \frac{1}{\sqrt{\pi}} + \frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)}{\sqrt{\pi}}\right| \]

   7. *-rgt-identity 99.4%

      \[\leadsto \left|\left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right) \cdot \frac{1}{\sqrt{\pi}} + \frac{\color{blue}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) \cdot 1}}{\sqrt{\pi}}\right| \]

   8. *-lft-identity 99.4%

      \[\leadsto \left|\left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right) \cdot \frac{1}{\sqrt{\pi}} + \frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) \cdot 1}{\color{blue}{1 \cdot \sqrt{\pi}}}\right| \]

   9. times-frac 99.9%

      \[\leadsto \left|\left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right) \cdot \frac{1}{\sqrt{\pi}} + \color{blue}{\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)}{1} \cdot \frac{1}{\sqrt{\pi}}}\right| \]

   10. /-rgt-identity 99.9%

       \[\leadsto \left|\left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right) \cdot \frac{1}{\sqrt{\pi}} + \color{blue}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)} \cdot \frac{1}{\sqrt{\pi}}\right| \]

5. Simplified 99.4%

   \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}}\right| \]

6. Taylor expanded in x around 0 68.1%

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

   1. *-commutative 68.1%

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

8. Simplified 68.1%

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

9. Final simplification 68.1%

   \[\leadsto \left|\frac{x \cdot 2}{\sqrt{\pi}}\right| \]

Reproduce

herbie shell --seed 2023174 
(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)))))))