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

Percentage Accurate: 99.8% → 99.9%

Time: 8.5s

Alternatives: 8

Speedup: 3.7×

• 29.3% 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, 3.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (* x (pow PI -0.5))
   (+
    (+ (* 0.6666666666666666 (* x x)) 2.0)
    (+ (* 0.2 (pow x 4.0)) (* 0.047619047619047616 (pow x 6.0)))))))

C

double code(double x) {
	return fabs(((x * pow(((double) M_PI), -0.5)) * (((0.6666666666666666 * (x * x)) + 2.0) + ((0.2 * pow(x, 4.0)) + (0.047619047619047616 * pow(x, 6.0))))));
}

Java

public static double code(double x) {
	return Math.abs(((x * Math.pow(Math.PI, -0.5)) * (((0.6666666666666666 * (x * x)) + 2.0) + ((0.2 * Math.pow(x, 4.0)) + (0.047619047619047616 * Math.pow(x, 6.0))))));
}

Python

def code(x):
	return math.fabs(((x * math.pow(math.pi, -0.5)) * (((0.6666666666666666 * (x * x)) + 2.0) + ((0.2 * math.pow(x, 4.0)) + (0.047619047619047616 * math.pow(x, 6.0))))))

Julia

function code(x)
	return abs(Float64(Float64(x * (pi ^ -0.5)) * Float64(Float64(Float64(0.6666666666666666 * Float64(x * x)) + 2.0) + Float64(Float64(0.2 * (x ^ 4.0)) + Float64(0.047619047619047616 * (x ^ 6.0))))))
end

MATLAB

function tmp = code(x)
	tmp = abs(((x * (pi ^ -0.5)) * (((0.6666666666666666 * (x * x)) + 2.0) + ((0.2 * (x ^ 4.0)) + (0.047619047619047616 * (x ^ 6.0))))));
end

Wolfram

code[x_] := N[Abs[N[(N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] + N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

   1. div-inv 99.8%

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

5. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.4%

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

   2. *-rgt-identity 99.4%

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

   3. unpow1 99.4%

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

   4. sqr-pow 32.5%

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

   5. fabs-sqr 32.5%

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

   6. sqr-pow 99.4%

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

   7. unpow1 99.4%

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

7. Simplified 99.4%

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

   1. fma-udef 99.4%

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

9. Applied egg-rr 99.4%

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

    1. div-inv 99.8%

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

    2. pow1/2 99.8%

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

    3. pow-flip 99.8%

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

    4. metadata-eval 99.8%

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

11. Applied egg-rr 99.8%

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

12. Taylor expanded in x around 0 99.8%

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

13. Final simplification 99.8%

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

Alternative 2: 99.3% accurate, 4.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (* x (pow PI -0.5))
   (+
    (+ (* 0.6666666666666666 (* x x)) 2.0)
    (* 0.047619047619047616 (pow x 6.0))))))

C

double code(double x) {
	return fabs(((x * pow(((double) M_PI), -0.5)) * (((0.6666666666666666 * (x * x)) + 2.0) + (0.047619047619047616 * pow(x, 6.0)))));
}

Java

public static double code(double x) {
	return Math.abs(((x * Math.pow(Math.PI, -0.5)) * (((0.6666666666666666 * (x * x)) + 2.0) + (0.047619047619047616 * Math.pow(x, 6.0)))));
}

Python

def code(x):
	return math.fabs(((x * math.pow(math.pi, -0.5)) * (((0.6666666666666666 * (x * x)) + 2.0) + (0.047619047619047616 * math.pow(x, 6.0)))))

Julia

function code(x)
	return abs(Float64(Float64(x * (pi ^ -0.5)) * Float64(Float64(Float64(0.6666666666666666 * Float64(x * x)) + 2.0) + Float64(0.047619047619047616 * (x ^ 6.0)))))
end

MATLAB

function tmp = code(x)
	tmp = abs(((x * (pi ^ -0.5)) * (((0.6666666666666666 * (x * x)) + 2.0) + (0.047619047619047616 * (x ^ 6.0)))));
end

Wolfram

code[x_] := N[Abs[N[(N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

   1. div-inv 99.8%

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

5. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.4%

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

   2. *-rgt-identity 99.4%

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

   3. unpow1 99.4%

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

   4. sqr-pow 32.5%

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

   5. fabs-sqr 32.5%

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

   6. sqr-pow 99.4%

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

   7. unpow1 99.4%

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

7. Simplified 99.4%

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

   1. fma-udef 99.4%

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

9. Applied egg-rr 99.4%

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

    1. div-inv 99.8%

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

    2. pow1/2 99.8%

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

    3. pow-flip 99.8%

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

    4. metadata-eval 99.8%

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

11. Applied egg-rr 99.8%

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

12. Taylor expanded in x around inf 99.3%

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

13. Final simplification 99.3%

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

Alternative 3: 74.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.6666666666666666 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq 2.1:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \frac{4 - t_0 \cdot t_0}{2 - t_0}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(x \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.6666666666666666 (* x x))))
   (if (<= x 2.1)
     (fabs (* (sqrt (/ 1.0 PI)) (* x (/ (- 4.0 (* t_0 t_0)) (- 2.0 t_0)))))
     (fabs (* (pow PI -0.5) (* x (* 0.047619047619047616 (pow x 6.0))))))))

C

double code(double x) {
	double t_0 = 0.6666666666666666 * (x * x);
	double tmp;
	if (x <= 2.1) {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * (x * ((4.0 - (t_0 * t_0)) / (2.0 - t_0)))));
	} else {
		tmp = fabs((pow(((double) M_PI), -0.5) * (x * (0.047619047619047616 * pow(x, 6.0)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 0.6666666666666666 * (x * x);
	double tmp;
	if (x <= 2.1) {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * (x * ((4.0 - (t_0 * t_0)) / (2.0 - t_0)))));
	} else {
		tmp = Math.abs((Math.pow(Math.PI, -0.5) * (x * (0.047619047619047616 * Math.pow(x, 6.0)))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.6666666666666666 * (x * x)
	tmp = 0
	if x <= 2.1:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * (x * ((4.0 - (t_0 * t_0)) / (2.0 - t_0)))))
	else:
		tmp = math.fabs((math.pow(math.pi, -0.5) * (x * (0.047619047619047616 * math.pow(x, 6.0)))))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.6666666666666666 * Float64(x * x))
	tmp = 0.0
	if (x <= 2.1)
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(x * Float64(Float64(4.0 - Float64(t_0 * t_0)) / Float64(2.0 - t_0)))));
	else
		tmp = abs(Float64((pi ^ -0.5) * Float64(x * Float64(0.047619047619047616 * (x ^ 6.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.6666666666666666 * (x * x);
	tmp = 0.0;
	if (x <= 2.1)
		tmp = abs((sqrt((1.0 / pi)) * (x * ((4.0 - (t_0 * t_0)) / (2.0 - t_0)))));
	else
		tmp = abs(((pi ^ -0.5) * (x * (0.047619047619047616 * (x ^ 6.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.1], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x * N[(N[(4.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(2.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x * N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.10000000000000009

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 91.5%

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

      1. associate-*r* 91.5%

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

      2. unpow2 91.5%

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

      3. associate-*r* 91.5%

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

      4. distribute-rgt-out 91.5%

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

      5. +-commutative 91.5%

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

      6. associate-*r* 91.5%

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

      7. *-commutative 91.5%

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

      8. associate-*l* 91.5%

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

      9. *-commutative 91.5%

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

      10. distribute-lft-in 91.5%

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

      11. fma-def 91.5%

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

   6. Simplified 91.5%

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

      1. fma-udef 91.5%

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

      2. +-commutative 91.5%

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

      3. flip-+ 73.4%

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

      4. metadata-eval 73.4%

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

   8. Applied egg-rr 73.4%

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

   if 2.10000000000000009 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 36.0%

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

      1. associate-*r* 36.0%

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

      2. *-commutative 36.0%

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

      3. *-commutative 36.0%

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

      4. associate-*l* 36.0%

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

      5. *-commutative 36.0%

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

   6. Simplified 36.0%

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

      1. expm1-log1p-u 35.8%

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

      2. expm1-udef 35.7%

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

      3. inv-pow 35.7%

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

      4. sqrt-pow1 35.7%

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

      5. metadata-eval 35.7%

         \[\leadsto \left|e^{\mathsf{log1p}\left({\pi}^{\color{blue}{-0.5}} \cdot \left(\left|x\right| \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)\right)} - 1\right| \]

      6. add-sqr-sqrt 1.8%

         \[\leadsto \left|e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)\right)} - 1\right| \]

      7. fabs-sqr 1.8%

         \[\leadsto \left|e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)\right)} - 1\right| \]

      8. add-sqr-sqrt 3.7%

         \[\leadsto \left|e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(\color{blue}{x} \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)\right)} - 1\right| \]

   8. Applied egg-rr 3.7%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)\right)} - 1}\right| \]
   9. Step-by-step derivation

      1. expm1-def 3.8%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)\right)\right)}\right| \]

      2. expm1-log1p 36.0%

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

   10. Simplified 36.0%

       \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(x \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \frac{4 - \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)}{2 - 0.6666666666666666 \cdot \left(x \cdot x\right)}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(x \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)\right|\\ \end{array} \]

Alternative 4: 74.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.6666666666666666 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq 2.1:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \frac{4 - t_0 \cdot t_0}{2 - t_0}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{7} \cdot \left({\pi}^{-0.5} \cdot 0.047619047619047616\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.6666666666666666 (* x x))))
   (if (<= x 2.1)
     (fabs (* (sqrt (/ 1.0 PI)) (* x (/ (- 4.0 (* t_0 t_0)) (- 2.0 t_0)))))
     (fabs (* (pow x 7.0) (* (pow PI -0.5) 0.047619047619047616))))))

C

double code(double x) {
	double t_0 = 0.6666666666666666 * (x * x);
	double tmp;
	if (x <= 2.1) {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * (x * ((4.0 - (t_0 * t_0)) / (2.0 - t_0)))));
	} else {
		tmp = fabs((pow(x, 7.0) * (pow(((double) M_PI), -0.5) * 0.047619047619047616)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 0.6666666666666666 * (x * x);
	double tmp;
	if (x <= 2.1) {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * (x * ((4.0 - (t_0 * t_0)) / (2.0 - t_0)))));
	} else {
		tmp = Math.abs((Math.pow(x, 7.0) * (Math.pow(Math.PI, -0.5) * 0.047619047619047616)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.6666666666666666 * (x * x)
	tmp = 0
	if x <= 2.1:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * (x * ((4.0 - (t_0 * t_0)) / (2.0 - t_0)))))
	else:
		tmp = math.fabs((math.pow(x, 7.0) * (math.pow(math.pi, -0.5) * 0.047619047619047616)))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.6666666666666666 * Float64(x * x))
	tmp = 0.0
	if (x <= 2.1)
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(x * Float64(Float64(4.0 - Float64(t_0 * t_0)) / Float64(2.0 - t_0)))));
	else
		tmp = abs(Float64((x ^ 7.0) * Float64((pi ^ -0.5) * 0.047619047619047616)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.6666666666666666 * (x * x);
	tmp = 0.0;
	if (x <= 2.1)
		tmp = abs((sqrt((1.0 / pi)) * (x * ((4.0 - (t_0 * t_0)) / (2.0 - t_0)))));
	else
		tmp = abs(((x ^ 7.0) * ((pi ^ -0.5) * 0.047619047619047616)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.1], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x * N[(N[(4.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(2.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Power[x, 7.0], $MachinePrecision] * N[(N[Power[Pi, -0.5], $MachinePrecision] * 0.047619047619047616), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.10000000000000009

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 91.5%

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

      1. associate-*r* 91.5%

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

      2. unpow2 91.5%

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

      3. associate-*r* 91.5%

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

      4. distribute-rgt-out 91.5%

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

      5. +-commutative 91.5%

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

      6. associate-*r* 91.5%

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

      7. *-commutative 91.5%

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

      8. associate-*l* 91.5%

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

      9. *-commutative 91.5%

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

      10. distribute-lft-in 91.5%

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

      11. fma-def 91.5%

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

   6. Simplified 91.5%

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

      1. fma-udef 91.5%

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

      2. +-commutative 91.5%

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

      3. flip-+ 73.4%

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

      4. metadata-eval 73.4%

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

   8. Applied egg-rr 73.4%

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

   if 2.10000000000000009 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 36.0%

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

      1. associate-*r* 36.0%

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

      2. *-commutative 36.0%

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

      3. *-commutative 36.0%

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

      4. associate-*l* 36.0%

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

      5. *-commutative 36.0%

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

   6. Simplified 36.0%

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

      1. expm1-log1p-u 35.8%

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

      2. expm1-udef 35.7%

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

      3. inv-pow 35.7%

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

      4. sqrt-pow1 35.7%

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

      5. metadata-eval 35.7%

         \[\leadsto \left|e^{\mathsf{log1p}\left({\pi}^{\color{blue}{-0.5}} \cdot \left(\left|x\right| \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)\right)} - 1\right| \]

      6. add-sqr-sqrt 1.8%

         \[\leadsto \left|e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)\right)} - 1\right| \]

      7. fabs-sqr 1.8%

         \[\leadsto \left|e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)\right)} - 1\right| \]

      8. add-sqr-sqrt 3.7%

         \[\leadsto \left|e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(\color{blue}{x} \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)\right)} - 1\right| \]

   8. Applied egg-rr 3.7%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)\right)} - 1}\right| \]
   9. Step-by-step derivation

      1. expm1-def 3.8%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)\right)\right)}\right| \]

      2. expm1-log1p 36.0%

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

      3. *-commutative 36.0%

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

      4. associate-*l* 36.0%

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

      5. pow-plus 36.0%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \color{blue}{{x}^{\left(6 + 1\right)}}\right)\right| \]

      6. metadata-eval 36.0%

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

   10. Simplified 36.0%

       \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)}\right| \]
   11. Step-by-step derivation

       1. expm1-log1p-u 3.8%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right)\right)}\right| \]

       2. expm1-udef 3.7%

          \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right)} - 1}\right| \]

   12. Applied egg-rr 3.7%

       \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right)} - 1}\right| \]
   13. Step-by-step derivation

       1. expm1-def 3.8%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right)\right)}\right| \]

       2. expm1-log1p 36.0%

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

       3. associate-*r* 36.0%

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

   14. Simplified 36.0%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \frac{4 - \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)}{2 - 0.6666666666666666 \cdot \left(x \cdot x\right)}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{7} \cdot \left({\pi}^{-0.5} \cdot 0.047619047619047616\right)\right|\\ \end{array} \]

Alternative 5: 74.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.6666666666666666 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq 2.1:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \frac{4 - t_0 \cdot t_0}{2 - t_0}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.6666666666666666 (* x x))))
   (if (<= x 2.1)
     (fabs (* (sqrt (/ 1.0 PI)) (* x (/ (- 4.0 (* t_0 t_0)) (- 2.0 t_0)))))
     (fabs (* (pow PI -0.5) (* 0.047619047619047616 (pow x 7.0)))))))

C

double code(double x) {
	double t_0 = 0.6666666666666666 * (x * x);
	double tmp;
	if (x <= 2.1) {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * (x * ((4.0 - (t_0 * t_0)) / (2.0 - t_0)))));
	} else {
		tmp = fabs((pow(((double) M_PI), -0.5) * (0.047619047619047616 * pow(x, 7.0))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 0.6666666666666666 * (x * x);
	double tmp;
	if (x <= 2.1) {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * (x * ((4.0 - (t_0 * t_0)) / (2.0 - t_0)))));
	} else {
		tmp = Math.abs((Math.pow(Math.PI, -0.5) * (0.047619047619047616 * Math.pow(x, 7.0))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.6666666666666666 * (x * x)
	tmp = 0
	if x <= 2.1:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * (x * ((4.0 - (t_0 * t_0)) / (2.0 - t_0)))))
	else:
		tmp = math.fabs((math.pow(math.pi, -0.5) * (0.047619047619047616 * math.pow(x, 7.0))))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.6666666666666666 * Float64(x * x))
	tmp = 0.0
	if (x <= 2.1)
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(x * Float64(Float64(4.0 - Float64(t_0 * t_0)) / Float64(2.0 - t_0)))));
	else
		tmp = abs(Float64((pi ^ -0.5) * Float64(0.047619047619047616 * (x ^ 7.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.6666666666666666 * (x * x);
	tmp = 0.0;
	if (x <= 2.1)
		tmp = abs((sqrt((1.0 / pi)) * (x * ((4.0 - (t_0 * t_0)) / (2.0 - t_0)))));
	else
		tmp = abs(((pi ^ -0.5) * (0.047619047619047616 * (x ^ 7.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.1], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x * N[(N[(4.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(2.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.10000000000000009

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 91.5%

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

      1. associate-*r* 91.5%

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

      2. unpow2 91.5%

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

      3. associate-*r* 91.5%

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

      4. distribute-rgt-out 91.5%

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

      5. +-commutative 91.5%

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

      6. associate-*r* 91.5%

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

      7. *-commutative 91.5%

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

      8. associate-*l* 91.5%

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

      9. *-commutative 91.5%

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

      10. distribute-lft-in 91.5%

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

      11. fma-def 91.5%

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

   6. Simplified 91.5%

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

      1. fma-udef 91.5%

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

      2. +-commutative 91.5%

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

      3. flip-+ 73.4%

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

      4. metadata-eval 73.4%

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

   8. Applied egg-rr 73.4%

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

   if 2.10000000000000009 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 36.0%

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

      1. associate-*r* 36.0%

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

      2. *-commutative 36.0%

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

      3. *-commutative 36.0%

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

      4. associate-*l* 36.0%

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

      5. *-commutative 36.0%

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

   6. Simplified 36.0%

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

      1. expm1-log1p-u 35.8%

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

      2. expm1-udef 35.7%

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

      3. inv-pow 35.7%

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

      4. sqrt-pow1 35.7%

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

      5. metadata-eval 35.7%

         \[\leadsto \left|e^{\mathsf{log1p}\left({\pi}^{\color{blue}{-0.5}} \cdot \left(\left|x\right| \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)\right)} - 1\right| \]

      6. add-sqr-sqrt 1.8%

         \[\leadsto \left|e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)\right)} - 1\right| \]

      7. fabs-sqr 1.8%

         \[\leadsto \left|e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)\right)} - 1\right| \]

      8. add-sqr-sqrt 3.7%

         \[\leadsto \left|e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(\color{blue}{x} \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)\right)} - 1\right| \]

   8. Applied egg-rr 3.7%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)\right)} - 1}\right| \]
   9. Step-by-step derivation

      1. expm1-def 3.8%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)\right)\right)}\right| \]

      2. expm1-log1p 36.0%

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

      3. *-commutative 36.0%

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

      4. associate-*l* 36.0%

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

      5. pow-plus 36.0%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \color{blue}{{x}^{\left(6 + 1\right)}}\right)\right| \]

      6. metadata-eval 36.0%

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

   10. Simplified 36.0%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \frac{4 - \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)}{2 - 0.6666666666666666 \cdot \left(x \cdot x\right)}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \end{array} \]

Alternative 6: 89.3% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around 0 91.5%

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

   1. associate-*r* 91.5%

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

   2. unpow2 91.5%

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

   3. associate-*r* 91.5%

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

   4. distribute-rgt-out 91.5%

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

   5. +-commutative 91.5%

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

   6. associate-*r* 91.5%

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

   7. *-commutative 91.5%

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

   8. associate-*l* 91.5%

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

   9. *-commutative 91.5%

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

   10. distribute-lft-in 91.5%

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

   11. fma-def 91.5%

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

6. Simplified 91.5%

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

   1. fma-udef 99.4%

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

8. Applied egg-rr 91.5%

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

9. Final simplification 91.5%

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

Alternative 7: 67.3% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.72:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \left({\pi}^{-0.5} \cdot \left(x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.72)
   (fabs (* x (/ 2.0 (sqrt PI))))
   (fabs (* x (* (pow PI -0.5) (* x (* x 0.6666666666666666)))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.72) {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((x * (pow(((double) M_PI), -0.5) * (x * (x * 0.6666666666666666)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.72) {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs((x * (Math.pow(Math.PI, -0.5) * (x * (x * 0.6666666666666666)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.72:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs((x * (math.pow(math.pi, -0.5) * (x * (x * 0.6666666666666666)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.72)
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(Float64(x * Float64((pi ^ -0.5) * Float64(x * Float64(x * 0.6666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.72)
		tmp = abs((x * (2.0 / sqrt(pi))));
	else
		tmp = abs((x * ((pi ^ -0.5) * (x * (x * 0.6666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.72], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x * N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.71999999999999997

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 69.0%

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

      1. *-commutative 69.0%

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

      2. associate-*l* 69.0%

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

      3. unpow1 69.0%

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

      4. sqr-pow 32.5%

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

      5. fabs-sqr 32.5%

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

      6. sqr-pow 69.0%

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

      7. unpow1 69.0%

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

   6. Simplified 69.0%

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

      1. expm1-log1p-u 67.0%

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

      2. expm1-udef 4.5%

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

      3. associate-*r* 4.5%

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

      4. sqrt-div 4.5%

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

      5. metadata-eval 4.5%

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

      6. div-inv 4.5%

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

   8. Applied egg-rr 4.5%

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

      1. expm1-def 66.6%

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

      2. expm1-log1p 68.6%

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

      3. associate-*l/ 68.6%

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

   10. Simplified 68.6%

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

       1. expm1-log1p-u 66.6%

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

       2. expm1-udef 4.5%

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

       3. associate-/l* 4.5%

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

   12. Applied egg-rr 4.5%

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

       1. expm1-def 66.6%

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

       2. expm1-log1p 68.6%

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

       3. associate-/l* 68.6%

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

       4. associate-*r/ 69.0%

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

   14. Simplified 69.0%

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

   if 1.71999999999999997 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 91.5%

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

      1. associate-*r* 91.5%

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

      2. unpow2 91.5%

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

      3. associate-*r* 91.5%

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

      4. distribute-rgt-out 91.5%

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

      5. +-commutative 91.5%

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

      6. associate-*r* 91.5%

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

      7. *-commutative 91.5%

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

      8. associate-*l* 91.5%

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

      9. *-commutative 91.5%

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

      10. distribute-lft-in 91.5%

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

      11. fma-def 91.5%

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

   6. Simplified 91.5%

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

      1. fma-udef 91.5%

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

      2. +-commutative 91.5%

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

      3. flip-+ 73.4%

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

      4. metadata-eval 73.4%

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

   8. Applied egg-rr 73.4%

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

   9. Taylor expanded in x around inf 28.5%

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

       1. unpow2 28.5%

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

       2. *-commutative 28.5%

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

       3. associate-*l* 28.5%

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

   11. Simplified 28.5%

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

       1. expm1-log1p-u 4.1%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)\right)\right)}\right| \]

       2. expm1-udef 3.7%

          \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)\right)} - 1}\right| \]

       3. associate-*r* 3.7%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.6666666666666666\right)\right)}\right)} - 1\right| \]

       4. pow1/2 3.7%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\left(\color{blue}{{\left(\frac{1}{\pi}\right)}^{0.5}} \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)} - 1\right| \]

       5. inv-pow 3.7%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\left({\color{blue}{\left({\pi}^{-1}\right)}}^{0.5} \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)} - 1\right| \]

       6. pow-pow 3.7%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\left(\color{blue}{{\pi}^{\left(-1 \cdot 0.5\right)}} \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)} - 1\right| \]

       7. metadata-eval 3.7%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\left({\pi}^{\color{blue}{-0.5}} \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)} - 1\right| \]

       8. *-commutative 3.7%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\left(x \cdot {\pi}^{-0.5}\right)} \cdot \left(x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)} - 1\right| \]

   13. Applied egg-rr 3.7%

       \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)} - 1}\right| \]
   14. Step-by-step derivation

       1. expm1-def 4.1%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)\right)}\right| \]

       2. expm1-log1p 28.5%

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

       3. associate-*l* 28.5%

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

   15. Simplified 28.5%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.72:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \left({\pi}^{-0.5} \cdot \left(x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)\right|\\ \end{array} \]

Alternative 8: 67.3% accurate, 6.4× speedup

Math

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

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(x * Float64(2.0 / sqrt(pi))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around 0 69.0%

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

   1. *-commutative 69.0%

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

   2. associate-*l* 69.0%

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

   3. unpow1 69.0%

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

   4. sqr-pow 32.5%

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

   5. fabs-sqr 32.5%

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

   6. sqr-pow 69.0%

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

   7. unpow1 69.0%

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

6. Simplified 69.0%

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

   1. expm1-log1p-u 67.0%

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

   2. expm1-udef 4.5%

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

   3. associate-*r* 4.5%

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

   4. sqrt-div 4.5%

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

   5. metadata-eval 4.5%

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

   6. div-inv 4.5%

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

8. Applied egg-rr 4.5%

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

   1. expm1-def 66.6%

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

   2. expm1-log1p 68.6%

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

   3. associate-*l/ 68.6%

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

10. Simplified 68.6%

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

    1. expm1-log1p-u 66.6%

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

    2. expm1-udef 4.5%

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

    3. associate-/l* 4.5%

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

12. Applied egg-rr 4.5%

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

    1. expm1-def 66.6%

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

    2. expm1-log1p 68.6%

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

    3. associate-/l* 68.6%

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

    4. associate-*r/ 69.0%

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

14. Simplified 69.0%

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

15. Final simplification 69.0%

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

Reproduce

herbie shell --seed 2023238 
(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)))))))