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

Percentage Accurate: 100.0% → 99.6%

Time: 13.3s

Alternatives: 5

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: 99.6% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{{x}^{3}}{\sqrt{\pi}}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* 0.5 (log1p (expm1 (/ (pow x 3.0) (sqrt PI))))))

C

double code(double x) {
	return 0.5 * log1p(expm1((pow(x, 3.0) / sqrt(((double) M_PI)))));
}

Java

public static double code(double x) {
	return 0.5 * Math.log1p(Math.expm1((Math.pow(x, 3.0) / Math.sqrt(Math.PI))));
}

Python

def code(x):
	return 0.5 * math.log1p(math.expm1((math.pow(x, 3.0) / math.sqrt(math.pi))))

Julia

function code(x)
	return Float64(0.5 * log1p(expm1(Float64((x ^ 3.0) / sqrt(pi)))))
end

Wolfram

code[x_] := N[(0.5 * N[Log[1 + N[(Exp[N[(N[Power[x, 3.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]] - 1), $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 1.8%

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

   1. associate-*l/ 1.8%

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

   2. *-lft-identity 1.8%

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

7. Simplified 1.8%

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

   1. /-rgt-identity 1.8%

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

   2. expm1-log1p-u 1.8%

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

   3. /-rgt-identity 1.8%

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

   4. add-sqr-sqrt 1.8%

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

   5. fabs-sqr 1.8%

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

   6. add-sqr-sqrt 1.8%

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

   7. pow-plus 1.8%

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

   8. metadata-eval 1.8%

      \[\leadsto 0.5 \cdot \frac{\sqrt{\frac{1}{\pi}}}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\color{blue}{3}}\right)\right)} \]

9. Applied egg-rr 1.8%

   \[\leadsto 0.5 \cdot \frac{\sqrt{\frac{1}{\pi}}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{3}\right)\right)}} \]
10. Step-by-step derivation

    1. log1p-expm1-u 1.8%

       \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sqrt{\frac{1}{\pi}}}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{3}\right)\right)}\right)\right)} \]

    2. div-inv 1.8%

       \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{3}\right)\right)}}\right)\right) \]

    3. inv-pow 1.8%

       \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{\frac{1}{\pi}} \cdot \color{blue}{{\left(\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{3}\right)\right)\right)}^{-1}}\right)\right) \]

    4. expm1-log1p-u 1.8%

       \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{\frac{1}{\pi}} \cdot {\color{blue}{\left({x}^{3}\right)}}^{-1}\right)\right) \]

    5. pow-to-exp 1.8%

       \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{\frac{1}{\pi}} \cdot {\color{blue}{\left(e^{\log x \cdot 3}\right)}}^{-1}\right)\right) \]

    6. *-commutative 1.8%

       \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{\frac{1}{\pi}} \cdot {\left(e^{\color{blue}{3 \cdot \log x}}\right)}^{-1}\right)\right) \]

    7. exp-prod 1.8%

       \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{\frac{1}{\pi}} \cdot \color{blue}{e^{\left(3 \cdot \log x\right) \cdot -1}}\right)\right) \]

    8. sqrt-div 1.8%

       \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}} \cdot e^{\left(3 \cdot \log x\right) \cdot -1}\right)\right) \]

    9. metadata-eval 1.8%

       \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{1}}{\sqrt{\pi}} \cdot e^{\left(3 \cdot \log x\right) \cdot -1}\right)\right) \]

    10. associate-*l/ 1.8%

        \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{1 \cdot e^{\left(3 \cdot \log x\right) \cdot -1}}{\sqrt{\pi}}}\right)\right) \]

    11. *-un-lft-identity 1.8%

        \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{e^{\left(3 \cdot \log x\right) \cdot -1}}}{\sqrt{\pi}}\right)\right) \]

11. Applied egg-rr 100.0%

    \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{{x}^{3}}{\sqrt{\pi}}\right)\right)} \]
12. Add Preprocessing

Alternative 2: 82.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{\sqrt{\frac{1}{\pi}}}{\sqrt{{x}^{-6}}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* 0.5 (/ (sqrt (/ 1.0 PI)) (sqrt (pow x -6.0)))))

C

double code(double x) {
	return 0.5 * (sqrt((1.0 / ((double) M_PI))) / sqrt(pow(x, -6.0)));
}

Java

public static double code(double x) {
	return 0.5 * (Math.sqrt((1.0 / Math.PI)) / Math.sqrt(Math.pow(x, -6.0)));
}

Python

def code(x):
	return 0.5 * (math.sqrt((1.0 / math.pi)) / math.sqrt(math.pow(x, -6.0)))

Julia

function code(x)
	return Float64(0.5 * Float64(sqrt(Float64(1.0 / pi)) / sqrt((x ^ -6.0))))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * (sqrt((1.0 / pi)) / sqrt((x ^ -6.0)));
end

Wolfram

code[x_] := N[(0.5 * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[Power[x, -6.0], $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. Taylor expanded in x around 0 1.8%

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

   1. associate-*l/ 1.8%

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

   2. *-lft-identity 1.8%

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

7. Simplified 1.8%

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

   1. /-rgt-identity 1.8%

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

   2. expm1-log1p-u 1.8%

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

   3. /-rgt-identity 1.8%

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

   4. add-sqr-sqrt 1.8%

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

   5. fabs-sqr 1.8%

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

   6. add-sqr-sqrt 1.8%

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

   7. pow-plus 1.8%

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

   8. metadata-eval 1.8%

      \[\leadsto 0.5 \cdot \frac{\sqrt{\frac{1}{\pi}}}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\color{blue}{3}}\right)\right)} \]

9. Applied egg-rr 1.8%

   \[\leadsto 0.5 \cdot \frac{\sqrt{\frac{1}{\pi}}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{3}\right)\right)}} \]
10. Step-by-step derivation

    1. expm1-log1p-u 1.8%

       \[\leadsto 0.5 \cdot \frac{\sqrt{\frac{1}{\pi}}}{\color{blue}{{x}^{3}}} \]

    2. pow-to-exp 1.8%

       \[\leadsto 0.5 \cdot \frac{\sqrt{\frac{1}{\pi}}}{\color{blue}{e^{\log x \cdot 3}}} \]

    3. *-commutative 1.8%

       \[\leadsto 0.5 \cdot \frac{\sqrt{\frac{1}{\pi}}}{e^{\color{blue}{3 \cdot \log x}}} \]

    4. add-sqr-sqrt 1.8%

       \[\leadsto 0.5 \cdot \frac{\sqrt{\frac{1}{\pi}}}{e^{\color{blue}{\sqrt{3 \cdot \log x} \cdot \sqrt{3 \cdot \log x}}}} \]

    5. sqrt-unprod 1.8%

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

    6. *-un-lft-identity 1.8%

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

    7. metadata-eval 1.8%

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

    8. swap-sqr 1.8%

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

    9. *-commutative 1.8%

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

    10. *-commutative 1.8%

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

    11. sqrt-unprod 0.0%

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

    12. add-sqr-sqrt 72.0%

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

    13. add-sqr-sqrt 72.0%

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

    14. sqrt-unprod 85.0%

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

11. Applied egg-rr 85.0%

    \[\leadsto 0.5 \cdot \frac{\sqrt{\frac{1}{\pi}}}{\color{blue}{\sqrt{{x}^{-6}}}} \]
12. Add Preprocessing

Alternative 3: 68.0% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot {x}^{3}}{\sqrt{\pi}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(0.5 * N[Power[x, 3.0], $MachinePrecision]), $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%

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

5. Taylor expanded in x around 0 1.8%

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

   1. add-exp-log 1.8%

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

   2. associate-*r* 1.8%

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

   3. sqrt-div 1.8%

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

   4. metadata-eval 1.8%

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

   5. un-div-inv 1.8%

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

7. Applied egg-rr 1.8%

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

   1. rem-exp-log 1.8%

      \[\leadsto \color{blue}{\frac{{x}^{-3} \cdot 0.5}{\sqrt{\pi}}} \]

9. Applied egg-rr 72.0%

   \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{3}}{\sqrt{\pi}}} \]
10. Add Preprocessing

Alternative 4: 1.8% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{{x}^{-3}}{\sqrt{\pi}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(0.5 * N[(N[Power[x, -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 1.8%

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

   1. associate-*l/ 1.8%

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

   2. *-lft-identity 1.8%

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

7. Simplified 1.8%

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

9. Applied egg-rr 1.8%

   \[\leadsto 0.5 \cdot \color{blue}{\left(0 + \frac{{x}^{-3}}{\sqrt{\pi}}\right)} \]
10. Step-by-step derivation

    1. +-lft-identity 1.8%

       \[\leadsto 0.5 \cdot \color{blue}{\frac{{x}^{-3}}{\sqrt{\pi}}} \]

11. Simplified 1.8%

    \[\leadsto 0.5 \cdot \color{blue}{\frac{{x}^{-3}}{\sqrt{\pi}}} \]
12. Add Preprocessing

Alternative 5: 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 1.8%

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

   1. expm1-log1p-u 1.8%

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

   2. expm1-undefine 1.6%

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

7. Applied egg-rr 1.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{x}^{-3} \cdot 0.5}{\sqrt{\pi}}\right)} - 1} \]

8. Taylor expanded in x around inf 1.6%

   \[\leadsto \color{blue}{1} - 1 \]

9. Final simplification 1.6%

   \[\leadsto 0 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024111 
(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)))))))