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

Percentage Accurate: 100.0% → 99.7%

Time: 11.2s

Alternatives: 3

Speedup: N/A×

• 99.6% 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: 99.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(0.75 / N[Power[x, 5.0], $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. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right), \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. inv-pow 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. pow-pow 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 7\right)}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   9. fabs-sqr 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 0 + {\color{blue}{x}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 0 + {x}^{\color{blue}{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{0 + {x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
7. Step-by-step derivation

   1. +-lft-identity 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around inf 100.0%

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

    1. associate-*r/ 100.0%

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

    2. metadata-eval 100.0%

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

11. Simplified 100.0%

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

    1. frac-2neg 100.0%

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

    2. div-inv 100.0%

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

    3. metadata-eval 100.0%

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

    4. add-sqr-sqrt 100.0%

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

    5. fabs-sqr 100.0%

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

    6. add-sqr-sqrt 100.0%

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

13. Applied egg-rr 100.0%

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

    1. associate-*r/ 100.0%

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

    2. metadata-eval 100.0%

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

    3. distribute-neg-frac2 100.0%

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

    4. distribute-neg-frac 100.0%

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

    5. metadata-eval 100.0%

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

15. Simplified 100.0%

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

16. Final simplification 100.0%

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

Alternative 2: 14.1% accurate, 6.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (* (/ (exp (* x x)) (sqrt PI)) (* (pow x -7.0) 1.875)))

C

double code(double x) {
	return (exp((x * x)) / sqrt(((double) M_PI))) * (pow(x, -7.0) * 1.875);
}

Java

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

Python

def code(x):
	return (math.exp((x * x)) / math.sqrt(math.pi)) * (math.pow(x, -7.0) * 1.875)

Julia

function code(x)
	return Float64(Float64(exp(Float64(x * x)) / sqrt(pi)) * Float64((x ^ -7.0) * 1.875))
end

MATLAB

function tmp = code(x)
	tmp = (exp((x * x)) / sqrt(pi)) * ((x ^ -7.0) * 1.875);
end

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Power[x, -7.0], $MachinePrecision] * 1.875), $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. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right), \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. inv-pow 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. pow-pow 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 7\right)}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   9. fabs-sqr 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 0 + {\color{blue}{x}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 0 + {x}^{\color{blue}{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{0 + {x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
7. Step-by-step derivation

   1. +-lft-identity 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around 0 14.8%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1.875}{{x}^{7}}} \]
10. Step-by-step derivation

    1. clear-num 14.8%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1}{\frac{{x}^{7}}{1.875}}} \]

    2. associate-/r/ 14.8%

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

    3. pow-flip 16.0%

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

    4. metadata-eval 16.0%

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

11. Applied egg-rr 16.0%

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

12. Final simplification 16.0%

    \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left({x}^{-7} \cdot 1.875\right) \]
13. Add Preprocessing

Alternative 3: 1.7% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ 1.875 \cdot \frac{{x}^{-7}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 1.875 (/ (pow x -7.0) (sqrt PI))))

C

double code(double x) {
	return 1.875 * (pow(x, -7.0) / sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return 1.875 * (Math.pow(x, -7.0) / Math.sqrt(Math.PI));
}

Python

def code(x):
	return 1.875 * (math.pow(x, -7.0) / math.sqrt(math.pi))

Julia

function code(x)
	return Float64(1.875 * Float64((x ^ -7.0) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = 1.875 * ((x ^ -7.0) / sqrt(pi));
end

Wolfram

code[x_] := N[(1.875 * N[(N[Power[x, -7.0], $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. Step-by-step derivation

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{7}}\right), \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. inv-pow 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. pow-pow 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 7\right)}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   9. fabs-sqr 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 0 + {\color{blue}{x}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, 0 + {x}^{\color{blue}{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{0 + {x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
7. Step-by-step derivation

   1. +-lft-identity 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around 0 14.8%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1.875}{{x}^{7}}} \]

10. Taylor expanded in x around 0 1.7%

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

    1. expm1-log1p-u 1.7%

       \[\leadsto 1.875 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]

    2. expm1-undefine 1.6%

       \[\leadsto 1.875 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right)} - 1\right)} \]

    3. sqrt-div 1.6%

       \[\leadsto 1.875 \cdot \left(e^{\mathsf{log1p}\left(\frac{1}{{x}^{7}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right)} - 1\right) \]

    4. metadata-eval 1.6%

       \[\leadsto 1.875 \cdot \left(e^{\mathsf{log1p}\left(\frac{1}{{x}^{7}} \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right)} - 1\right) \]

    5. un-div-inv 1.6%

       \[\leadsto 1.875 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{{x}^{7}}}{\sqrt{\pi}}}\right)} - 1\right) \]

    6. pow-flip 1.6%

       \[\leadsto 1.875 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{x}^{\left(-7\right)}}}{\sqrt{\pi}}\right)} - 1\right) \]

    7. metadata-eval 1.6%

       \[\leadsto 1.875 \cdot \left(e^{\mathsf{log1p}\left(\frac{{x}^{\color{blue}{-7}}}{\sqrt{\pi}}\right)} - 1\right) \]

12. Applied egg-rr 1.6%

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

    1. sub-neg 1.6%

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

    2. metadata-eval 1.6%

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

    3. +-commutative 1.6%

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

    4. log1p-undefine 1.6%

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

    5. rem-exp-log 1.6%

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

    6. associate-+r+ 1.7%

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

    7. metadata-eval 1.7%

       \[\leadsto 1.875 \cdot \left(\color{blue}{0} + \frac{{x}^{-7}}{\sqrt{\pi}}\right) \]

    8. div0 1.7%

       \[\leadsto 1.875 \cdot \left(\color{blue}{\frac{0}{\sqrt{\pi}}} + \frac{{x}^{-7}}{\sqrt{\pi}}\right) \]

    9. exp-to-pow 1.7%

       \[\leadsto 1.875 \cdot \left(\frac{0}{\sqrt{\pi}} + \frac{\color{blue}{e^{\log x \cdot -7}}}{\sqrt{\pi}}\right) \]

    10. *-commutative 1.7%

        \[\leadsto 1.875 \cdot \left(\frac{0}{\sqrt{\pi}} + \frac{e^{\color{blue}{-7 \cdot \log x}}}{\sqrt{\pi}}\right) \]

    11. metadata-eval 1.7%

        \[\leadsto 1.875 \cdot \left(\frac{0}{\sqrt{\pi}} + \frac{e^{\color{blue}{\left(-7\right)} \cdot \log x}}{\sqrt{\pi}}\right) \]

    12. distribute-lft-neg-in 1.7%

        \[\leadsto 1.875 \cdot \left(\frac{0}{\sqrt{\pi}} + \frac{e^{\color{blue}{-7 \cdot \log x}}}{\sqrt{\pi}}\right) \]

    13. rec-exp 1.7%

        \[\leadsto 1.875 \cdot \left(\frac{0}{\sqrt{\pi}} + \frac{\color{blue}{\frac{1}{e^{7 \cdot \log x}}}}{\sqrt{\pi}}\right) \]

    14. metadata-eval 1.7%

        \[\leadsto 1.875 \cdot \left(\frac{0}{\sqrt{\pi}} + \frac{\frac{\color{blue}{--1}}{e^{7 \cdot \log x}}}{\sqrt{\pi}}\right) \]

    15. *-commutative 1.7%

        \[\leadsto 1.875 \cdot \left(\frac{0}{\sqrt{\pi}} + \frac{\frac{--1}{e^{\color{blue}{\log x \cdot 7}}}}{\sqrt{\pi}}\right) \]

    16. exp-to-pow 1.7%

        \[\leadsto 1.875 \cdot \left(\frac{0}{\sqrt{\pi}} + \frac{\frac{--1}{\color{blue}{{x}^{7}}}}{\sqrt{\pi}}\right) \]

    17. distribute-neg-frac 1.7%

        \[\leadsto 1.875 \cdot \left(\frac{0}{\sqrt{\pi}} + \frac{\color{blue}{-\frac{-1}{{x}^{7}}}}{\sqrt{\pi}}\right) \]

    18. distribute-frac-neg 1.7%

        \[\leadsto 1.875 \cdot \left(\frac{0}{\sqrt{\pi}} + \color{blue}{\left(-\frac{\frac{-1}{{x}^{7}}}{\sqrt{\pi}}\right)}\right) \]

    19. sub-neg 1.7%

        \[\leadsto 1.875 \cdot \color{blue}{\left(\frac{0}{\sqrt{\pi}} - \frac{\frac{-1}{{x}^{7}}}{\sqrt{\pi}}\right)} \]

14. Simplified 1.7%

    \[\leadsto 1.875 \cdot \color{blue}{\frac{{x}^{-7}}{\sqrt{\pi}}} \]

15. Final simplification 1.7%

    \[\leadsto 1.875 \cdot \frac{{x}^{-7}}{\sqrt{\pi}} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024067 
(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)))))))