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

Percentage Accurate: 99.8% → 99.9%

Time: 14.7s

Alternatives: 9

Speedup: 2.6×

• 28.8% 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, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 99.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 99.5%

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

6. Final simplification 99.5%

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

Alternative 3: 34.4% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 99.5%

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

   1. add-sqr-sqrt 98.9%

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

   2. pow2 98.9%

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

7. Applied egg-rr 32.1%

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

   1. unpow2 32.1%

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

   2. add-sqr-sqrt 33.9%

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

   3. *-commutative 33.9%

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

   4. div-inv 33.9%

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

   5. metadata-eval 33.9%

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

   6. sqrt-div 33.9%

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

   7. associate-*l* 33.9%

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

   8. inv-pow 33.9%

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

   9. sqrt-pow1 33.9%

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

   10. metadata-eval 33.9%

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

9. Applied egg-rr 33.9%

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

    1. *-commutative 33.9%

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

    2. *-commutative 33.9%

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

11. Simplified 33.9%

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

12. Final simplification 33.9%

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

Alternative 4: 98.8% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 99.5%

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

6. Taylor expanded in x around inf 99.2%

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

7. Final simplification 99.2%

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

Alternative 5: 34.4% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ t_1 := 0.2 \cdot {x}^{5}\\ \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;t\_0 \cdot \left(t\_1 + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 + 0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))) (t_1 (* 0.2 (pow x 5.0))))
   (if (<= x 1.9)
     (* t_0 (+ t_1 (* x 2.0)))
     (* t_0 (+ t_1 (* 0.047619047619047616 (pow x 7.0)))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	double t_1 = 0.2 * pow(x, 5.0);
	double tmp;
	if (x <= 1.9) {
		tmp = t_0 * (t_1 + (x * 2.0));
	} else {
		tmp = t_0 * (t_1 + (0.047619047619047616 * pow(x, 7.0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	double t_1 = 0.2 * Math.pow(x, 5.0);
	double tmp;
	if (x <= 1.9) {
		tmp = t_0 * (t_1 + (x * 2.0));
	} else {
		tmp = t_0 * (t_1 + (0.047619047619047616 * Math.pow(x, 7.0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / math.pi))
	t_1 = 0.2 * math.pow(x, 5.0)
	tmp = 0
	if x <= 1.9:
		tmp = t_0 * (t_1 + (x * 2.0))
	else:
		tmp = t_0 * (t_1 + (0.047619047619047616 * math.pow(x, 7.0)))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / pi))
	t_1 = Float64(0.2 * (x ^ 5.0))
	tmp = 0.0
	if (x <= 1.9)
		tmp = Float64(t_0 * Float64(t_1 + Float64(x * 2.0)));
	else
		tmp = Float64(t_0 * Float64(t_1 + Float64(0.047619047619047616 * (x ^ 7.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 / pi));
	t_1 = 0.2 * (x ^ 5.0);
	tmp = 0.0;
	if (x <= 1.9)
		tmp = t_0 * (t_1 + (x * 2.0));
	else
		tmp = t_0 * (t_1 + (0.047619047619047616 * (x ^ 7.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.9], N[(t$95$0 * N[(t$95$1 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(t$95$1 + N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. add-sqr-sqrt 98.9%

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

      2. pow2 98.9%

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

   7. Applied egg-rr 32.1%

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

      1. unpow2 32.1%

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

      2. add-sqr-sqrt 33.9%

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

      3. expm1-log1p-u 33.8%

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

      4. expm1-undefine 4.5%

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

   9. Applied egg-rr 4.5%

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

       1. expm1-define 33.8%

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

       2. associate-*r/ 33.6%

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

       3. associate-*l/ 33.6%

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

       4. *-commutative 33.6%

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

       5. *-commutative 33.6%

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

   11. Simplified 33.6%

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

   12. Taylor expanded in x around 0 33.9%

       \[\leadsto \color{blue}{0.2 \cdot \left({x}^{5} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   13. Step-by-step derivation

       1. associate-*r* 33.9%

          \[\leadsto \color{blue}{\left(0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}}} + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right) \]

       2. associate-*r* 33.9%

          \[\leadsto \left(0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} \]

       3. distribute-rgt-out 33.9%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 2 \cdot x\right)} \]

       4. *-commutative 33.9%

          \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{{x}^{5} \cdot 0.2} + 2 \cdot x\right) \]

       5. *-commutative 33.9%

          \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left({x}^{5} \cdot 0.2 + \color{blue}{x \cdot 2}\right) \]

   14. Simplified 33.9%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left({x}^{5} \cdot 0.2 + x \cdot 2\right)} \]

   if 1.8999999999999999 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. add-sqr-sqrt 98.9%

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

      2. pow2 98.9%

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

   7. Applied egg-rr 32.1%

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

      1. unpow2 32.1%

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

      2. add-sqr-sqrt 33.9%

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

      3. expm1-log1p-u 33.8%

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

      4. expm1-undefine 4.5%

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

   9. Applied egg-rr 4.5%

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

       1. expm1-define 33.8%

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

       2. associate-*r/ 33.6%

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

       3. associate-*l/ 33.6%

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

       4. *-commutative 33.6%

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

       5. *-commutative 33.6%

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

   11. Simplified 33.6%

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

   12. Taylor expanded in x around inf 3.8%

       \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.2 \cdot \left({x}^{5} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   13. Step-by-step derivation

       1. +-commutative 3.8%

          \[\leadsto \color{blue}{0.2 \cdot \left({x}^{5} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)} \]

       2. associate-*r* 3.8%

          \[\leadsto \color{blue}{\left(0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}}} + 0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right) \]

       3. associate-*r* 3.8%

          \[\leadsto \left(0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}} \]

       4. distribute-rgt-out 3.8%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)} \]

       5. *-commutative 3.8%

          \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{{x}^{5} \cdot 0.2} + 0.047619047619047616 \cdot {x}^{7}\right) \]

   14. Simplified 3.8%

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

4. Final simplification 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 34.4% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \mathbf{if}\;x \leq 2.3:\\ \;\;\;\;t\_0 \cdot \left(0.2 \cdot {x}^{5} + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (if (<= x 2.3)
     (* t_0 (+ (* 0.2 (pow x 5.0)) (* x 2.0)))
     (* t_0 (* 0.047619047619047616 (pow x 7.0))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	double tmp;
	if (x <= 2.3) {
		tmp = t_0 * ((0.2 * pow(x, 5.0)) + (x * 2.0));
	} else {
		tmp = t_0 * (0.047619047619047616 * pow(x, 7.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	double tmp;
	if (x <= 2.3) {
		tmp = t_0 * ((0.2 * Math.pow(x, 5.0)) + (x * 2.0));
	} else {
		tmp = t_0 * (0.047619047619047616 * Math.pow(x, 7.0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / math.pi))
	tmp = 0
	if x <= 2.3:
		tmp = t_0 * ((0.2 * math.pow(x, 5.0)) + (x * 2.0))
	else:
		tmp = t_0 * (0.047619047619047616 * math.pow(x, 7.0))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / pi))
	tmp = 0.0
	if (x <= 2.3)
		tmp = Float64(t_0 * Float64(Float64(0.2 * (x ^ 5.0)) + Float64(x * 2.0)));
	else
		tmp = Float64(t_0 * Float64(0.047619047619047616 * (x ^ 7.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 / pi));
	tmp = 0.0;
	if (x <= 2.3)
		tmp = t_0 * ((0.2 * (x ^ 5.0)) + (x * 2.0));
	else
		tmp = t_0 * (0.047619047619047616 * (x ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 2.3], N[(t$95$0 * N[(N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2999999999999998

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. add-sqr-sqrt 98.9%

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

      2. pow2 98.9%

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

   7. Applied egg-rr 32.1%

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

      1. unpow2 32.1%

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

      2. add-sqr-sqrt 33.9%

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

      3. expm1-log1p-u 33.8%

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

      4. expm1-undefine 4.5%

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

   9. Applied egg-rr 4.5%

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

       1. expm1-define 33.8%

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

       2. associate-*r/ 33.6%

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

       3. associate-*l/ 33.6%

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

       4. *-commutative 33.6%

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

       5. *-commutative 33.6%

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

   11. Simplified 33.6%

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

   12. Taylor expanded in x around 0 33.9%

       \[\leadsto \color{blue}{0.2 \cdot \left({x}^{5} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   13. Step-by-step derivation

       1. associate-*r* 33.9%

          \[\leadsto \color{blue}{\left(0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}}} + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right) \]

       2. associate-*r* 33.9%

          \[\leadsto \left(0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} \]

       3. distribute-rgt-out 33.9%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 2 \cdot x\right)} \]

       4. *-commutative 33.9%

          \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{{x}^{5} \cdot 0.2} + 2 \cdot x\right) \]

       5. *-commutative 33.9%

          \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left({x}^{5} \cdot 0.2 + \color{blue}{x \cdot 2}\right) \]

   14. Simplified 33.9%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left({x}^{5} \cdot 0.2 + x \cdot 2\right)} \]

   if 2.2999999999999998 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. add-sqr-sqrt 98.9%

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

      2. pow2 98.9%

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

   7. Applied egg-rr 32.1%

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

      1. unpow2 32.1%

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

      2. add-sqr-sqrt 33.9%

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

      3. expm1-log1p-u 33.8%

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

      4. expm1-undefine 4.5%

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

   9. Applied egg-rr 4.5%

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

       1. expm1-define 33.8%

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

       2. associate-*r/ 33.6%

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

       3. associate-*l/ 33.6%

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

       4. *-commutative 33.6%

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

       5. *-commutative 33.6%

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

   11. Simplified 33.6%

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

   12. Taylor expanded in x around inf 3.8%

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

       1. associate-*r* 3.8%

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

   14. Simplified 3.8%

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

4. Final simplification 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.3:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.3% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;\mathsf{expm1}\left(t\_0 \cdot \left(x \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left(t\_0 \cdot {x}^{7}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (if (<= x 2.4)
     (expm1 (* t_0 (* x 2.0)))
     (* 0.047619047619047616 (* t_0 (pow x 7.0))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	double tmp;
	if (x <= 2.4) {
		tmp = expm1((t_0 * (x * 2.0)));
	} else {
		tmp = 0.047619047619047616 * (t_0 * pow(x, 7.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	double tmp;
	if (x <= 2.4) {
		tmp = Math.expm1((t_0 * (x * 2.0)));
	} else {
		tmp = 0.047619047619047616 * (t_0 * Math.pow(x, 7.0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / math.pi))
	tmp = 0
	if x <= 2.4:
		tmp = math.expm1((t_0 * (x * 2.0)))
	else:
		tmp = 0.047619047619047616 * (t_0 * math.pow(x, 7.0))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / pi))
	tmp = 0.0
	if (x <= 2.4)
		tmp = expm1(Float64(t_0 * Float64(x * 2.0)));
	else
		tmp = Float64(0.047619047619047616 * Float64(t_0 * (x ^ 7.0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 2.4], N[(Exp[N[(t$95$0 * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], N[(0.047619047619047616 * N[(t$95$0 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. add-sqr-sqrt 98.9%

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

      2. pow2 98.9%

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

   7. Applied egg-rr 32.1%

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

      1. unpow2 32.1%

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

      2. add-sqr-sqrt 33.9%

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

      3. expm1-log1p-u 33.8%

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

      4. expm1-undefine 4.5%

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

   9. Applied egg-rr 4.5%

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

       1. expm1-define 33.8%

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

       2. associate-*r/ 33.6%

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

       3. associate-*l/ 33.6%

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

       4. *-commutative 33.6%

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

       5. *-commutative 33.6%

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

   11. Simplified 33.6%

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

   12. Taylor expanded in x around 0 33.7%

       \[\leadsto \mathsf{expm1}\left(\color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)}\right) \]
   13. Step-by-step derivation

       1. associate-*r* 33.7%

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

       2. *-commutative 33.7%

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

       3. *-commutative 33.7%

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

   14. Simplified 33.7%

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

   if 2.39999999999999991 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. add-sqr-sqrt 98.9%

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

      2. pow2 98.9%

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

   7. Applied egg-rr 32.1%

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

      1. unpow2 32.1%

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

      2. add-sqr-sqrt 33.9%

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

      3. expm1-log1p-u 33.8%

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

      4. expm1-undefine 4.5%

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

   9. Applied egg-rr 4.5%

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

       1. expm1-define 33.8%

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

       2. associate-*r/ 33.6%

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

       3. associate-*l/ 33.6%

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

       4. *-commutative 33.6%

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

       5. *-commutative 33.6%

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

   11. Simplified 33.6%

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

   12. Taylor expanded in x around inf 3.8%

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

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;\mathsf{expm1}\left(\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.3% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;\mathsf{expm1}\left(t\_0 \cdot \left(x \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (if (<= x 2.4)
     (expm1 (* t_0 (* x 2.0)))
     (* t_0 (* 0.047619047619047616 (pow x 7.0))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	double tmp;
	if (x <= 2.4) {
		tmp = expm1((t_0 * (x * 2.0)));
	} else {
		tmp = t_0 * (0.047619047619047616 * pow(x, 7.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	double tmp;
	if (x <= 2.4) {
		tmp = Math.expm1((t_0 * (x * 2.0)));
	} else {
		tmp = t_0 * (0.047619047619047616 * Math.pow(x, 7.0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / math.pi))
	tmp = 0
	if x <= 2.4:
		tmp = math.expm1((t_0 * (x * 2.0)))
	else:
		tmp = t_0 * (0.047619047619047616 * math.pow(x, 7.0))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / pi))
	tmp = 0.0
	if (x <= 2.4)
		tmp = expm1(Float64(t_0 * Float64(x * 2.0)));
	else
		tmp = Float64(t_0 * Float64(0.047619047619047616 * (x ^ 7.0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 2.4], N[(Exp[N[(t$95$0 * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], N[(t$95$0 * N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. add-sqr-sqrt 98.9%

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

      2. pow2 98.9%

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

   7. Applied egg-rr 32.1%

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

      1. unpow2 32.1%

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

      2. add-sqr-sqrt 33.9%

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

      3. expm1-log1p-u 33.8%

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

      4. expm1-undefine 4.5%

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

   9. Applied egg-rr 4.5%

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

       1. expm1-define 33.8%

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

       2. associate-*r/ 33.6%

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

       3. associate-*l/ 33.6%

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

       4. *-commutative 33.6%

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

       5. *-commutative 33.6%

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

   11. Simplified 33.6%

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

   12. Taylor expanded in x around 0 33.7%

       \[\leadsto \mathsf{expm1}\left(\color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)}\right) \]
   13. Step-by-step derivation

       1. associate-*r* 33.7%

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

       2. *-commutative 33.7%

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

       3. *-commutative 33.7%

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

   14. Simplified 33.7%

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

   if 2.39999999999999991 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. add-sqr-sqrt 98.9%

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

      2. pow2 98.9%

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

   7. Applied egg-rr 32.1%

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

      1. unpow2 32.1%

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

      2. add-sqr-sqrt 33.9%

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

      3. expm1-log1p-u 33.8%

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

      4. expm1-undefine 4.5%

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

   9. Applied egg-rr 4.5%

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

       1. expm1-define 33.8%

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

       2. associate-*r/ 33.6%

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

       3. associate-*l/ 33.6%

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

       4. *-commutative 33.6%

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

       5. *-commutative 33.6%

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

   11. Simplified 33.6%

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

   12. Taylor expanded in x around inf 3.8%

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

       1. associate-*r* 3.8%

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

   14. Simplified 3.8%

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

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;\mathsf{expm1}\left(\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.3% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (expm1 (* (sqrt (/ 1.0 PI)) (* x 2.0))))

C

double code(double x) {
	return expm1((sqrt((1.0 / ((double) M_PI))) * (x * 2.0)));
}

Java

public static double code(double x) {
	return Math.expm1((Math.sqrt((1.0 / Math.PI)) * (x * 2.0)));
}

Python

def code(x):
	return math.expm1((math.sqrt((1.0 / math.pi)) * (x * 2.0)))

Julia

function code(x)
	return expm1(Float64(sqrt(Float64(1.0 / pi)) * Float64(x * 2.0)))
end

Wolfram

code[x_] := N[(Exp[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 99.5%

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

   1. add-sqr-sqrt 98.9%

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

   2. pow2 98.9%

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

7. Applied egg-rr 32.1%

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

   1. unpow2 32.1%

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

   2. add-sqr-sqrt 33.9%

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

   3. expm1-log1p-u 33.8%

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

   4. expm1-undefine 4.5%

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

9. Applied egg-rr 4.5%

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

    1. expm1-define 33.8%

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

    2. associate-*r/ 33.6%

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

    3. associate-*l/ 33.6%

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

    4. *-commutative 33.6%

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

    5. *-commutative 33.6%

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

11. Simplified 33.6%

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

12. Taylor expanded in x around 0 33.7%

    \[\leadsto \mathsf{expm1}\left(\color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)}\right) \]
13. Step-by-step derivation

    1. associate-*r* 33.7%

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

    2. *-commutative 33.7%

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

    3. *-commutative 33.7%

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

14. Simplified 33.7%

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

15. Final simplification 33.7%

    \[\leadsto \mathsf{expm1}\left(\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2\right)\right) \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024050 
(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)))))))