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

Percentage Accurate: 100.0% → 100.0%

Time: 5.4s

Alternatives: 7

Speedup: 2.5×

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

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(1.875 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -0.75), $MachinePrecision] / N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / N[Abs[x], $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. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{3}{4} + \frac{15}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}}, \frac{1}{\left|x\right|}, \frac{\frac{\frac{1}{2}}{x \cdot x} - -1}{\left|x\right|}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{3}{4} + \frac{15}{8} \cdot \frac{1}{{x}^{2}}}{\color{blue}{{x}^{4}}}, \frac{1}{\left|x\right|}, \frac{\frac{\frac{1}{2}}{x \cdot x} - -1}{\left|x\right|}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   2. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{15}{8} \cdot \frac{1}{{x}^{2}} + \frac{3}{4}}{{\color{blue}{x}}^{4}}, \frac{1}{\left|x\right|}, \frac{\frac{\frac{1}{2}}{x \cdot x} - -1}{\left|x\right|}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   3. lower-+.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{15}{8} \cdot \frac{1}{{x}^{2}} + \frac{3}{4}}{{\color{blue}{x}}^{4}}, \frac{1}{\left|x\right|}, \frac{\frac{\frac{1}{2}}{x \cdot x} - -1}{\left|x\right|}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   4. mult-flip-rev N/A

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

   5. lower-/.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{15}{8}}{x \cdot x} + \frac{3}{4}}{{x}^{\left(2 + \color{blue}{2}\right)}}, \frac{1}{\left|x\right|}, \frac{\frac{\frac{1}{2}}{x \cdot x} - -1}{\left|x\right|}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   9. pow-prod-up N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{15}{8}}{x \cdot x} + \frac{3}{4}}{{x}^{2} \cdot \color{blue}{{x}^{2}}}, \frac{1}{\left|x\right|}, \frac{\frac{\frac{1}{2}}{x \cdot x} - -1}{\left|x\right|}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{15}{8}}{x \cdot x} + \frac{3}{4}}{{x}^{2} \cdot \color{blue}{{x}^{2}}}, \frac{1}{\left|x\right|}, \frac{\frac{\frac{1}{2}}{x \cdot x} - -1}{\left|x\right|}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   11. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{15}{8}}{x \cdot x} + \frac{3}{4}}{\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}}, \frac{1}{\left|x\right|}, \frac{\frac{\frac{1}{2}}{x \cdot x} - -1}{\left|x\right|}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   12. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{15}{8}}{x \cdot x} + \frac{3}{4}}{\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}}, \frac{1}{\left|x\right|}, \frac{\frac{\frac{1}{2}}{x \cdot x} - -1}{\left|x\right|}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   13. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{15}{8}}{x \cdot x} + \frac{3}{4}}{\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)}, \frac{1}{\left|x\right|}, \frac{\frac{\frac{1}{2}}{x \cdot x} - -1}{\left|x\right|}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   14. lift-*.f64 100.0

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

8. Applied rewrites 100.0%

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

10. Applied rewrites 100.0%

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

Alternative 2: 99.6% accurate, 2.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma
   (/ 0.75 (* (* x x) (* x x)))
   (/ 1.0 (fabs x))
   (/ (- (/ 0.5 (* x x)) -1.0) (fabs x)))
  (/ (exp (* x x)) (sqrt PI))))

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(0.75 / N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(x * x), $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. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{3}{4} + \frac{15}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}}, \frac{1}{\left|x\right|}, \frac{\frac{\frac{1}{2}}{x \cdot x} - -1}{\left|x\right|}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{3}{4} + \frac{15}{8} \cdot \frac{1}{{x}^{2}}}{\color{blue}{{x}^{4}}}, \frac{1}{\left|x\right|}, \frac{\frac{\frac{1}{2}}{x \cdot x} - -1}{\left|x\right|}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   2. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{15}{8} \cdot \frac{1}{{x}^{2}} + \frac{3}{4}}{{\color{blue}{x}}^{4}}, \frac{1}{\left|x\right|}, \frac{\frac{\frac{1}{2}}{x \cdot x} - -1}{\left|x\right|}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   3. lower-+.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{15}{8} \cdot \frac{1}{{x}^{2}} + \frac{3}{4}}{{\color{blue}{x}}^{4}}, \frac{1}{\left|x\right|}, \frac{\frac{\frac{1}{2}}{x \cdot x} - -1}{\left|x\right|}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   4. mult-flip-rev N/A

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

   5. lower-/.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{15}{8}}{x \cdot x} + \frac{3}{4}}{{x}^{\left(2 + \color{blue}{2}\right)}}, \frac{1}{\left|x\right|}, \frac{\frac{\frac{1}{2}}{x \cdot x} - -1}{\left|x\right|}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   9. pow-prod-up N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{15}{8}}{x \cdot x} + \frac{3}{4}}{{x}^{2} \cdot \color{blue}{{x}^{2}}}, \frac{1}{\left|x\right|}, \frac{\frac{\frac{1}{2}}{x \cdot x} - -1}{\left|x\right|}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{15}{8}}{x \cdot x} + \frac{3}{4}}{{x}^{2} \cdot \color{blue}{{x}^{2}}}, \frac{1}{\left|x\right|}, \frac{\frac{\frac{1}{2}}{x \cdot x} - -1}{\left|x\right|}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   11. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{15}{8}}{x \cdot x} + \frac{3}{4}}{\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}}, \frac{1}{\left|x\right|}, \frac{\frac{\frac{1}{2}}{x \cdot x} - -1}{\left|x\right|}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   12. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{15}{8}}{x \cdot x} + \frac{3}{4}}{\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}}, \frac{1}{\left|x\right|}, \frac{\frac{\frac{1}{2}}{x \cdot x} - -1}{\left|x\right|}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   13. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{15}{8}}{x \cdot x} + \frac{3}{4}}{\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)}, \frac{1}{\left|x\right|}, \frac{\frac{\frac{1}{2}}{x \cdot x} - -1}{\left|x\right|}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   14. lift-*.f64 100.0

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

8. Applied rewrites 100.0%

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

9. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(\frac{\frac{3}{4}}{\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot x\right)}, \frac{1}{\left|x\right|}, \frac{\frac{\frac{1}{2}}{x \cdot x} - -1}{\left|x\right|}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]
10. Step-by-step derivation

11. Applied rewrites 99.6%

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

Alternative 3: 99.6% accurate, 4.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (* (+ (/ 0.5 (* (* x x) x)) (/ 1.0 (fabs x))) (/ (exp (* x x)) (sqrt PI))))

C

double code(double x) {
	return ((0.5 / ((x * x) * x)) + (1.0 / fabs(x))) * (exp((x * x)) / sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return ((0.5 / ((x * x) * x)) + (1.0 / Math.abs(x))) * (Math.exp((x * x)) / Math.sqrt(Math.PI));
}

Python

def code(x):
	return ((0.5 / ((x * x) * x)) + (1.0 / math.fabs(x))) * (math.exp((x * x)) / math.sqrt(math.pi))

Julia

function code(x)
	return Float64(Float64(Float64(0.5 / Float64(Float64(x * x) * x)) + Float64(1.0 / abs(x))) * Float64(exp(Float64(x * x)) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = ((0.5 / ((x * x) * x)) + (1.0 / abs(x))) * (exp((x * x)) / sqrt(pi));
end

Wolfram

code[x_] := N[(N[(N[(0.5 / N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(x * x), $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. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 99.6%

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

Alternative 4: 99.6% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Abs[N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / x), $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. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{e^{{x}^{2}}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

8. Applied rewrites 99.5%

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

   1. lift-/.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. lift-sqrt.f64 N/A

      \[\leadsto \frac{e^{{x}^{2}}}{\left|x\right| \cdot \sqrt{\pi}} \]

   8. lift-PI.f64 N/A

      \[\leadsto \frac{e^{{x}^{2}}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}} \]

   9. associate-/r* N/A

      \[\leadsto \frac{\frac{e^{{x}^{2}}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{\frac{e^{{x}^{2}}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   11. exp-fabs N/A

       \[\leadsto \frac{\frac{\left|e^{{x}^{2}}\right|}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   12. div-fabs N/A

       \[\leadsto \frac{\left|\frac{e^{{x}^{2}}}{x}\right|}{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}} \]

   13. lower-fabs.f64 N/A

       \[\leadsto \frac{\left|\frac{e^{{x}^{2}}}{x}\right|}{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}} \]

   14. lower-/.f64 N/A

       \[\leadsto \frac{\left|\frac{e^{{x}^{2}}}{x}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

   15. lower-exp.f64 N/A

       \[\leadsto \frac{\left|\frac{e^{{x}^{2}}}{x}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

   16. pow2 N/A

       \[\leadsto \frac{\left|\frac{e^{x \cdot x}}{x}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

   17. lift-*.f64 N/A

       \[\leadsto \frac{\left|\frac{e^{x \cdot x}}{x}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

   18. lift-PI.f64 N/A

       \[\leadsto \frac{\left|\frac{e^{x \cdot x}}{x}\right|}{\sqrt{\pi}} \]

   19. lift-sqrt.f64 99.6

       \[\leadsto \frac{\left|\frac{e^{x \cdot x}}{x}\right|}{\sqrt{\pi}} \]

10. Applied rewrites 99.6%

    \[\leadsto \frac{\left|\frac{e^{x \cdot x}}{x}\right|}{\color{blue}{\sqrt{\pi}}} \]
11. Add Preprocessing

Alternative 5: 99.5% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[(N[Abs[x], $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. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{e^{{x}^{2}}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

8. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\pi}}} \]
9. Add Preprocessing

Alternative 6: 52.1% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(x, x, 1\right)}{\left|x\right| \cdot \sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (fma x x 1.0) (* (fabs x) (sqrt PI))))

C

double code(double x) {
	return fma(x, x, 1.0) / (fabs(x) * sqrt(((double) M_PI)));
}

Julia

function code(x)
	return Float64(fma(x, x, 1.0) / Float64(abs(x) * sqrt(pi)))
end

Wolfram

code[x_] := N[(N[(x * x + 1.0), $MachinePrecision] / N[(N[Abs[x], $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. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{e^{{x}^{2}}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

8. Applied rewrites 99.5%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. pow2 N/A

       \[\leadsto \frac{x \cdot x + 1}{\left|x\right| \cdot \sqrt{\pi}} \]

    3. lower-fma.f64 52.1

       \[\leadsto \frac{\mathsf{fma}\left(x, x, 1\right)}{\left|x\right| \cdot \sqrt{\pi}} \]

11. Applied rewrites 52.1%

    \[\leadsto \frac{\mathsf{fma}\left(x, x, 1\right)}{\color{blue}{\left|x\right|} \cdot \sqrt{\pi}} \]
12. Add Preprocessing

Alternative 7: 2.3% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left|x\right| \cdot \sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (* (fabs x) (sqrt PI))))

C

double code(double x) {
	return 1.0 / (fabs(x) * sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return 1.0 / (Math.abs(x) * Math.sqrt(Math.PI));
}

Python

def code(x):
	return 1.0 / (math.fabs(x) * math.sqrt(math.pi))

Julia

function code(x)
	return Float64(1.0 / Float64(abs(x) * sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (abs(x) * sqrt(pi));
end

Wolfram

code[x_] := N[(1.0 / N[(N[Abs[x], $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. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{e^{{x}^{2}}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

8. Applied rewrites 99.5%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 2.3%

    \[\leadsto \frac{1}{\color{blue}{\left|x\right|} \cdot \sqrt{\pi}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025140 
(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)))))))