Jmat.Real.erfi, branch x greater than or equal to 5

Percentage Accurate: 100.0% → 100.0%

Time: 13.6s

Alternatives: 11

Speedup: 2.3×

• 99.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot t\_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot t\_1\right) + \frac{3}{4} \cdot t\_2\right) + \frac{15}{8} \cdot \left(\left(t\_2 \cdot t\_0\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x)))
        (t_1 (* (* t_0 t_0) t_0))
        (t_2 (* (* t_1 t_0) t_0)))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
    (+
     (+ (+ t_0 (* (/ 1.0 2.0) t_1)) (* (/ 3.0 4.0) t_2))
     (* (/ 15.0 8.0) (* (* t_2 t_0) t_0))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / sqrt(((double) M_PI))) * exp((fabs(x) * fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((Math.abs(x) * Math.abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	t_1 = (t_0 * t_0) * t_0
	t_2 = (t_1 * t_0) * t_0
	return ((1.0 / math.sqrt(math.pi)) * math.exp((math.fabs(x) * math.fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	t_1 = Float64(Float64(t_0 * t_0) * t_0)
	t_2 = Float64(Float64(t_1 * t_0) * t_0)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(Float64(Float64(t_0 + Float64(Float64(1.0 / 2.0) * t_1)) + Float64(Float64(3.0 / 4.0) * t_2)) + Float64(Float64(15.0 / 8.0) * Float64(Float64(t_2 * t_0) * t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	t_1 = (t_0 * t_0) * t_0;
	t_2 = (t_1 * t_0) * t_0;
	tmp = ((1.0 / sqrt(pi)) * exp((abs(x) * abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 + N[(N[(1.0 / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(t$95$2 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot t\_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot t\_1\right) + \frac{3}{4} \cdot t\_2\right) + \frac{15}{8} \cdot \left(\left(t\_2 \cdot t\_0\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x)))
        (t_1 (* (* t_0 t_0) t_0))
        (t_2 (* (* t_1 t_0) t_0)))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
    (+
     (+ (+ t_0 (* (/ 1.0 2.0) t_1)) (* (/ 3.0 4.0) t_2))
     (* (/ 15.0 8.0) (* (* t_2 t_0) t_0))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / sqrt(((double) M_PI))) * exp((fabs(x) * fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((Math.abs(x) * Math.abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	t_1 = (t_0 * t_0) * t_0
	t_2 = (t_1 * t_0) * t_0
	return ((1.0 / math.sqrt(math.pi)) * math.exp((math.fabs(x) * math.fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	t_1 = Float64(Float64(t_0 * t_0) * t_0)
	t_2 = Float64(Float64(t_1 * t_0) * t_0)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(Float64(Float64(t_0 + Float64(Float64(1.0 / 2.0) * t_1)) + Float64(Float64(3.0 / 4.0) * t_2)) + Float64(Float64(15.0 / 8.0) * Float64(Float64(t_2 * t_0) * t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	t_1 = (t_0 * t_0) * t_0;
	t_2 = (t_1 * t_0) * t_0;
	tmp = ((1.0 / sqrt(pi)) * exp((abs(x) * abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 + N[(N[(1.0 / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(t$95$2 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Alternative 1: 100.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{5}} + 0.5 \cdot {\left(\left|x\right|\right)}^{-3}\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (sqrt PI))
  (+
   (/ 1.875 (pow (fabs x) 7.0))
   (+
    (/ 1.0 (fabs x))
    (+ (/ 0.75 (pow (fabs x) 5.0)) (* 0.5 (pow (fabs x) -3.0)))))))

C

double code(double x) {
	return (pow(exp(x), x) / sqrt(((double) M_PI))) * ((1.875 / pow(fabs(x), 7.0)) + ((1.0 / fabs(x)) + ((0.75 / pow(fabs(x), 5.0)) + (0.5 * pow(fabs(x), -3.0)))));
}

Java

public static double code(double x) {
	return (Math.pow(Math.exp(x), x) / Math.sqrt(Math.PI)) * ((1.875 / Math.pow(Math.abs(x), 7.0)) + ((1.0 / Math.abs(x)) + ((0.75 / Math.pow(Math.abs(x), 5.0)) + (0.5 * Math.pow(Math.abs(x), -3.0)))));
}

Python

def code(x):
	return (math.pow(math.exp(x), x) / math.sqrt(math.pi)) * ((1.875 / math.pow(math.fabs(x), 7.0)) + ((1.0 / math.fabs(x)) + ((0.75 / math.pow(math.fabs(x), 5.0)) + (0.5 * math.pow(math.fabs(x), -3.0)))))

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / sqrt(pi)) * Float64(Float64(1.875 / (abs(x) ^ 7.0)) + Float64(Float64(1.0 / abs(x)) + Float64(Float64(0.75 / (abs(x) ^ 5.0)) + Float64(0.5 * (abs(x) ^ -3.0))))))
end

MATLAB

function tmp = code(x)
	tmp = ((exp(x) ^ x) / sqrt(pi)) * ((1.875 / (abs(x) ^ 7.0)) + ((1.0 / abs(x)) + ((0.75 / (abs(x) ^ 5.0)) + (0.5 * (abs(x) ^ -3.0)))));
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(1.875 / N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(0.75 / N[Power[N[Abs[x], $MachinePrecision], 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Power[N[Abs[x], $MachinePrecision], -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \color{blue}{\left(1 + \left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{4}} + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{6}}\right)\right)}\right) \]
6. Step-by-step derivation

   1. associate-+r+ 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \color{blue}{\left(\left(1 + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{4}}\right) + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{6}}\right)}\right) \]

   2. +-commutative 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \color{blue}{\left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{6}} + \left(1 + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{4}}\right)\right)}\right) \]

   3. associate-*r/ 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(\color{blue}{\frac{1.875 \cdot 1}{{\left(\left|x\right|\right)}^{6}}} + \left(1 + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{4}}\right)\right)\right) \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(\frac{\color{blue}{1.875}}{{\left(\left|x\right|\right)}^{6}} + \left(1 + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{4}}\right)\right)\right) \]

   5. associate-*r/ 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(\frac{1.875}{{\left(\left|x\right|\right)}^{6}} + \left(1 + \color{blue}{\frac{0.75 \cdot 1}{{\left(\left|x\right|\right)}^{4}}}\right)\right)\right) \]

   6. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(\frac{1.875}{{\left(\left|x\right|\right)}^{6}} + \left(1 + \frac{\color{blue}{0.75}}{{\left(\left|x\right|\right)}^{4}}\right)\right)\right) \]

7. Simplified 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \color{blue}{\left(\frac{1.875}{{\left(\left|x\right|\right)}^{6}} + \left(1 + \frac{0.75}{{\left(\left|x\right|\right)}^{4}}\right)\right)}\right) \]

8. Taylor expanded in x around 0 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{\left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right)} \]
9. Step-by-step derivation

   1. associate-*r/ 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{\left(\left|x\right|\right)}^{3}}} + \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \]

   2. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{\color{blue}{0.5}}{{\left(\left|x\right|\right)}^{3}} + \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \]

   3. associate-*r/ 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{\left(\left|x\right|\right)}^{3}} + \left(\color{blue}{\frac{1.875 \cdot 1}{{\left(\left|x\right|\right)}^{7}}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{\left(\left|x\right|\right)}^{3}} + \left(\frac{\color{blue}{1.875}}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \]

   5. associate-*r/ 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{\left(\left|x\right|\right)}^{3}} + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \color{blue}{\frac{0.75 \cdot 1}{{\left(\left|x\right|\right)}^{5}}}\right)\right)\right) \]

   6. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{\left(\left|x\right|\right)}^{3}} + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{\color{blue}{0.75}}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \]

10. Simplified 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{0.5}{{\left(\left|x\right|\right)}^{3}} + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right)} \]
11. Step-by-step derivation

    1. add-cube-cbrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\color{blue}{\left(\sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} \cdot \sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}\right) \cdot \sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}} + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \]

    2. pow2 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\color{blue}{{\left(\sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}\right)}^{2}} \cdot \sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \]

    3. cbrt-div 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left({\color{blue}{\left(\frac{\sqrt[3]{0.5}}{\sqrt[3]{{\left(\left|x\right|\right)}^{3}}}\right)}}^{2} \cdot \sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \]

    4. rem-cbrt-cube 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left({\left(\frac{\sqrt[3]{0.5}}{\color{blue}{\left|x\right|}}\right)}^{2} \cdot \sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \]

    5. cbrt-div 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left({\left(\frac{\sqrt[3]{0.5}}{\left|x\right|}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{0.5}}{\sqrt[3]{{\left(\left|x\right|\right)}^{3}}}} + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \]

    6. rem-cbrt-cube 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left({\left(\frac{\sqrt[3]{0.5}}{\left|x\right|}\right)}^{2} \cdot \frac{\sqrt[3]{0.5}}{\color{blue}{\left|x\right|}} + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \]

12. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\color{blue}{{\left(\frac{\sqrt[3]{0.5}}{\left|x\right|}\right)}^{2} \cdot \frac{\sqrt[3]{0.5}}{\left|x\right|}} + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \]
13. Step-by-step derivation

    1. unpow2 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\color{blue}{\left(\frac{\sqrt[3]{0.5}}{\left|x\right|} \cdot \frac{\sqrt[3]{0.5}}{\left|x\right|}\right)} \cdot \frac{\sqrt[3]{0.5}}{\left|x\right|} + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \]

    2. unpow3 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\color{blue}{{\left(\frac{\sqrt[3]{0.5}}{\left|x\right|}\right)}^{3}} + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \]

14. Simplified 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\color{blue}{{\left(\frac{\sqrt[3]{0.5}}{\left|x\right|}\right)}^{3}} + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \]

15. Taylor expanded in x around 0 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{\left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \frac{{\left(\sqrt[3]{0.5}\right)}^{3}}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)} \]

17. Simplified 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{5}} + 0.5 \cdot {\left(\left|x\right|\right)}^{-3}\right)\right)\right)} \]
18. Add Preprocessing

Alternative 2: 100.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{\left(\left|x\right|\right)}^{3}} + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (sqrt PI))
  (+
   (/ 0.5 (pow (fabs x) 3.0))
   (+
    (/ 1.875 (pow (fabs x) 7.0))
    (+ (/ 1.0 (fabs x)) (/ 0.75 (pow (fabs x) 5.0)))))))

C

double code(double x) {
	return (pow(exp(x), x) / sqrt(((double) M_PI))) * ((0.5 / pow(fabs(x), 3.0)) + ((1.875 / pow(fabs(x), 7.0)) + ((1.0 / fabs(x)) + (0.75 / pow(fabs(x), 5.0)))));
}

Java

public static double code(double x) {
	return (Math.pow(Math.exp(x), x) / Math.sqrt(Math.PI)) * ((0.5 / Math.pow(Math.abs(x), 3.0)) + ((1.875 / Math.pow(Math.abs(x), 7.0)) + ((1.0 / Math.abs(x)) + (0.75 / Math.pow(Math.abs(x), 5.0)))));
}

Python

def code(x):
	return (math.pow(math.exp(x), x) / math.sqrt(math.pi)) * ((0.5 / math.pow(math.fabs(x), 3.0)) + ((1.875 / math.pow(math.fabs(x), 7.0)) + ((1.0 / math.fabs(x)) + (0.75 / math.pow(math.fabs(x), 5.0)))))

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / sqrt(pi)) * Float64(Float64(0.5 / (abs(x) ^ 3.0)) + Float64(Float64(1.875 / (abs(x) ^ 7.0)) + Float64(Float64(1.0 / abs(x)) + Float64(0.75 / (abs(x) ^ 5.0))))))
end

MATLAB

function tmp = code(x)
	tmp = ((exp(x) ^ x) / sqrt(pi)) * ((0.5 / (abs(x) ^ 3.0)) + ((1.875 / (abs(x) ^ 7.0)) + ((1.0 / abs(x)) + (0.75 / (abs(x) ^ 5.0)))));
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / N[Power[N[Abs[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.875 / N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(0.75 / N[Power[N[Abs[x], $MachinePrecision], 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \color{blue}{\left(1 + \left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{4}} + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{6}}\right)\right)}\right) \]
6. Step-by-step derivation

   1. associate-+r+ 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \color{blue}{\left(\left(1 + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{4}}\right) + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{6}}\right)}\right) \]

   2. +-commutative 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \color{blue}{\left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{6}} + \left(1 + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{4}}\right)\right)}\right) \]

   3. associate-*r/ 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(\color{blue}{\frac{1.875 \cdot 1}{{\left(\left|x\right|\right)}^{6}}} + \left(1 + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{4}}\right)\right)\right) \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(\frac{\color{blue}{1.875}}{{\left(\left|x\right|\right)}^{6}} + \left(1 + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{4}}\right)\right)\right) \]

   5. associate-*r/ 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(\frac{1.875}{{\left(\left|x\right|\right)}^{6}} + \left(1 + \color{blue}{\frac{0.75 \cdot 1}{{\left(\left|x\right|\right)}^{4}}}\right)\right)\right) \]

   6. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(\frac{1.875}{{\left(\left|x\right|\right)}^{6}} + \left(1 + \frac{\color{blue}{0.75}}{{\left(\left|x\right|\right)}^{4}}\right)\right)\right) \]

7. Simplified 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \color{blue}{\left(\frac{1.875}{{\left(\left|x\right|\right)}^{6}} + \left(1 + \frac{0.75}{{\left(\left|x\right|\right)}^{4}}\right)\right)}\right) \]

8. Taylor expanded in x around 0 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{\left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right)} \]
9. Step-by-step derivation

   1. associate-*r/ 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{\left(\left|x\right|\right)}^{3}}} + \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \]

   2. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{\color{blue}{0.5}}{{\left(\left|x\right|\right)}^{3}} + \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \]

   3. associate-*r/ 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{\left(\left|x\right|\right)}^{3}} + \left(\color{blue}{\frac{1.875 \cdot 1}{{\left(\left|x\right|\right)}^{7}}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{\left(\left|x\right|\right)}^{3}} + \left(\frac{\color{blue}{1.875}}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \]

   5. associate-*r/ 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{\left(\left|x\right|\right)}^{3}} + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \color{blue}{\frac{0.75 \cdot 1}{{\left(\left|x\right|\right)}^{5}}}\right)\right)\right) \]

   6. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.5}{{\left(\left|x\right|\right)}^{3}} + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{\color{blue}{0.75}}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \]

10. Simplified 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{0.5}{{\left(\left|x\right|\right)}^{3}} + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right)} \]
11. Add Preprocessing

Alternative 3: 100.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{x \cdot x}\right) \cdot \left(\frac{0.75}{{\left(\left|x\right|\right)}^{5}} + \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* (/ 1.0 (sqrt PI)) (exp (* x x)))
  (+
   (/ 0.75 (pow (fabs x) 5.0))
   (+
    (* 1.875 (/ 1.0 (pow (fabs x) 7.0)))
    (+ (/ 1.0 (fabs x)) (* 0.5 (/ 1.0 (pow (fabs x) 3.0))))))))

C

double code(double x) {
	return ((1.0 / sqrt(((double) M_PI))) * exp((x * x))) * ((0.75 / pow(fabs(x), 5.0)) + ((1.875 * (1.0 / pow(fabs(x), 7.0))) + ((1.0 / fabs(x)) + (0.5 * (1.0 / pow(fabs(x), 3.0))))));
}

Java

public static double code(double x) {
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((x * x))) * ((0.75 / Math.pow(Math.abs(x), 5.0)) + ((1.875 * (1.0 / Math.pow(Math.abs(x), 7.0))) + ((1.0 / Math.abs(x)) + (0.5 * (1.0 / Math.pow(Math.abs(x), 3.0))))));
}

Python

def code(x):
	return ((1.0 / math.sqrt(math.pi)) * math.exp((x * x))) * ((0.75 / math.pow(math.fabs(x), 5.0)) + ((1.875 * (1.0 / math.pow(math.fabs(x), 7.0))) + ((1.0 / math.fabs(x)) + (0.5 * (1.0 / math.pow(math.fabs(x), 3.0))))))

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(x * x))) * Float64(Float64(0.75 / (abs(x) ^ 5.0)) + Float64(Float64(1.875 * Float64(1.0 / (abs(x) ^ 7.0))) + Float64(Float64(1.0 / abs(x)) + Float64(0.5 * Float64(1.0 / (abs(x) ^ 3.0)))))))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / sqrt(pi)) * exp((x * x))) * ((0.75 / (abs(x) ^ 5.0)) + ((1.875 * (1.0 / (abs(x) ^ 7.0))) + ((1.0 / abs(x)) + (0.5 * (1.0 / (abs(x) ^ 3.0))))));
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.75 / N[Power[N[Abs[x], $MachinePrecision], 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.875 * N[(1.0 / N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Power[N[Abs[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{\pi}} \cdot e^{x \cdot x}\right) \cdot \left(\left(\left(0.5 \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) + \frac{1}{\left|x\right|}\right) + 0.75 \cdot \left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right)\right) + 1.875 \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{x \cdot x}\right) \cdot \color{blue}{\left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)} \]

6. Taylor expanded in x around 0 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{x \cdot x}\right) \cdot \left(\color{blue}{\frac{0.75}{{\left(\left|x\right|\right)}^{5}}} + \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]
7. Add Preprocessing

Alternative 4: 100.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\left|x\right|\right)}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (exp (* x x)) (sqrt PI))
  (fma
   0.75
   (pow (/ 1.0 (fabs x)) 5.0)
   (fma 1.875 (pow (fabs x) -7.0) (/ (+ 1.0 (/ 0.5 (* x x))) (fabs x))))))

C

double code(double x) {
	return (exp((x * x)) / sqrt(((double) M_PI))) * fma(0.75, pow((1.0 / fabs(x)), 5.0), fma(1.875, pow(fabs(x), -7.0), ((1.0 + (0.5 / (x * x))) / fabs(x))));
}

Julia

function code(x)
	return Float64(Float64(exp(Float64(x * x)) / sqrt(pi)) * fma(0.75, (Float64(1.0 / abs(x)) ^ 5.0), fma(1.875, (abs(x) ^ -7.0), Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x)))))
end

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(0.75 * N[Power[N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision], 5.0], $MachinePrecision] + N[(1.875 * N[Power[N[Abs[x], $MachinePrecision], -7.0], $MachinePrecision] + N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-un-lft-identity 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{1 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   2. inv-pow 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 1 \cdot {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   3. pow-pow 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 1 \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 7\right)}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 1 \cdot {\left(\left|x\right|\right)}^{\color{blue}{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{1 \cdot {\left(\left|x\right|\right)}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
7. Step-by-step derivation

   1. *-lft-identity 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{{\left(\left|x\right|\right)}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

8. Simplified 100.0%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{{\left(\left|x\right|\right)}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
9. Add Preprocessing

Alternative 5: 99.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left|x\right|\right)}^{-3}\\ \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{+107}:\\ \;\;\;\;\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, 0.5 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;e^{{\left({\log t\_0}^{3}\right)}^{0.3333333333333333}} \cdot \left(0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (fabs x) -3.0)))
   (if (<= (fabs x) 2e+107)
     (*
      (/ (exp (* x x)) (sqrt PI))
      (fma 0.75 (pow (/ 1.0 (fabs x)) 5.0) (* 0.5 t_0)))
     (*
      (exp (pow (pow (log t_0) 3.0) 0.3333333333333333))
      (* 0.5 (sqrt (/ 1.0 PI)))))))

C

double code(double x) {
	double t_0 = pow(fabs(x), -3.0);
	double tmp;
	if (fabs(x) <= 2e+107) {
		tmp = (exp((x * x)) / sqrt(((double) M_PI))) * fma(0.75, pow((1.0 / fabs(x)), 5.0), (0.5 * t_0));
	} else {
		tmp = exp(pow(pow(log(t_0), 3.0), 0.3333333333333333)) * (0.5 * sqrt((1.0 / ((double) M_PI))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = abs(x) ^ -3.0
	tmp = 0.0
	if (abs(x) <= 2e+107)
		tmp = Float64(Float64(exp(Float64(x * x)) / sqrt(pi)) * fma(0.75, (Float64(1.0 / abs(x)) ^ 5.0), Float64(0.5 * t_0)));
	else
		tmp = Float64(exp(((log(t_0) ^ 3.0) ^ 0.3333333333333333)) * Float64(0.5 * sqrt(Float64(1.0 / pi))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Abs[x], $MachinePrecision], -3.0], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 2e+107], N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(0.75 * N[Power[N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision], 5.0], $MachinePrecision] + N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[Power[N[Power[N[Log[t$95$0], $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]], $MachinePrecision] * N[(0.5 * N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.9999999999999999e107

   1. Initial program 99.9%

      \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 92.7%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \color{blue}{\frac{0.5}{{x}^{2} \cdot \left|x\right|}}\right) \]
   6. Step-by-step derivation

      1. metadata-eval 92.7%

         \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \frac{\color{blue}{0.5 \cdot 1}}{{x}^{2} \cdot \left|x\right|}\right) \]

      2. associate-*r/ 92.7%

         \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \color{blue}{0.5 \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}}\right) \]

      3. *-commutative 92.7%

         \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, 0.5 \cdot \frac{1}{\color{blue}{\left|x\right| \cdot {x}^{2}}}\right) \]

      4. unpow2 92.7%

         \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, 0.5 \cdot \frac{1}{\left|x\right| \cdot \color{blue}{\left(x \cdot x\right)}}\right) \]

      5. sqr-abs 92.7%

         \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, 0.5 \cdot \frac{1}{\left|x\right| \cdot \color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)}}\right) \]

      6. cube-mult 92.7%

         \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, 0.5 \cdot \frac{1}{\color{blue}{{\left(\left|x\right|\right)}^{3}}}\right) \]

      7. exp-to-pow 92.7%

         \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, 0.5 \cdot \frac{1}{\color{blue}{e^{\log \left(\left|x\right|\right) \cdot 3}}}\right) \]

      8. *-commutative 92.7%

         \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, 0.5 \cdot \frac{1}{e^{\color{blue}{3 \cdot \log \left(\left|x\right|\right)}}}\right) \]

      9. exp-neg 98.2%

         \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, 0.5 \cdot \color{blue}{e^{-3 \cdot \log \left(\left|x\right|\right)}}\right) \]

      10. distribute-lft-neg-in 98.2%

          \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, 0.5 \cdot e^{\color{blue}{\left(-3\right) \cdot \log \left(\left|x\right|\right)}}\right) \]

      11. metadata-eval 98.2%

          \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, 0.5 \cdot e^{\color{blue}{-3} \cdot \log \left(\left|x\right|\right)}\right) \]

      12. *-commutative 98.2%

          \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, 0.5 \cdot e^{\color{blue}{\log \left(\left|x\right|\right) \cdot -3}}\right) \]

      13. exp-to-pow 98.2%

          \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, 0.5 \cdot \color{blue}{{\left(\left|x\right|\right)}^{-3}}\right) \]

   7. Simplified 98.2%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \color{blue}{0.5 \cdot {\left(\left|x\right|\right)}^{-3}}\right) \]

   if 1.9999999999999999e107 < (fabs.f64 x)

   1. Initial program 100.0%

      \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 1.6%

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

      1. *-commutative 1.6%

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

      2. associate-*l* 1.6%

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

      3. *-commutative 1.6%

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

      4. unpow2 1.6%

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

      5. sqr-abs 1.6%

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

      6. cube-mult 1.6%

         \[\leadsto \frac{1}{\color{blue}{{\left(\left|x\right|\right)}^{3}}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

      7. exp-to-pow 1.6%

         \[\leadsto \frac{1}{\color{blue}{e^{\log \left(\left|x\right|\right) \cdot 3}}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

      8. *-commutative 1.6%

         \[\leadsto \frac{1}{e^{\color{blue}{3 \cdot \log \left(\left|x\right|\right)}}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

      9. exp-neg 1.6%

         \[\leadsto \color{blue}{e^{-3 \cdot \log \left(\left|x\right|\right)}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

      10. distribute-lft-neg-in 1.6%

          \[\leadsto e^{\color{blue}{\left(-3\right) \cdot \log \left(\left|x\right|\right)}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

      11. metadata-eval 1.6%

          \[\leadsto e^{\color{blue}{-3} \cdot \log \left(\left|x\right|\right)} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

      12. *-commutative 1.6%

          \[\leadsto e^{\color{blue}{\log \left(\left|x\right|\right) \cdot -3}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

      13. exp-to-pow 1.6%

          \[\leadsto \color{blue}{{\left(\left|x\right|\right)}^{-3}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

   7. Simplified 1.6%

      \[\leadsto \color{blue}{{\left(\left|x\right|\right)}^{-3} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right)} \]
   8. Step-by-step derivation

      1. add-exp-log 1.6%

         \[\leadsto \color{blue}{e^{\log \left({\left(\left|x\right|\right)}^{-3}\right)}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

      2. log-pow 1.6%

         \[\leadsto e^{\color{blue}{-3 \cdot \log \left(\left|x\right|\right)}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

   9. Applied egg-rr 1.6%

      \[\leadsto \color{blue}{e^{-3 \cdot \log \left(\left|x\right|\right)}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]
   10. Step-by-step derivation

       1. add-cbrt-cube 1.6%

          \[\leadsto e^{-3 \cdot \color{blue}{\sqrt[3]{\left(\log \left(\left|x\right|\right) \cdot \log \left(\left|x\right|\right)\right) \cdot \log \left(\left|x\right|\right)}}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

       2. pow3 1.6%

          \[\leadsto e^{-3 \cdot \sqrt[3]{\color{blue}{{\log \left(\left|x\right|\right)}^{3}}}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

   11. Applied egg-rr 1.6%

       \[\leadsto e^{-3 \cdot \color{blue}{\sqrt[3]{{\log \left(\left|x\right|\right)}^{3}}}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]
   12. Step-by-step derivation

       1. rem-cbrt-cube 1.6%

          \[\leadsto e^{-3 \cdot \color{blue}{\log \left(\left|x\right|\right)}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

       2. add-cbrt-cube 1.6%

          \[\leadsto e^{\color{blue}{\sqrt[3]{\left(\left(-3 \cdot \log \left(\left|x\right|\right)\right) \cdot \left(-3 \cdot \log \left(\left|x\right|\right)\right)\right) \cdot \left(-3 \cdot \log \left(\left|x\right|\right)\right)}}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

       3. pow1/3 0.0%

          \[\leadsto e^{\color{blue}{{\left(\left(\left(-3 \cdot \log \left(\left|x\right|\right)\right) \cdot \left(-3 \cdot \log \left(\left|x\right|\right)\right)\right) \cdot \left(-3 \cdot \log \left(\left|x\right|\right)\right)\right)}^{0.3333333333333333}}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

       4. rem-cbrt-cube 0.0%

          \[\leadsto e^{{\left(\left(\left(-3 \cdot \color{blue}{\sqrt[3]{{\log \left(\left|x\right|\right)}^{3}}}\right) \cdot \left(-3 \cdot \log \left(\left|x\right|\right)\right)\right) \cdot \left(-3 \cdot \log \left(\left|x\right|\right)\right)\right)}^{0.3333333333333333}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

       5. rem-cbrt-cube 0.0%

          \[\leadsto e^{{\left(\left(\left(-3 \cdot \sqrt[3]{{\log \left(\left|x\right|\right)}^{3}}\right) \cdot \left(-3 \cdot \color{blue}{\sqrt[3]{{\log \left(\left|x\right|\right)}^{3}}}\right)\right) \cdot \left(-3 \cdot \log \left(\left|x\right|\right)\right)\right)}^{0.3333333333333333}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

       6. rem-cbrt-cube 0.0%

          \[\leadsto e^{{\left(\left(\left(-3 \cdot \sqrt[3]{{\log \left(\left|x\right|\right)}^{3}}\right) \cdot \left(-3 \cdot \sqrt[3]{{\log \left(\left|x\right|\right)}^{3}}\right)\right) \cdot \left(-3 \cdot \color{blue}{\sqrt[3]{{\log \left(\left|x\right|\right)}^{3}}}\right)\right)}^{0.3333333333333333}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

       7. pow3 0.0%

          \[\leadsto e^{{\color{blue}{\left({\left(-3 \cdot \sqrt[3]{{\log \left(\left|x\right|\right)}^{3}}\right)}^{3}\right)}}^{0.3333333333333333}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

       8. rem-cbrt-cube 0.0%

          \[\leadsto e^{{\left({\left(-3 \cdot \color{blue}{\log \left(\left|x\right|\right)}\right)}^{3}\right)}^{0.3333333333333333}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

       9. log-pow 100.0%

          \[\leadsto e^{{\left({\color{blue}{\log \left({\left(\left|x\right|\right)}^{-3}\right)}}^{3}\right)}^{0.3333333333333333}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

   13. Applied egg-rr 100.0%

       \[\leadsto e^{\color{blue}{{\left({\log \left({\left(\left|x\right|\right)}^{-3}\right)}^{3}\right)}^{0.3333333333333333}}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{+107}:\\ \;\;\;\;\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, 0.5 \cdot {\left(\left|x\right|\right)}^{-3}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{{\left({\log \left({\left(\left|x\right|\right)}^{-3}\right)}^{3}\right)}^{0.3333333333333333}} \cdot \left(0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{0.75}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \frac{1}{\left|x\right|}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (sqrt PI))
  (+
   (/ 0.75 (pow (fabs x) 5.0))
   (+ (/ 1.875 (pow (fabs x) 7.0)) (/ 1.0 (fabs x))))))

C

double code(double x) {
	return (pow(exp(x), x) / sqrt(((double) M_PI))) * ((0.75 / pow(fabs(x), 5.0)) + ((1.875 / pow(fabs(x), 7.0)) + (1.0 / fabs(x))));
}

Java

public static double code(double x) {
	return (Math.pow(Math.exp(x), x) / Math.sqrt(Math.PI)) * ((0.75 / Math.pow(Math.abs(x), 5.0)) + ((1.875 / Math.pow(Math.abs(x), 7.0)) + (1.0 / Math.abs(x))));
}

Python

def code(x):
	return (math.pow(math.exp(x), x) / math.sqrt(math.pi)) * ((0.75 / math.pow(math.fabs(x), 5.0)) + ((1.875 / math.pow(math.fabs(x), 7.0)) + (1.0 / math.fabs(x))))

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / sqrt(pi)) * Float64(Float64(0.75 / (abs(x) ^ 5.0)) + Float64(Float64(1.875 / (abs(x) ^ 7.0)) + Float64(1.0 / abs(x)))))
end

MATLAB

function tmp = code(x)
	tmp = ((exp(x) ^ x) / sqrt(pi)) * ((0.75 / (abs(x) ^ 5.0)) + ((1.875 / (abs(x) ^ 7.0)) + (1.0 / abs(x))));
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(0.75 / N[Power[N[Abs[x], $MachinePrecision], 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.875 / N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 99.4%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 99.4%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\color{blue}{\frac{0.75 \cdot 1}{{\left(\left|x\right|\right)}^{5}}} + \left(\frac{1}{\left|x\right|} + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right) \]

   2. metadata-eval 99.4%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{\color{blue}{0.75}}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right) \]

   3. +-commutative 99.4%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{0.75}{{\left(\left|x\right|\right)}^{5}} + \color{blue}{\left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \frac{1}{\left|x\right|}\right)}\right) \]

   4. associate-*r/ 99.4%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{0.75}{{\left(\left|x\right|\right)}^{5}} + \left(\color{blue}{\frac{1.875 \cdot 1}{{\left(\left|x\right|\right)}^{7}}} + \frac{1}{\left|x\right|}\right)\right) \]

   5. metadata-eval 99.4%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{0.75}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{\color{blue}{1.875}}{{\left(\left|x\right|\right)}^{7}} + \frac{1}{\left|x\right|}\right)\right) \]

7. Simplified 99.4%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{0.75}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \frac{1}{\left|x\right|}\right)\right)} \]
8. Step-by-step derivation

   1. pow-exp 99.4%

      \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{\sqrt{\pi}} \cdot \left(\frac{0.75}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \frac{1}{\left|x\right|}\right)\right) \]

9. Applied egg-rr 99.4%

   \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{\sqrt{\pi}} \cdot \left(\frac{0.75}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \frac{1}{\left|x\right|}\right)\right) \]
10. Add Preprocessing

Alternative 7: 99.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{0.75}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \frac{1}{\left|x\right|}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (exp (* x x)) (sqrt PI))
  (+
   (/ 0.75 (pow (fabs x) 5.0))
   (+ (/ 1.875 (pow (fabs x) 7.0)) (/ 1.0 (fabs x))))))

C

double code(double x) {
	return (exp((x * x)) / sqrt(((double) M_PI))) * ((0.75 / pow(fabs(x), 5.0)) + ((1.875 / pow(fabs(x), 7.0)) + (1.0 / fabs(x))));
}

Java

public static double code(double x) {
	return (Math.exp((x * x)) / Math.sqrt(Math.PI)) * ((0.75 / Math.pow(Math.abs(x), 5.0)) + ((1.875 / Math.pow(Math.abs(x), 7.0)) + (1.0 / Math.abs(x))));
}

Python

def code(x):
	return (math.exp((x * x)) / math.sqrt(math.pi)) * ((0.75 / math.pow(math.fabs(x), 5.0)) + ((1.875 / math.pow(math.fabs(x), 7.0)) + (1.0 / math.fabs(x))))

Julia

function code(x)
	return Float64(Float64(exp(Float64(x * x)) / sqrt(pi)) * Float64(Float64(0.75 / (abs(x) ^ 5.0)) + Float64(Float64(1.875 / (abs(x) ^ 7.0)) + Float64(1.0 / abs(x)))))
end

MATLAB

function tmp = code(x)
	tmp = (exp((x * x)) / sqrt(pi)) * ((0.75 / (abs(x) ^ 5.0)) + ((1.875 / (abs(x) ^ 7.0)) + (1.0 / abs(x))));
end

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(0.75 / N[Power[N[Abs[x], $MachinePrecision], 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.875 / N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 99.4%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 99.4%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\color{blue}{\frac{0.75 \cdot 1}{{\left(\left|x\right|\right)}^{5}}} + \left(\frac{1}{\left|x\right|} + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right) \]

   2. metadata-eval 99.4%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{\color{blue}{0.75}}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right) \]

   3. +-commutative 99.4%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{0.75}{{\left(\left|x\right|\right)}^{5}} + \color{blue}{\left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \frac{1}{\left|x\right|}\right)}\right) \]

   4. associate-*r/ 99.4%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{0.75}{{\left(\left|x\right|\right)}^{5}} + \left(\color{blue}{\frac{1.875 \cdot 1}{{\left(\left|x\right|\right)}^{7}}} + \frac{1}{\left|x\right|}\right)\right) \]

   5. metadata-eval 99.4%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{0.75}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{\color{blue}{1.875}}{{\left(\left|x\right|\right)}^{7}} + \frac{1}{\left|x\right|}\right)\right) \]

7. Simplified 99.4%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{0.75}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \frac{1}{\left|x\right|}\right)\right)} \]
8. Add Preprocessing

Alternative 8: 65.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ e^{{\left({\log \left({\left(\left|x\right|\right)}^{-3}\right)}^{3}\right)}^{0.3333333333333333}} \cdot \left(0.5 \cdot \sqrt{\frac{1}{\pi}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (exp (pow (pow (log (pow (fabs x) -3.0)) 3.0) 0.3333333333333333))
  (* 0.5 (sqrt (/ 1.0 PI)))))

C

double code(double x) {
	return exp(pow(pow(log(pow(fabs(x), -3.0)), 3.0), 0.3333333333333333)) * (0.5 * sqrt((1.0 / ((double) M_PI))));
}

Java

public static double code(double x) {
	return Math.exp(Math.pow(Math.pow(Math.log(Math.pow(Math.abs(x), -3.0)), 3.0), 0.3333333333333333)) * (0.5 * Math.sqrt((1.0 / Math.PI)));
}

Python

def code(x):
	return math.exp(math.pow(math.pow(math.log(math.pow(math.fabs(x), -3.0)), 3.0), 0.3333333333333333)) * (0.5 * math.sqrt((1.0 / math.pi)))

Julia

function code(x)
	return Float64(exp(((log((abs(x) ^ -3.0)) ^ 3.0) ^ 0.3333333333333333)) * Float64(0.5 * sqrt(Float64(1.0 / pi))))
end

MATLAB

function tmp = code(x)
	tmp = exp(((log((abs(x) ^ -3.0)) ^ 3.0) ^ 0.3333333333333333)) * (0.5 * sqrt((1.0 / pi)));
end

Wolfram

code[x_] := N[(N[Exp[N[Power[N[Power[N[Log[N[Power[N[Abs[x], $MachinePrecision], -3.0], $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]], $MachinePrecision] * N[(0.5 * N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 1.9%

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

   1. *-commutative 1.9%

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

   2. associate-*l* 1.9%

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

   3. *-commutative 1.9%

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

   4. unpow2 1.9%

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

   5. sqr-abs 1.9%

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

   6. cube-mult 1.9%

      \[\leadsto \frac{1}{\color{blue}{{\left(\left|x\right|\right)}^{3}}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

   7. exp-to-pow 1.9%

      \[\leadsto \frac{1}{\color{blue}{e^{\log \left(\left|x\right|\right) \cdot 3}}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

   8. *-commutative 1.9%

      \[\leadsto \frac{1}{e^{\color{blue}{3 \cdot \log \left(\left|x\right|\right)}}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

   9. exp-neg 1.9%

      \[\leadsto \color{blue}{e^{-3 \cdot \log \left(\left|x\right|\right)}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

   10. distribute-lft-neg-in 1.9%

       \[\leadsto e^{\color{blue}{\left(-3\right) \cdot \log \left(\left|x\right|\right)}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

   11. metadata-eval 1.9%

       \[\leadsto e^{\color{blue}{-3} \cdot \log \left(\left|x\right|\right)} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

   12. *-commutative 1.9%

       \[\leadsto e^{\color{blue}{\log \left(\left|x\right|\right) \cdot -3}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

   13. exp-to-pow 1.9%

       \[\leadsto \color{blue}{{\left(\left|x\right|\right)}^{-3}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

7. Simplified 1.9%

   \[\leadsto \color{blue}{{\left(\left|x\right|\right)}^{-3} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right)} \]
8. Step-by-step derivation

   1. add-exp-log 1.9%

      \[\leadsto \color{blue}{e^{\log \left({\left(\left|x\right|\right)}^{-3}\right)}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

   2. log-pow 1.9%

      \[\leadsto e^{\color{blue}{-3 \cdot \log \left(\left|x\right|\right)}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

9. Applied egg-rr 1.9%

   \[\leadsto \color{blue}{e^{-3 \cdot \log \left(\left|x\right|\right)}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]
10. Step-by-step derivation

    1. add-cbrt-cube 1.9%

       \[\leadsto e^{-3 \cdot \color{blue}{\sqrt[3]{\left(\log \left(\left|x\right|\right) \cdot \log \left(\left|x\right|\right)\right) \cdot \log \left(\left|x\right|\right)}}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

    2. pow3 1.9%

       \[\leadsto e^{-3 \cdot \sqrt[3]{\color{blue}{{\log \left(\left|x\right|\right)}^{3}}}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

11. Applied egg-rr 1.9%

    \[\leadsto e^{-3 \cdot \color{blue}{\sqrt[3]{{\log \left(\left|x\right|\right)}^{3}}}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]
12. Step-by-step derivation

    1. rem-cbrt-cube 1.9%

       \[\leadsto e^{-3 \cdot \color{blue}{\log \left(\left|x\right|\right)}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

    2. add-cbrt-cube 1.9%

       \[\leadsto e^{\color{blue}{\sqrt[3]{\left(\left(-3 \cdot \log \left(\left|x\right|\right)\right) \cdot \left(-3 \cdot \log \left(\left|x\right|\right)\right)\right) \cdot \left(-3 \cdot \log \left(\left|x\right|\right)\right)}}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

    3. pow1/3 0.0%

       \[\leadsto e^{\color{blue}{{\left(\left(\left(-3 \cdot \log \left(\left|x\right|\right)\right) \cdot \left(-3 \cdot \log \left(\left|x\right|\right)\right)\right) \cdot \left(-3 \cdot \log \left(\left|x\right|\right)\right)\right)}^{0.3333333333333333}}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

    4. rem-cbrt-cube 0.0%

       \[\leadsto e^{{\left(\left(\left(-3 \cdot \color{blue}{\sqrt[3]{{\log \left(\left|x\right|\right)}^{3}}}\right) \cdot \left(-3 \cdot \log \left(\left|x\right|\right)\right)\right) \cdot \left(-3 \cdot \log \left(\left|x\right|\right)\right)\right)}^{0.3333333333333333}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

    5. rem-cbrt-cube 0.0%

       \[\leadsto e^{{\left(\left(\left(-3 \cdot \sqrt[3]{{\log \left(\left|x\right|\right)}^{3}}\right) \cdot \left(-3 \cdot \color{blue}{\sqrt[3]{{\log \left(\left|x\right|\right)}^{3}}}\right)\right) \cdot \left(-3 \cdot \log \left(\left|x\right|\right)\right)\right)}^{0.3333333333333333}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

    6. rem-cbrt-cube 0.0%

       \[\leadsto e^{{\left(\left(\left(-3 \cdot \sqrt[3]{{\log \left(\left|x\right|\right)}^{3}}\right) \cdot \left(-3 \cdot \sqrt[3]{{\log \left(\left|x\right|\right)}^{3}}\right)\right) \cdot \left(-3 \cdot \color{blue}{\sqrt[3]{{\log \left(\left|x\right|\right)}^{3}}}\right)\right)}^{0.3333333333333333}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

    7. pow3 0.0%

       \[\leadsto e^{{\color{blue}{\left({\left(-3 \cdot \sqrt[3]{{\log \left(\left|x\right|\right)}^{3}}\right)}^{3}\right)}}^{0.3333333333333333}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

    8. rem-cbrt-cube 0.0%

       \[\leadsto e^{{\left({\left(-3 \cdot \color{blue}{\log \left(\left|x\right|\right)}\right)}^{3}\right)}^{0.3333333333333333}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

    9. log-pow 64.1%

       \[\leadsto e^{{\left({\color{blue}{\log \left({\left(\left|x\right|\right)}^{-3}\right)}}^{3}\right)}^{0.3333333333333333}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

13. Applied egg-rr 64.1%

    \[\leadsto e^{\color{blue}{{\left({\log \left({\left(\left|x\right|\right)}^{-3}\right)}^{3}\right)}^{0.3333333333333333}}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.5\right) \]

14. Final simplification 64.1%

    \[\leadsto e^{{\left({\log \left({\left(\left|x\right|\right)}^{-3}\right)}^{3}\right)}^{0.3333333333333333}} \cdot \left(0.5 \cdot \sqrt{\frac{1}{\pi}}\right) \]
15. Add Preprocessing

Alternative 9: 2.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{\pi}} \cdot \left(\frac{0.75}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \frac{1}{\left|x\right|}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ 1.0 (sqrt PI))
  (+
   (/ 0.75 (pow (fabs x) 5.0))
   (+ (/ 1.875 (pow (fabs x) 7.0)) (/ 1.0 (fabs x))))))

C

double code(double x) {
	return (1.0 / sqrt(((double) M_PI))) * ((0.75 / pow(fabs(x), 5.0)) + ((1.875 / pow(fabs(x), 7.0)) + (1.0 / fabs(x))));
}

Java

public static double code(double x) {
	return (1.0 / Math.sqrt(Math.PI)) * ((0.75 / Math.pow(Math.abs(x), 5.0)) + ((1.875 / Math.pow(Math.abs(x), 7.0)) + (1.0 / Math.abs(x))));
}

Python

def code(x):
	return (1.0 / math.sqrt(math.pi)) * ((0.75 / math.pow(math.fabs(x), 5.0)) + ((1.875 / math.pow(math.fabs(x), 7.0)) + (1.0 / math.fabs(x))))

Julia

function code(x)
	return Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(0.75 / (abs(x) ^ 5.0)) + Float64(Float64(1.875 / (abs(x) ^ 7.0)) + Float64(1.0 / abs(x)))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / sqrt(pi)) * ((0.75 / (abs(x) ^ 5.0)) + ((1.875 / (abs(x) ^ 7.0)) + (1.0 / abs(x))));
end

Wolfram

code[x_] := N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(0.75 / N[Power[N[Abs[x], $MachinePrecision], 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.875 / N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 99.4%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 99.4%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\color{blue}{\frac{0.75 \cdot 1}{{\left(\left|x\right|\right)}^{5}}} + \left(\frac{1}{\left|x\right|} + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right) \]

   2. metadata-eval 99.4%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{\color{blue}{0.75}}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right) \]

   3. +-commutative 99.4%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{0.75}{{\left(\left|x\right|\right)}^{5}} + \color{blue}{\left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \frac{1}{\left|x\right|}\right)}\right) \]

   4. associate-*r/ 99.4%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{0.75}{{\left(\left|x\right|\right)}^{5}} + \left(\color{blue}{\frac{1.875 \cdot 1}{{\left(\left|x\right|\right)}^{7}}} + \frac{1}{\left|x\right|}\right)\right) \]

   5. metadata-eval 99.4%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{0.75}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{\color{blue}{1.875}}{{\left(\left|x\right|\right)}^{7}} + \frac{1}{\left|x\right|}\right)\right) \]

7. Simplified 99.4%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{0.75}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \frac{1}{\left|x\right|}\right)\right)} \]

8. Taylor expanded in x around 0 2.4%

   \[\leadsto \frac{\color{blue}{1}}{\sqrt{\pi}} \cdot \left(\frac{0.75}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \frac{1}{\left|x\right|}\right)\right) \]
9. Add Preprocessing

Alternative 10: 1.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{{\left(\left|x\right|\right)}^{-3}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (/ (pow (fabs x) -3.0) (sqrt PI))))

C

double code(double x) {
	return 0.5 * (pow(fabs(x), -3.0) / sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return 0.5 * (Math.pow(Math.abs(x), -3.0) / Math.sqrt(Math.PI));
}

Python

def code(x):
	return 0.5 * (math.pow(math.fabs(x), -3.0) / math.sqrt(math.pi))

Julia

function code(x)
	return Float64(0.5 * Float64((abs(x) ^ -3.0) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * ((abs(x) ^ -3.0) / sqrt(pi));
end

Wolfram

code[x_] := N[(0.5 * N[(N[Power[N[Abs[x], $MachinePrecision], -3.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 1.9%

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

   1. *-commutative 1.9%

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

   2. sqrt-div 1.9%

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

   3. metadata-eval 1.9%

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

   4. inv-pow 1.9%

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

   5. inv-pow 1.9%

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

   6. pow2 1.9%

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

   7. pow-prod-down 1.9%

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

   8. *-commutative 1.9%

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

   9. pow2 1.9%

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

7. Applied egg-rr 1.9%

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

   1. unpow-1 1.9%

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

   2. *-commutative 1.9%

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

   3. associate-/r* 1.9%

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

   4. unpow2 1.9%

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

   5. sqr-abs 1.9%

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

   6. cube-mult 1.9%

      \[\leadsto 0.5 \cdot \frac{\frac{1}{\color{blue}{{\left(\left|x\right|\right)}^{3}}}}{\sqrt{\pi}} \]

   7. exp-to-pow 1.9%

      \[\leadsto 0.5 \cdot \frac{\frac{1}{\color{blue}{e^{\log \left(\left|x\right|\right) \cdot 3}}}}{\sqrt{\pi}} \]

   8. *-commutative 1.9%

      \[\leadsto 0.5 \cdot \frac{\frac{1}{e^{\color{blue}{3 \cdot \log \left(\left|x\right|\right)}}}}{\sqrt{\pi}} \]

   9. exp-neg 1.9%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{e^{-3 \cdot \log \left(\left|x\right|\right)}}}{\sqrt{\pi}} \]

   10. distribute-lft-neg-in 1.9%

       \[\leadsto 0.5 \cdot \frac{e^{\color{blue}{\left(-3\right) \cdot \log \left(\left|x\right|\right)}}}{\sqrt{\pi}} \]

   11. metadata-eval 1.9%

       \[\leadsto 0.5 \cdot \frac{e^{\color{blue}{-3} \cdot \log \left(\left|x\right|\right)}}{\sqrt{\pi}} \]

   12. *-commutative 1.9%

       \[\leadsto 0.5 \cdot \frac{e^{\color{blue}{\log \left(\left|x\right|\right) \cdot -3}}}{\sqrt{\pi}} \]

   13. exp-to-pow 1.9%

       \[\leadsto 0.5 \cdot \frac{\color{blue}{{\left(\left|x\right|\right)}^{-3}}}{\sqrt{\pi}} \]

9. Simplified 1.9%

   \[\leadsto 0.5 \cdot \color{blue}{\frac{{\left(\left|x\right|\right)}^{-3}}{\sqrt{\pi}}} \]
10. Add Preprocessing

Alternative 11: 1.6% accurate, 2083.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 1.9%

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

   1. add-exp-log 1.9%

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

   2. *-commutative 1.9%

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

   3. pow1/2 1.9%

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

   4. inv-pow 1.9%

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

   5. pow-pow 1.9%

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

   6. metadata-eval 1.9%

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

   7. pow2 1.9%

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

   8. associate-/r* 1.9%

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

   9. pow2 1.9%

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

   10. pow-flip 1.9%

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

   11. metadata-eval 1.9%

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

7. Applied egg-rr 1.9%

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

   1. rem-exp-log 1.9%

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

   2. expm1-log1p-u 1.9%

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

   3. expm1-undefine 1.7%

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

9. Applied egg-rr 1.7%

   \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \frac{{x}^{-2}}{\left|x\right|}\right)} - 1\right)} \]
10. Step-by-step derivation

    1. log1p-undefine 1.7%

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

    2. rem-exp-log 1.7%

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

    3. associate-+r- 1.7%

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

    4. fma-neg 1.7%

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

    5. metadata-eval 1.7%

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

11. Simplified 1.7%

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

12. Taylor expanded in x around inf 1.6%

    \[\leadsto 0.5 \cdot \left(1 + \color{blue}{-1}\right) \]

13. Final simplification 1.6%

    \[\leadsto 0 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024147 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x greater than or equal to 5"
  :precision binary64
  :pre (>= x 0.5)
  (* (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x)))) (+ (+ (+ (/ 1.0 (fabs x)) (* (/ 1.0 2.0) (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 3.0 4.0) (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 15.0 8.0) (* (* (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))))