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

Percentage Accurate: 99.8% → 99.8%

Time: 10.9s

Alternatives: 8

Speedup: 3.6×

• 28.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Abs[x], $MachinePrecision] * N[Abs[N[(N[(N[(N[Power[x, 4.0], $MachinePrecision] * N[(0.2 + N[(0.047619047619047616 * N[(x * x), $MachinePrecision]), $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 99.8%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 99.9%

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

   1. pow2 99.9%

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

7. Applied egg-rr 99.9%

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

Alternative 2: 98.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2:\\ \;\;\;\;\left|x\right| \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, 2\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(\left|x\right| \cdot \left({x}^{6} \cdot {\pi}^{-0.5}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 2.0)
   (* (fabs x) (/ (fma 0.2 (pow x 4.0) 2.0) (sqrt PI)))
   (fabs (* 0.047619047619047616 (* (fabs x) (* (pow x 6.0) (pow PI -0.5)))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 2.0) {
		tmp = fabs(x) * (fma(0.2, pow(x, 4.0), 2.0) / sqrt(((double) M_PI)));
	} else {
		tmp = fabs((0.047619047619047616 * (fabs(x) * (pow(x, 6.0) * pow(((double) M_PI), -0.5)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 2.0)
		tmp = Float64(abs(x) * Float64(fma(0.2, (x ^ 4.0), 2.0) / sqrt(pi)));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64(abs(x) * Float64((x ^ 6.0) * (pi ^ -0.5)))));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2

   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%

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

   5. Taylor expanded in x around 0 98.9%

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

   6. Taylor expanded in x around 0 97.9%

      \[\leadsto \left|x\right| \cdot \left|\frac{0.2 \cdot {x}^{4} + \color{blue}{2}}{\sqrt{\pi}}\right| \]
   7. Step-by-step derivation

      1. pow1 97.9%

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

      2. mul-fabs 97.9%

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

      3. *-commutative 97.9%

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

      4. fma-define 97.9%

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

   8. Applied egg-rr 97.9%

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

      1. unpow1 97.9%

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

      2. fabs-mul 97.9%

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

      3. rem-square-sqrt 96.2%

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

      4. fabs-sqr 96.2%

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

      5. rem-square-sqrt 97.9%

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

      6. fma-undefine 97.9%

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

      7. *-commutative 97.9%

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

      8. fma-define 97.9%

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

   10. Simplified 97.9%

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

   if 2 < (fabs.f64 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%

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

   5. Taylor expanded in x around 0 99.6%

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

   6. Taylor expanded in x around inf 98.6%

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

      1. associate-*l* 98.6%

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

      2. *-commutative 98.6%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in x around 0 98.6%

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

       1. *-commutative 98.6%

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

       2. associate-*l* 98.6%

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

       3. rem-exp-log 98.6%

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

       4. exp-neg 98.6%

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

       5. unpow1/2 98.6%

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

       6. exp-prod 98.6%

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

       7. distribute-lft-neg-out 98.6%

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

       8. distribute-rgt-neg-in 98.6%

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

       9. metadata-eval 98.6%

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

       10. exp-to-pow 98.6%

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

   11. Simplified 98.6%

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

Alternative 3: 99.0% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Abs[x], $MachinePrecision] * N[Abs[N[(N[(N[(N[Power[x, 4.0], $MachinePrecision] * N[(0.2 + N[(0.047619047619047616 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / 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 99.9%

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

5. Taylor expanded in x around 0 99.9%

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

   1. pow2 99.9%

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

7. Applied egg-rr 99.9%

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

8. Taylor expanded in x around 0 98.6%

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

Alternative 4: 93.2% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.20000000000000001

   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%

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

   5. Taylor expanded in x around 0 99.3%

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

   6. Taylor expanded in x around 0 98.4%

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

   7. Taylor expanded in x around 0 98.4%

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

      1. *-un-lft-identity 98.4%

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

      2. fabs-mul 98.4%

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

      3. metadata-eval 98.4%

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

      4. pow1/2 98.4%

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

      5. inv-pow 98.4%

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

      6. pow-pow 98.4%

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

      7. metadata-eval 98.4%

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

   9. Applied egg-rr 98.4%

      \[\leadsto \left|x\right| \cdot \color{blue}{\left(1 \cdot \left(2 \cdot \left|{\pi}^{-0.5}\right|\right)\right)} \]
   10. Step-by-step derivation

       1. *-lft-identity 98.4%

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

       2. rem-square-sqrt 98.4%

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

       3. fabs-sqr 98.4%

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

       4. rem-square-sqrt 98.4%

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

   11. Simplified 98.4%

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

   if 0.20000000000000001 < (fabs.f64 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 99.9%

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

   5. Taylor expanded in x around 0 85.0%

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

   6. Taylor expanded in x around 0 85.0%

      \[\leadsto \left|x\right| \cdot \left|\frac{0.2 \cdot {x}^{4} + \color{blue}{2}}{\sqrt{\pi}}\right| \]
   7. Step-by-step derivation

      1. pow1 85.0%

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

      2. mul-fabs 85.0%

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

      3. *-commutative 85.0%

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

      4. fma-define 85.0%

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

   8. Applied egg-rr 85.0%

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

      1. unpow1 85.0%

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

      2. associate-*r/ 85.0%

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

   10. Simplified 85.0%

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

   11. Taylor expanded in x around inf 85.0%

       \[\leadsto \left|\frac{\color{blue}{0.2 \cdot {x}^{5}}}{\sqrt{\pi}}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 94.8% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\mathsf{expm1}\left(\left|x \cdot \left(2 \cdot {\pi}^{-0.5}\right)\right|\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log1p (expm1 (fabs (* x (* 2.0 (pow PI -0.5)))))))

C

double code(double x) {
	return log1p(expm1(fabs((x * (2.0 * pow(((double) M_PI), -0.5))))));
}

Java

public static double code(double x) {
	return Math.log1p(Math.expm1(Math.abs((x * (2.0 * Math.pow(Math.PI, -0.5))))));
}

Python

def code(x):
	return math.log1p(math.expm1(math.fabs((x * (2.0 * math.pow(math.pi, -0.5))))))

Julia

function code(x)
	return log1p(expm1(abs(Float64(x * Float64(2.0 * (pi ^ -0.5))))))
end

Wolfram

code[x_] := N[Log[1 + N[(Exp[N[Abs[N[(x * N[(2.0 * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $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 99.9%

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

5. Taylor expanded in x around 0 94.2%

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

6. Taylor expanded in x around 0 93.6%

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

7. Taylor expanded in x around 0 65.5%

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

   1. log1p-expm1-u 94.2%

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

   2. mul-fabs 94.2%

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

   3. *-commutative 94.2%

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

   4. pow1/2 94.2%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\left|x \cdot \left(\color{blue}{{\left(\frac{1}{\pi}\right)}^{0.5}} \cdot 2\right)\right|\right)\right) \]

   5. inv-pow 94.2%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\left|x \cdot \left({\color{blue}{\left({\pi}^{-1}\right)}}^{0.5} \cdot 2\right)\right|\right)\right) \]

   6. pow-pow 94.2%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\left|x \cdot \left(\color{blue}{{\pi}^{\left(-1 \cdot 0.5\right)}} \cdot 2\right)\right|\right)\right) \]

   7. metadata-eval 94.2%

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

9. Applied egg-rr 94.2%

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

10. Final simplification 94.2%

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\left|x \cdot \left(2 \cdot {\pi}^{-0.5}\right)\right|\right)\right) \]
11. Add Preprocessing

Alternative 6: 92.8% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 94.2%

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

6. Taylor expanded in x around 0 93.6%

   \[\leadsto \left|x\right| \cdot \left|\frac{0.2 \cdot {x}^{4} + \color{blue}{2}}{\sqrt{\pi}}\right| \]
7. Step-by-step derivation

   1. pow1 93.6%

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

   2. mul-fabs 93.6%

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

   3. *-commutative 93.6%

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

   4. fma-define 93.6%

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

8. Applied egg-rr 93.6%

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

   1. unpow1 93.6%

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

   2. associate-*r/ 93.2%

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

10. Simplified 93.2%

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

    1. fma-undefine 93.2%

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

    2. *-commutative 93.2%

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

12. Applied egg-rr 93.2%

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

13. Final simplification 93.2%

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

Alternative 7: 68.5% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Abs[x], $MachinePrecision] * N[(2.0 * N[Power[Pi, -0.5], $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 99.9%

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

5. Taylor expanded in x around 0 94.2%

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

6. Taylor expanded in x around 0 93.6%

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

7. Taylor expanded in x around 0 65.5%

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

   1. *-un-lft-identity 65.5%

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

   2. fabs-mul 65.5%

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

   3. metadata-eval 65.5%

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

   4. pow1/2 65.5%

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

   5. inv-pow 65.5%

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

   6. pow-pow 65.5%

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

   7. metadata-eval 65.5%

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

9. Applied egg-rr 65.5%

   \[\leadsto \left|x\right| \cdot \color{blue}{\left(1 \cdot \left(2 \cdot \left|{\pi}^{-0.5}\right|\right)\right)} \]
10. Step-by-step derivation

    1. *-lft-identity 65.5%

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

    2. rem-square-sqrt 65.5%

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

    3. fabs-sqr 65.5%

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

    4. rem-square-sqrt 65.5%

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

11. Simplified 65.5%

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

Alternative 8: 68.0% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 94.2%

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

6. Taylor expanded in x around 0 93.6%

   \[\leadsto \left|x\right| \cdot \left|\frac{0.2 \cdot {x}^{4} + \color{blue}{2}}{\sqrt{\pi}}\right| \]
7. Step-by-step derivation

   1. pow1 93.6%

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

   2. mul-fabs 93.6%

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

   3. *-commutative 93.6%

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

   4. fma-define 93.6%

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

8. Applied egg-rr 93.6%

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

   1. unpow1 93.6%

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

   2. associate-*r/ 93.2%

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

10. Simplified 93.2%

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

11. Taylor expanded in x around 0 65.1%

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

    1. *-commutative 65.1%

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

13. Simplified 65.1%

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

Reproduce

herbie shell --seed 2024139 
(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)))))))