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

Percentage Accurate: 100.0% → 100.0%

Time: 10.4s

Alternatives: 6

Speedup: 3.5×

• 99.5% 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, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. expm1-log1p-u 100.0%

      \[\leadsto e^{x \cdot x} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)}{\sqrt{\pi}}\right)\right)} \]

   2. expm1-udef 5.1%

      \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)}{\sqrt{\pi}}\right)} - 1\right)} \]

6. Applied egg-rr 5.1%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}\right)} - 1\right)} \]
7. Step-by-step derivation

   1. expm1-def 100.0%

      \[\leadsto e^{x \cdot x} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}\right)\right)} \]

   2. expm1-log1p 100.0%

      \[\leadsto e^{x \cdot x} \cdot \color{blue}{\frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}} \]

8. Simplified 100.0%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}} \]

9. Taylor expanded in x around 0 100.0%

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

    1. associate-*r/ 100.0%

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

    2. metadata-eval 100.0%

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

11. Simplified 100.0%

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

12. Final simplification 100.0%

    \[\leadsto e^{x \cdot x} \cdot \frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1}{x} + \frac{0.5}{{x}^{3}}\right)\right)}{\sqrt{\pi}} \]
13. Add Preprocessing

Alternative 2: 100.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Taylor expanded in x around 0 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 3: 99.6% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Taylor expanded in x around inf 99.7%

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

   1. associate-*r/ 99.7%

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

   2. metadata-eval 99.7%

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

   3. unpow1 99.7%

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

   4. sqr-pow 99.7%

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

   5. fabs-sqr 99.7%

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

   6. sqr-pow 99.7%

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

   7. unpow1 99.7%

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

   8. +-commutative 99.7%

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

   9. associate-*r/ 99.7%

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

   10. metadata-eval 99.7%

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

   11. unpow1 99.7%

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

   12. sqr-pow 99.7%

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

   13. fabs-sqr 99.7%

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

   14. sqr-pow 99.7%

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

   15. unpow1 99.7%

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

   16. unpow1 99.7%

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

   17. sqr-pow 99.7%

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

   18. fabs-sqr 99.7%

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

   19. sqr-pow 99.7%

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

   20. unpow1 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 4: 99.6% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ e^{x \cdot x} \cdot \frac{{\pi}^{-0.5}}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (exp (* x x)) (/ (pow PI -0.5) x)))

C

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

Java

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

Python

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

Julia

function code(x)
	return Float64(exp(Float64(x * x)) * Float64((pi ^ -0.5) / x))
end

MATLAB

function tmp = code(x)
	tmp = exp((x * x)) * ((pi ^ -0.5) / x);
end

Wolfram

code[x_] := N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(N[Power[Pi, -0.5], $MachinePrecision] / x), $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%

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

5. Taylor expanded in x around 0 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.7%

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

   1. associate-*l/ 99.7%

      \[\leadsto e^{x \cdot x} \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{\pi}}}{x}} \]

   2. *-lft-identity 99.7%

      \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\sqrt{\frac{1}{\pi}}}}{x} \]

10. Simplified 99.7%

    \[\leadsto e^{x \cdot x} \cdot \color{blue}{\frac{\sqrt{\frac{1}{\pi}}}{x}} \]
11. Step-by-step derivation

    1. expm1-log1p-u 99.7%

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

    2. expm1-udef 4.7%

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

    3. inv-pow 4.7%

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

    4. sqrt-pow1 4.7%

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

    5. metadata-eval 4.7%

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

12. Applied egg-rr 4.7%

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

    1. expm1-def 99.7%

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

    2. expm1-log1p 99.7%

       \[\leadsto e^{x \cdot x} \cdot \color{blue}{\frac{{\pi}^{-0.5}}{x}} \]

14. Simplified 99.7%

    \[\leadsto e^{x \cdot x} \cdot \color{blue}{\frac{{\pi}^{-0.5}}{x}} \]

15. Final simplification 99.7%

    \[\leadsto e^{x \cdot x} \cdot \frac{{\pi}^{-0.5}}{x} \]
16. Add Preprocessing

Alternative 5: 2.3% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \frac{2.1875}{x \cdot \sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.1875 (* x (sqrt PI))))

C

double code(double x) {
	return 2.1875 / (x * sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return 2.1875 / (x * Math.sqrt(Math.PI));
}

Python

def code(x):
	return 2.1875 / (x * math.sqrt(math.pi))

Julia

function code(x)
	return Float64(2.1875 / Float64(x * sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = 2.1875 / (x * sqrt(pi));
end

Wolfram

code[x_] := N[(2.1875 / N[(x * 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%

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

   1. expm1-log1p-u 100.0%

      \[\leadsto e^{x \cdot x} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)}{\sqrt{\pi}}\right)\right)} \]

   2. expm1-udef 5.1%

      \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)}{\sqrt{\pi}}\right)} - 1\right)} \]

6. Applied egg-rr 5.1%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}\right)} - 1\right)} \]
7. Step-by-step derivation

   1. expm1-def 100.0%

      \[\leadsto e^{x \cdot x} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}\right)\right)} \]

   2. expm1-log1p 100.0%

      \[\leadsto e^{x \cdot x} \cdot \color{blue}{\frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}} \]

8. Simplified 100.0%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}} \]

9. Taylor expanded in x around 0 2.4%

   \[\leadsto \color{blue}{1.875 \cdot \left(\frac{1}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right) + \left(2.1875 \cdot \left(\frac{1}{x} \cdot \sqrt{\frac{1}{\pi}}\right) + \left(2.1875 \cdot \left(\frac{1}{{x}^{3}} \cdot \sqrt{\frac{1}{\pi}}\right) + 2.625 \cdot \left(\frac{1}{{x}^{5}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)} \]
10. Step-by-step derivation

    1. associate-+r+ 2.4%

       \[\leadsto \color{blue}{\left(1.875 \cdot \left(\frac{1}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right) + 2.1875 \cdot \left(\frac{1}{x} \cdot \sqrt{\frac{1}{\pi}}\right)\right) + \left(2.1875 \cdot \left(\frac{1}{{x}^{3}} \cdot \sqrt{\frac{1}{\pi}}\right) + 2.625 \cdot \left(\frac{1}{{x}^{5}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]

    2. +-commutative 2.4%

       \[\leadsto \color{blue}{\left(2.1875 \cdot \left(\frac{1}{{x}^{3}} \cdot \sqrt{\frac{1}{\pi}}\right) + 2.625 \cdot \left(\frac{1}{{x}^{5}} \cdot \sqrt{\frac{1}{\pi}}\right)\right) + \left(1.875 \cdot \left(\frac{1}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right) + 2.1875 \cdot \left(\frac{1}{x} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]

    3. +-commutative 2.4%

       \[\leadsto \color{blue}{\left(2.625 \cdot \left(\frac{1}{{x}^{5}} \cdot \sqrt{\frac{1}{\pi}}\right) + 2.1875 \cdot \left(\frac{1}{{x}^{3}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} + \left(1.875 \cdot \left(\frac{1}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right) + 2.1875 \cdot \left(\frac{1}{x} \cdot \sqrt{\frac{1}{\pi}}\right)\right) \]

    4. associate-*r* 2.4%

       \[\leadsto \left(\color{blue}{\left(2.625 \cdot \frac{1}{{x}^{5}}\right) \cdot \sqrt{\frac{1}{\pi}}} + 2.1875 \cdot \left(\frac{1}{{x}^{3}} \cdot \sqrt{\frac{1}{\pi}}\right)\right) + \left(1.875 \cdot \left(\frac{1}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right) + 2.1875 \cdot \left(\frac{1}{x} \cdot \sqrt{\frac{1}{\pi}}\right)\right) \]

    5. associate-*r* 2.4%

       \[\leadsto \left(\left(2.625 \cdot \frac{1}{{x}^{5}}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(2.1875 \cdot \frac{1}{{x}^{3}}\right) \cdot \sqrt{\frac{1}{\pi}}}\right) + \left(1.875 \cdot \left(\frac{1}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right) + 2.1875 \cdot \left(\frac{1}{x} \cdot \sqrt{\frac{1}{\pi}}\right)\right) \]

    6. distribute-rgt-out 2.4%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(2.625 \cdot \frac{1}{{x}^{5}} + 2.1875 \cdot \frac{1}{{x}^{3}}\right)} + \left(1.875 \cdot \left(\frac{1}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right) + 2.1875 \cdot \left(\frac{1}{x} \cdot \sqrt{\frac{1}{\pi}}\right)\right) \]

11. Simplified 2.4%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{2.625}{{x}^{5}} + \frac{2.1875}{{x}^{3}}\right) + \left(\frac{1.875}{{x}^{7}} + \frac{2.1875}{x}\right)\right)} \]

12. Taylor expanded in x around inf 2.3%

    \[\leadsto \color{blue}{2.1875 \cdot \left(\frac{1}{x} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
13. Step-by-step derivation

    1. associate-*r* 2.3%

       \[\leadsto \color{blue}{\left(2.1875 \cdot \frac{1}{x}\right) \cdot \sqrt{\frac{1}{\pi}}} \]

    2. *-commutative 2.3%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(2.1875 \cdot \frac{1}{x}\right)} \]

    3. associate-*r/ 2.3%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \color{blue}{\frac{2.1875 \cdot 1}{x}} \]

    4. metadata-eval 2.3%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{\color{blue}{2.1875}}{x} \]

14. Simplified 2.3%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \frac{2.1875}{x}} \]
15. Step-by-step derivation

    1. sqrt-div 2.3%

       \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}} \cdot \frac{2.1875}{x} \]

    2. metadata-eval 2.3%

       \[\leadsto \frac{\color{blue}{1}}{\sqrt{\pi}} \cdot \frac{2.1875}{x} \]

    3. frac-times 2.3%

       \[\leadsto \color{blue}{\frac{1 \cdot 2.1875}{\sqrt{\pi} \cdot x}} \]

    4. metadata-eval 2.3%

       \[\leadsto \frac{\color{blue}{2.1875}}{\sqrt{\pi} \cdot x} \]

16. Applied egg-rr 2.3%

    \[\leadsto \color{blue}{\frac{2.1875}{\sqrt{\pi} \cdot x}} \]

17. Final simplification 2.3%

    \[\leadsto \frac{2.1875}{x \cdot \sqrt{\pi}} \]
18. Add Preprocessing

Alternative 6: 2.3% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{2.1875}{x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ 2.1875 x) (sqrt PI)))

C

double code(double x) {
	return (2.1875 / x) / sqrt(((double) M_PI));
}

Java

public static double code(double x) {
	return (2.1875 / x) / Math.sqrt(Math.PI);
}

Python

def code(x):
	return (2.1875 / x) / math.sqrt(math.pi)

Julia

function code(x)
	return Float64(Float64(2.1875 / x) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	tmp = (2.1875 / x) / sqrt(pi);
end

Wolfram

code[x_] := N[(N[(2.1875 / x), $MachinePrecision] / N[Sqrt[Pi], $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%

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

   1. expm1-log1p-u 100.0%

      \[\leadsto e^{x \cdot x} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)}{\sqrt{\pi}}\right)\right)} \]

   2. expm1-udef 5.1%

      \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)}{\sqrt{\pi}}\right)} - 1\right)} \]

6. Applied egg-rr 5.1%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}\right)} - 1\right)} \]
7. Step-by-step derivation

   1. expm1-def 100.0%

      \[\leadsto e^{x \cdot x} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}\right)\right)} \]

   2. expm1-log1p 100.0%

      \[\leadsto e^{x \cdot x} \cdot \color{blue}{\frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}} \]

8. Simplified 100.0%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}} \]

9. Taylor expanded in x around 0 2.4%

   \[\leadsto \color{blue}{1.875 \cdot \left(\frac{1}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right) + \left(2.1875 \cdot \left(\frac{1}{x} \cdot \sqrt{\frac{1}{\pi}}\right) + \left(2.1875 \cdot \left(\frac{1}{{x}^{3}} \cdot \sqrt{\frac{1}{\pi}}\right) + 2.625 \cdot \left(\frac{1}{{x}^{5}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)} \]
10. Step-by-step derivation

    1. associate-+r+ 2.4%

       \[\leadsto \color{blue}{\left(1.875 \cdot \left(\frac{1}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right) + 2.1875 \cdot \left(\frac{1}{x} \cdot \sqrt{\frac{1}{\pi}}\right)\right) + \left(2.1875 \cdot \left(\frac{1}{{x}^{3}} \cdot \sqrt{\frac{1}{\pi}}\right) + 2.625 \cdot \left(\frac{1}{{x}^{5}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]

    2. +-commutative 2.4%

       \[\leadsto \color{blue}{\left(2.1875 \cdot \left(\frac{1}{{x}^{3}} \cdot \sqrt{\frac{1}{\pi}}\right) + 2.625 \cdot \left(\frac{1}{{x}^{5}} \cdot \sqrt{\frac{1}{\pi}}\right)\right) + \left(1.875 \cdot \left(\frac{1}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right) + 2.1875 \cdot \left(\frac{1}{x} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]

    3. +-commutative 2.4%

       \[\leadsto \color{blue}{\left(2.625 \cdot \left(\frac{1}{{x}^{5}} \cdot \sqrt{\frac{1}{\pi}}\right) + 2.1875 \cdot \left(\frac{1}{{x}^{3}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} + \left(1.875 \cdot \left(\frac{1}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right) + 2.1875 \cdot \left(\frac{1}{x} \cdot \sqrt{\frac{1}{\pi}}\right)\right) \]

    4. associate-*r* 2.4%

       \[\leadsto \left(\color{blue}{\left(2.625 \cdot \frac{1}{{x}^{5}}\right) \cdot \sqrt{\frac{1}{\pi}}} + 2.1875 \cdot \left(\frac{1}{{x}^{3}} \cdot \sqrt{\frac{1}{\pi}}\right)\right) + \left(1.875 \cdot \left(\frac{1}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right) + 2.1875 \cdot \left(\frac{1}{x} \cdot \sqrt{\frac{1}{\pi}}\right)\right) \]

    5. associate-*r* 2.4%

       \[\leadsto \left(\left(2.625 \cdot \frac{1}{{x}^{5}}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(2.1875 \cdot \frac{1}{{x}^{3}}\right) \cdot \sqrt{\frac{1}{\pi}}}\right) + \left(1.875 \cdot \left(\frac{1}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right) + 2.1875 \cdot \left(\frac{1}{x} \cdot \sqrt{\frac{1}{\pi}}\right)\right) \]

    6. distribute-rgt-out 2.4%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(2.625 \cdot \frac{1}{{x}^{5}} + 2.1875 \cdot \frac{1}{{x}^{3}}\right)} + \left(1.875 \cdot \left(\frac{1}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right) + 2.1875 \cdot \left(\frac{1}{x} \cdot \sqrt{\frac{1}{\pi}}\right)\right) \]

11. Simplified 2.4%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{2.625}{{x}^{5}} + \frac{2.1875}{{x}^{3}}\right) + \left(\frac{1.875}{{x}^{7}} + \frac{2.1875}{x}\right)\right)} \]

12. Taylor expanded in x around inf 2.3%

    \[\leadsto \color{blue}{2.1875 \cdot \left(\frac{1}{x} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
13. Step-by-step derivation

    1. associate-*r* 2.3%

       \[\leadsto \color{blue}{\left(2.1875 \cdot \frac{1}{x}\right) \cdot \sqrt{\frac{1}{\pi}}} \]

    2. *-commutative 2.3%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(2.1875 \cdot \frac{1}{x}\right)} \]

    3. associate-*r/ 2.3%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \color{blue}{\frac{2.1875 \cdot 1}{x}} \]

    4. metadata-eval 2.3%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{\color{blue}{2.1875}}{x} \]

14. Simplified 2.3%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \frac{2.1875}{x}} \]
15. Step-by-step derivation

    1. sqrt-div 2.3%

       \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}} \cdot \frac{2.1875}{x} \]

    2. metadata-eval 2.3%

       \[\leadsto \frac{\color{blue}{1}}{\sqrt{\pi}} \cdot \frac{2.1875}{x} \]

    3. associate-*l/ 2.3%

       \[\leadsto \color{blue}{\frac{1 \cdot \frac{2.1875}{x}}{\sqrt{\pi}}} \]

    4. *-un-lft-identity 2.3%

       \[\leadsto \frac{\color{blue}{\frac{2.1875}{x}}}{\sqrt{\pi}} \]

16. Applied egg-rr 2.3%

    \[\leadsto \color{blue}{\frac{\frac{2.1875}{x}}{\sqrt{\pi}}} \]

17. Final simplification 2.3%

    \[\leadsto \frac{\frac{2.1875}{x}}{\sqrt{\pi}} \]
18. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(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)))))))