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

Percentage Accurate: 99.8% → 99.9%

Time: 9.2s

Alternatives: 9

Speedup: 3.1×

• 29.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \left|\left({\pi}^{-0.5} \cdot x\right) \cdot \left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (* (pow PI -0.5) x)
   (+
    (+ (* 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(((pow(((double) M_PI), -0.5) * x) * (((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((pi ^ -0.5) * x) * 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[(N[Power[Pi, -0.5], $MachinePrecision] * x), $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 99.9%

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

3. Simplified 99.4%

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

   1. clear-num 99.4%

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

   2. associate-/r/ 99.9%

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

   3. pow1/2 99.9%

      \[\leadsto \left|\left(\frac{1}{\color{blue}{{\pi}^{0.5}}} \cdot \left|x\right|\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   4. pow-flip 99.9%

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

   5. metadata-eval 99.9%

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

   6. add-sqr-sqrt 32.9%

      \[\leadsto \left|\left({\pi}^{-0.5} \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   7. fabs-sqr 32.9%

      \[\leadsto \left|\left({\pi}^{-0.5} \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   8. add-sqr-sqrt 99.9%

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

5. Applied egg-rr 99.9%

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

   1. metadata-eval 99.9%

      \[\leadsto \left|\left({\pi}^{-0.5} \cdot x\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{1}{5}}, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   2. fma-udef 99.9%

      \[\leadsto \left|\left({\pi}^{-0.5} \cdot x\right) \cdot \left(\color{blue}{\left(\frac{1}{5} \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   3. metadata-eval 99.9%

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

7. Applied egg-rr 99.9%

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

8. Final simplification 99.9%

   \[\leadsto \left|\left({\pi}^{-0.5} \cdot x\right) \cdot \left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

Alternative 2: 99.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

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. Step-by-step derivation

   1. expm1-log1p-u 97.9%

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

   2. expm1-udef 38.4%

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

   3. add-sqr-sqrt 2.2%

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

   4. fabs-sqr 2.2%

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

   5. add-sqr-sqrt 4.9%

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

5. Applied egg-rr 5.8%

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

   1. expm1-def 64.4%

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

   2. expm1-log1p 98.0%

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

7. Simplified 99.4%

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

   1. metadata-eval 99.9%

      \[\leadsto \left|\left({\pi}^{-0.5} \cdot x\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{1}{5}}, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   2. fma-udef 99.9%

      \[\leadsto \left|\left({\pi}^{-0.5} \cdot x\right) \cdot \left(\color{blue}{\left(\frac{1}{5} \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   3. metadata-eval 99.9%

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

9. Applied egg-rr 99.4%

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

10. Final simplification 99.4%

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

Alternative 3: 99.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \left|\left({\pi}^{-0.5} \cdot x\right) \cdot \left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + 2\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (* (pow PI -0.5) x)
   (+ (+ (* 0.2 (pow x 4.0)) (* 0.047619047619047616 (pow x 6.0))) 2.0))))

C

double code(double x) {
	return fabs(((pow(((double) M_PI), -0.5) * x) * (((0.2 * pow(x, 4.0)) + (0.047619047619047616 * pow(x, 6.0))) + 2.0)));
}

Java

public static double code(double x) {
	return Math.abs(((Math.pow(Math.PI, -0.5) * x) * (((0.2 * Math.pow(x, 4.0)) + (0.047619047619047616 * Math.pow(x, 6.0))) + 2.0)));
}

Python

def code(x):
	return math.fabs(((math.pow(math.pi, -0.5) * x) * (((0.2 * math.pow(x, 4.0)) + (0.047619047619047616 * math.pow(x, 6.0))) + 2.0)))

Julia

function code(x)
	return abs(Float64(Float64((pi ^ -0.5) * x) * Float64(Float64(Float64(0.2 * (x ^ 4.0)) + Float64(0.047619047619047616 * (x ^ 6.0))) + 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = abs((((pi ^ -0.5) * x) * (((0.2 * (x ^ 4.0)) + (0.047619047619047616 * (x ^ 6.0))) + 2.0)));
end

Wolfram

code[x_] := N[Abs[N[(N[(N[Power[Pi, -0.5], $MachinePrecision] * x), $MachinePrecision] * N[(N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

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. Step-by-step derivation

   1. clear-num 99.4%

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

   2. associate-/r/ 99.9%

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

   3. pow1/2 99.9%

      \[\leadsto \left|\left(\frac{1}{\color{blue}{{\pi}^{0.5}}} \cdot \left|x\right|\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   4. pow-flip 99.9%

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

   5. metadata-eval 99.9%

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

   6. add-sqr-sqrt 32.9%

      \[\leadsto \left|\left({\pi}^{-0.5} \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   7. fabs-sqr 32.9%

      \[\leadsto \left|\left({\pi}^{-0.5} \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   8. add-sqr-sqrt 99.9%

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

5. Applied egg-rr 99.9%

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

   1. metadata-eval 99.9%

      \[\leadsto \left|\left({\pi}^{-0.5} \cdot x\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{1}{5}}, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   2. fma-udef 99.9%

      \[\leadsto \left|\left({\pi}^{-0.5} \cdot x\right) \cdot \left(\color{blue}{\left(\frac{1}{5} \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   3. metadata-eval 99.9%

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

7. Applied egg-rr 99.9%

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

8. Taylor expanded in x around 0 98.8%

   \[\leadsto \left|\left({\pi}^{-0.5} \cdot x\right) \cdot \left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + \color{blue}{2}\right)\right| \]

9. Final simplification 98.8%

   \[\leadsto \left|\left({\pi}^{-0.5} \cdot x\right) \cdot \left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + 2\right)\right| \]

Alternative 4: 98.9% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

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. Taylor expanded in x around inf 98.4%

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

5. Taylor expanded in x around 0 98.0%

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

   1. expm1-log1p-u 97.8%

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

   2. expm1-udef 38.4%

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

   3. associate-*l/ 38.4%

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

   4. add-sqr-sqrt 2.2%

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

   5. fabs-sqr 2.2%

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

   6. add-sqr-sqrt 5.0%

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

   7. *-commutative 5.0%

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

   8. associate-*r/ 5.0%

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

   9. fma-def 5.0%

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

7. Applied egg-rr 5.0%

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

   1. expm1-def 64.4%

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

   2. expm1-log1p 98.0%

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

   3. associate-*r/ 98.0%

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

   4. associate-*l/ 98.5%

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

   5. *-commutative 98.5%

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

9. Simplified 98.5%

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

10. Final simplification 98.5%

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

Alternative 5: 98.4% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

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. Taylor expanded in x around inf 98.4%

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

5. Taylor expanded in x around 0 98.0%

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

   1. expm1-log1p-u 97.9%

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

   2. expm1-udef 38.4%

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

   3. add-sqr-sqrt 2.2%

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

   4. fabs-sqr 2.2%

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

   5. add-sqr-sqrt 4.9%

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

7. Applied egg-rr 4.9%

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

   1. expm1-def 64.4%

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

   2. expm1-log1p 98.0%

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

9. Simplified 98.0%

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

10. Final simplification 98.0%

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

Alternative 6: 67.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.86:\\ \;\;\;\;\left|\left({\pi}^{-0.5} \cdot x\right) \cdot 2\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{{x}^{14}}{\pi} \cdot 0.0022675736961451248}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.86)
   (fabs (* (* (pow PI -0.5) x) 2.0))
   (fabs (sqrt (* (/ (pow x 14.0) PI) 0.0022675736961451248)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.86) {
		tmp = fabs(((pow(((double) M_PI), -0.5) * x) * 2.0));
	} else {
		tmp = fabs(sqrt(((pow(x, 14.0) / ((double) M_PI)) * 0.0022675736961451248)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.86) {
		tmp = Math.abs(((Math.pow(Math.PI, -0.5) * x) * 2.0));
	} else {
		tmp = Math.abs(Math.sqrt(((Math.pow(x, 14.0) / Math.PI) * 0.0022675736961451248)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.86:
		tmp = math.fabs(((math.pow(math.pi, -0.5) * x) * 2.0))
	else:
		tmp = math.fabs(math.sqrt(((math.pow(x, 14.0) / math.pi) * 0.0022675736961451248)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.86)
		tmp = abs(Float64(Float64((pi ^ -0.5) * x) * 2.0));
	else
		tmp = abs(sqrt(Float64(Float64((x ^ 14.0) / pi) * 0.0022675736961451248)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.86)
		tmp = abs((((pi ^ -0.5) * x) * 2.0));
	else
		tmp = abs(sqrt((((x ^ 14.0) / pi) * 0.0022675736961451248)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.86], N[Abs[N[(N[(N[Power[Pi, -0.5], $MachinePrecision] * x), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Abs[N[Sqrt[N[(N[(N[Power[x, 14.0], $MachinePrecision] / Pi), $MachinePrecision] * 0.0022675736961451248), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8600000000000001

   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 99.4%

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

   4. Taylor expanded in x around 0 98.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7} + \left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)}\right| \]
   5. Step-by-step derivation

      1. fma-def 98.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)}\right| \]

      2. rem-square-sqrt 32.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right| \]

      3. fabs-sqr 32.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right| \]

      4. rem-square-sqrt 72.7%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {\color{blue}{x}}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right| \]

      5. fma-def 72.7%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right|\right)}\right)\right| \]

      6. rem-square-sqrt 32.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}, 2 \cdot \left|x\right|\right)\right)\right| \]

      7. fabs-sqr 32.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}, 2 \cdot \left|x\right|\right)\right)\right| \]

      8. rem-square-sqrt 98.7%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {\color{blue}{x}}^{3}, 2 \cdot \left|x\right|\right)\right)\right| \]

      9. *-commutative 98.7%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\left|x\right| \cdot 2}\right)\right)\right| \]

      10. rem-square-sqrt 32.7%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot 2\right)\right)\right| \]

      11. fabs-sqr 32.7%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot 2\right)\right)\right| \]

      12. rem-square-sqrt 98.9%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{x} \cdot 2\right)\right)\right| \]

   6. Simplified 98.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, x \cdot 2\right)\right)}\right| \]

   7. Taylor expanded in x around 0 66.8%

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

      1. *-commutative 66.8%

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

   9. Simplified 66.8%

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

       1. inv-pow 66.8%

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

       2. sqrt-pow1 66.8%

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

       3. metadata-eval 66.8%

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

       4. expm1-log1p-u 64.9%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)\right)} \cdot 2\right| \]

       5. *-commutative 64.9%

          \[\leadsto \left|\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\pi}^{-0.5} \cdot x}\right)\right) \cdot 2\right| \]

       6. expm1-udef 4.9%

          \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot x\right)} - 1\right)} \cdot 2\right| \]

       7. add-exp-log 2.2%

          \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{e^{\log \left({\pi}^{-0.5} \cdot x\right)}}\right)} - 1\right) \cdot 2\right| \]

       8. add-exp-log 4.9%

          \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{{\pi}^{-0.5} \cdot x}\right)} - 1\right) \cdot 2\right| \]

       9. metadata-eval 4.9%

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

       10. sqrt-pow1 4.9%

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

       11. inv-pow 4.9%

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

       12. sqrt-div 4.9%

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

       13. metadata-eval 4.9%

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

       14. associate-/r/ 4.9%

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

       15. clear-num 4.9%

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

   11. Applied egg-rr 4.9%

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

       1. expm1-def 64.4%

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

       2. expm1-log1p 66.3%

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

   13. Simplified 66.3%

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

       1. clear-num 66.3%

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

       2. associate-/r/ 66.8%

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

       3. pow1/2 66.8%

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

       4. pow-flip 66.8%

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

       5. metadata-eval 66.8%

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

   15. Applied egg-rr 66.8%

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

   if 1.8600000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 98.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7} + \left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)}\right| \]
   5. Step-by-step derivation

      1. fma-def 98.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)}\right| \]

      2. rem-square-sqrt 32.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right| \]

      3. fabs-sqr 32.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right| \]

      4. rem-square-sqrt 72.7%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {\color{blue}{x}}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right| \]

      5. fma-def 72.7%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right|\right)}\right)\right| \]

      6. rem-square-sqrt 32.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}, 2 \cdot \left|x\right|\right)\right)\right| \]

      7. fabs-sqr 32.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}, 2 \cdot \left|x\right|\right)\right)\right| \]

      8. rem-square-sqrt 98.7%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {\color{blue}{x}}^{3}, 2 \cdot \left|x\right|\right)\right)\right| \]

      9. *-commutative 98.7%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\left|x\right| \cdot 2}\right)\right)\right| \]

      10. rem-square-sqrt 32.7%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot 2\right)\right)\right| \]

      11. fabs-sqr 32.7%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot 2\right)\right)\right| \]

      12. rem-square-sqrt 98.9%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{x} \cdot 2\right)\right)\right| \]

   6. Simplified 98.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, x \cdot 2\right)\right)}\right| \]

   7. Taylor expanded in x around inf 37.3%

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

      1. add-sqr-sqrt 3.2%

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

      2. sqrt-unprod 36.5%

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

      3. *-commutative 36.5%

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

      4. *-commutative 36.5%

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

      5. swap-sqr 36.5%

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

      6. *-commutative 36.5%

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

      7. *-commutative 36.5%

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

      8. swap-sqr 36.5%

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

      9. add-sqr-sqrt 36.5%

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

      10. pow-prod-up 36.5%

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

      11. metadata-eval 36.5%

          \[\leadsto \left|\sqrt{\left(\frac{1}{\pi} \cdot {x}^{\color{blue}{14}}\right) \cdot \left(0.047619047619047616 \cdot 0.047619047619047616\right)}\right| \]

      12. metadata-eval 36.5%

          \[\leadsto \left|\sqrt{\left(\frac{1}{\pi} \cdot {x}^{14}\right) \cdot \color{blue}{0.0022675736961451248}}\right| \]

   9. Applied egg-rr 36.5%

      \[\leadsto \left|\color{blue}{\sqrt{\left(\frac{1}{\pi} \cdot {x}^{14}\right) \cdot 0.0022675736961451248}}\right| \]
   10. Step-by-step derivation

       1. metadata-eval 36.5%

          \[\leadsto \left|\sqrt{\left(\frac{1}{\pi} \cdot {x}^{\color{blue}{\left(2 \cdot 7\right)}}\right) \cdot 0.0022675736961451248}\right| \]

       2. pow-sqr 36.5%

          \[\leadsto \left|\sqrt{\left(\frac{1}{\pi} \cdot \color{blue}{\left({x}^{7} \cdot {x}^{7}\right)}\right) \cdot 0.0022675736961451248}\right| \]

       3. associate-*l/ 36.5%

          \[\leadsto \left|\sqrt{\color{blue}{\frac{1 \cdot \left({x}^{7} \cdot {x}^{7}\right)}{\pi}} \cdot 0.0022675736961451248}\right| \]

       4. *-lft-identity 36.5%

          \[\leadsto \left|\sqrt{\frac{\color{blue}{{x}^{7} \cdot {x}^{7}}}{\pi} \cdot 0.0022675736961451248}\right| \]

       5. pow-sqr 36.5%

          \[\leadsto \left|\sqrt{\frac{\color{blue}{{x}^{\left(2 \cdot 7\right)}}}{\pi} \cdot 0.0022675736961451248}\right| \]

       6. metadata-eval 36.5%

          \[\leadsto \left|\sqrt{\frac{{x}^{\color{blue}{14}}}{\pi} \cdot 0.0022675736961451248}\right| \]

   11. Simplified 36.5%

       \[\leadsto \left|\color{blue}{\sqrt{\frac{{x}^{14}}{\pi} \cdot 0.0022675736961451248}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.86:\\ \;\;\;\;\left|\left({\pi}^{-0.5} \cdot x\right) \cdot 2\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{{x}^{14}}{\pi} \cdot 0.0022675736961451248}\right|\\ \end{array} \]

Alternative 7: 67.2% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.8600000000000001

   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 99.4%

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

   4. Taylor expanded in x around 0 98.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7} + \left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)}\right| \]
   5. Step-by-step derivation

      1. fma-def 98.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)}\right| \]

      2. rem-square-sqrt 32.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right| \]

      3. fabs-sqr 32.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right| \]

      4. rem-square-sqrt 72.7%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {\color{blue}{x}}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right| \]

      5. fma-def 72.7%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right|\right)}\right)\right| \]

      6. rem-square-sqrt 32.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}, 2 \cdot \left|x\right|\right)\right)\right| \]

      7. fabs-sqr 32.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}, 2 \cdot \left|x\right|\right)\right)\right| \]

      8. rem-square-sqrt 98.7%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {\color{blue}{x}}^{3}, 2 \cdot \left|x\right|\right)\right)\right| \]

      9. *-commutative 98.7%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\left|x\right| \cdot 2}\right)\right)\right| \]

      10. rem-square-sqrt 32.7%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot 2\right)\right)\right| \]

      11. fabs-sqr 32.7%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot 2\right)\right)\right| \]

      12. rem-square-sqrt 98.9%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{x} \cdot 2\right)\right)\right| \]

   6. Simplified 98.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, x \cdot 2\right)\right)}\right| \]

   7. Taylor expanded in x around 0 66.8%

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

      1. *-commutative 66.8%

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

   9. Simplified 66.8%

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

       1. inv-pow 66.8%

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

       2. sqrt-pow1 66.8%

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

       3. metadata-eval 66.8%

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

       4. expm1-log1p-u 64.9%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)\right)} \cdot 2\right| \]

       5. *-commutative 64.9%

          \[\leadsto \left|\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\pi}^{-0.5} \cdot x}\right)\right) \cdot 2\right| \]

       6. expm1-udef 4.9%

          \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot x\right)} - 1\right)} \cdot 2\right| \]

       7. add-exp-log 2.2%

          \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{e^{\log \left({\pi}^{-0.5} \cdot x\right)}}\right)} - 1\right) \cdot 2\right| \]

       8. add-exp-log 4.9%

          \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{{\pi}^{-0.5} \cdot x}\right)} - 1\right) \cdot 2\right| \]

       9. metadata-eval 4.9%

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

       10. sqrt-pow1 4.9%

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

       11. inv-pow 4.9%

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

       12. sqrt-div 4.9%

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

       13. metadata-eval 4.9%

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

       14. associate-/r/ 4.9%

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

       15. clear-num 4.9%

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

   11. Applied egg-rr 4.9%

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

       1. expm1-def 64.4%

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

       2. expm1-log1p 66.3%

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

   13. Simplified 66.3%

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

       1. clear-num 66.3%

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

       2. associate-/r/ 66.8%

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

       3. pow1/2 66.8%

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

       4. pow-flip 66.8%

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

       5. metadata-eval 66.8%

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

   15. Applied egg-rr 66.8%

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

   if 1.8600000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 98.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7} + \left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)}\right| \]
   5. Step-by-step derivation

      1. fma-def 98.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)}\right| \]

      2. rem-square-sqrt 32.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right| \]

      3. fabs-sqr 32.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right| \]

      4. rem-square-sqrt 72.7%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {\color{blue}{x}}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right| \]

      5. fma-def 72.7%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right|\right)}\right)\right| \]

      6. rem-square-sqrt 32.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}, 2 \cdot \left|x\right|\right)\right)\right| \]

      7. fabs-sqr 32.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}, 2 \cdot \left|x\right|\right)\right)\right| \]

      8. rem-square-sqrt 98.7%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {\color{blue}{x}}^{3}, 2 \cdot \left|x\right|\right)\right)\right| \]

      9. *-commutative 98.7%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\left|x\right| \cdot 2}\right)\right)\right| \]

      10. rem-square-sqrt 32.7%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot 2\right)\right)\right| \]

      11. fabs-sqr 32.7%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot 2\right)\right)\right| \]

      12. rem-square-sqrt 98.9%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{x} \cdot 2\right)\right)\right| \]

   6. Simplified 98.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, x \cdot 2\right)\right)}\right| \]

   7. Taylor expanded in x around inf 37.3%

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

      1. expm1-log1p-u 3.7%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)}\right| \]

      2. expm1-udef 3.5%

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

      3. sqrt-div 3.5%

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

      4. metadata-eval 3.5%

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

      5. un-div-inv 3.5%

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

   9. Applied egg-rr 3.5%

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

       1. expm1-def 3.7%

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

       2. expm1-log1p 37.3%

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

   11. Simplified 37.3%

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

4. Final simplification 66.8%

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

Alternative 8: 67.2% accurate, 6.4× speedup

Math

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

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(Float64((pi ^ -0.5) * 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[(N[Power[Pi, -0.5], $MachinePrecision] * x), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around 0 98.9%

   \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7} + \left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)}\right| \]
5. Step-by-step derivation

   1. fma-def 98.9%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)}\right| \]

   2. rem-square-sqrt 32.9%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right| \]

   3. fabs-sqr 32.9%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right| \]

   4. rem-square-sqrt 72.7%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {\color{blue}{x}}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right| \]

   5. fma-def 72.7%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right|\right)}\right)\right| \]

   6. rem-square-sqrt 32.9%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}, 2 \cdot \left|x\right|\right)\right)\right| \]

   7. fabs-sqr 32.9%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}, 2 \cdot \left|x\right|\right)\right)\right| \]

   8. rem-square-sqrt 98.7%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {\color{blue}{x}}^{3}, 2 \cdot \left|x\right|\right)\right)\right| \]

   9. *-commutative 98.7%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\left|x\right| \cdot 2}\right)\right)\right| \]

   10. rem-square-sqrt 32.7%

       \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot 2\right)\right)\right| \]

   11. fabs-sqr 32.7%

       \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot 2\right)\right)\right| \]

   12. rem-square-sqrt 98.9%

       \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{x} \cdot 2\right)\right)\right| \]

6. Simplified 98.9%

   \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, x \cdot 2\right)\right)}\right| \]

7. Taylor expanded in x around 0 66.8%

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

   1. *-commutative 66.8%

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

9. Simplified 66.8%

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

    1. inv-pow 66.8%

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

    2. sqrt-pow1 66.8%

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

    3. metadata-eval 66.8%

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

    4. expm1-log1p-u 64.9%

       \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)\right)} \cdot 2\right| \]

    5. *-commutative 64.9%

       \[\leadsto \left|\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\pi}^{-0.5} \cdot x}\right)\right) \cdot 2\right| \]

    6. expm1-udef 4.9%

       \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot x\right)} - 1\right)} \cdot 2\right| \]

    7. add-exp-log 2.2%

       \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{e^{\log \left({\pi}^{-0.5} \cdot x\right)}}\right)} - 1\right) \cdot 2\right| \]

    8. add-exp-log 4.9%

       \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{{\pi}^{-0.5} \cdot x}\right)} - 1\right) \cdot 2\right| \]

    9. metadata-eval 4.9%

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

    10. sqrt-pow1 4.9%

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

    11. inv-pow 4.9%

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

    12. sqrt-div 4.9%

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

    13. metadata-eval 4.9%

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

    14. associate-/r/ 4.9%

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

    15. clear-num 4.9%

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

11. Applied egg-rr 4.9%

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

    1. expm1-def 64.4%

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

    2. expm1-log1p 66.3%

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

13. Simplified 66.3%

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

    1. clear-num 66.3%

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

    2. associate-/r/ 66.8%

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

    3. pow1/2 66.8%

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

    4. pow-flip 66.8%

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

    5. metadata-eval 66.8%

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

15. Applied egg-rr 66.8%

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

16. Final simplification 66.8%

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

Alternative 9: 66.7% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around 0 98.9%

   \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7} + \left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)}\right| \]
5. Step-by-step derivation

   1. fma-def 98.9%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)}\right| \]

   2. rem-square-sqrt 32.9%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right| \]

   3. fabs-sqr 32.9%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right| \]

   4. rem-square-sqrt 72.7%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {\color{blue}{x}}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right| \]

   5. fma-def 72.7%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right|\right)}\right)\right| \]

   6. rem-square-sqrt 32.9%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}, 2 \cdot \left|x\right|\right)\right)\right| \]

   7. fabs-sqr 32.9%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}, 2 \cdot \left|x\right|\right)\right)\right| \]

   8. rem-square-sqrt 98.7%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {\color{blue}{x}}^{3}, 2 \cdot \left|x\right|\right)\right)\right| \]

   9. *-commutative 98.7%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\left|x\right| \cdot 2}\right)\right)\right| \]

   10. rem-square-sqrt 32.7%

       \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot 2\right)\right)\right| \]

   11. fabs-sqr 32.7%

       \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot 2\right)\right)\right| \]

   12. rem-square-sqrt 98.9%

       \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{x} \cdot 2\right)\right)\right| \]

6. Simplified 98.9%

   \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, x \cdot 2\right)\right)}\right| \]

7. Taylor expanded in x around 0 66.8%

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

   1. *-commutative 66.8%

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

9. Simplified 66.8%

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

    1. inv-pow 66.8%

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

    2. sqrt-pow1 66.8%

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

    3. metadata-eval 66.8%

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

    4. expm1-log1p-u 64.9%

       \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)\right)} \cdot 2\right| \]

    5. *-commutative 64.9%

       \[\leadsto \left|\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\pi}^{-0.5} \cdot x}\right)\right) \cdot 2\right| \]

    6. expm1-udef 4.9%

       \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot x\right)} - 1\right)} \cdot 2\right| \]

    7. add-exp-log 2.2%

       \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{e^{\log \left({\pi}^{-0.5} \cdot x\right)}}\right)} - 1\right) \cdot 2\right| \]

    8. add-exp-log 4.9%

       \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{{\pi}^{-0.5} \cdot x}\right)} - 1\right) \cdot 2\right| \]

    9. metadata-eval 4.9%

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

    10. sqrt-pow1 4.9%

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

    11. inv-pow 4.9%

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

    12. sqrt-div 4.9%

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

    13. metadata-eval 4.9%

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

    14. associate-/r/ 4.9%

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

    15. clear-num 4.9%

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

11. Applied egg-rr 4.9%

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

    1. expm1-def 64.4%

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

    2. expm1-log1p 66.3%

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

13. Simplified 66.3%

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

14. Final simplification 66.3%

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

Reproduce

herbie shell --seed 2023299 
(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)))))))