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

Average Error: 2.8 → 1.2

Time: 5.2s

Precision: binary64

\[x \geq 0.5\]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\sqrt{\pi}}\\ \left(\left(\frac{1}{x} + \frac{0.75}{{x}^{5}}\right) \cdot {\left(e^{x}\right)}^{x}\right) \cdot t_0 + t_0 \cdot \left(e^{x \cdot x} \cdot \frac{\frac{0.5}{x} + \frac{1.875}{{x}^{5}}}{x \cdot x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* (/ 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)))))))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt PI))))
   (+
    (* (* (+ (/ 1.0 x) (/ 0.75 (pow x 5.0))) (pow (exp x) x)) t_0)
    (*
     t_0
     (* (exp (* x x)) (/ (+ (/ 0.5 x) (/ 1.875 (pow x 5.0))) (* x x)))))))

C

double code(double x) {
	return ((1.0 / sqrt(((double) M_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)))));
}

↓

double code(double x) {
	double t_0 = 1.0 / sqrt(((double) M_PI));
	return ((((1.0 / x) + (0.75 / pow(x, 5.0))) * pow(exp(x), x)) * t_0) + (t_0 * (exp((x * x)) * (((0.5 / x) + (1.875 / pow(x, 5.0))) / (x * x))));
}

Java

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

↓

public static double code(double x) {
	double t_0 = 1.0 / Math.sqrt(Math.PI);
	return ((((1.0 / x) + (0.75 / Math.pow(x, 5.0))) * Math.pow(Math.exp(x), x)) * t_0) + (t_0 * (Math.exp((x * x)) * (((0.5 / x) + (1.875 / Math.pow(x, 5.0))) / (x * x))));
}

Python

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

↓

def code(x):
	t_0 = 1.0 / math.sqrt(math.pi)
	return ((((1.0 / x) + (0.75 / math.pow(x, 5.0))) * math.pow(math.exp(x), x)) * t_0) + (t_0 * (math.exp((x * x)) * (((0.5 / x) + (1.875 / math.pow(x, 5.0))) / (x * x))))

Julia

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

↓

function code(x)
	t_0 = Float64(1.0 / sqrt(pi))
	return Float64(Float64(Float64(Float64(Float64(1.0 / x) + Float64(0.75 / (x ^ 5.0))) * (exp(x) ^ x)) * t_0) + Float64(t_0 * Float64(exp(Float64(x * x)) * Float64(Float64(Float64(0.5 / x) + Float64(1.875 / (x ^ 5.0))) / Float64(x * x)))))
end

MATLAB

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

↓

function tmp = code(x)
	t_0 = 1.0 / sqrt(pi);
	tmp = ((((1.0 / x) + (0.75 / (x ^ 5.0))) * (exp(x) ^ x)) * t_0) + (t_0 * (exp((x * x)) * (((0.5 / x) + (1.875 / (x ^ 5.0))) / (x * x))));
end

Wolfram

code[x_] := 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[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 2.0), $MachinePrecision] * N[(N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * N[(N[(N[(N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(1.0 / x), $MachinePrecision] + N[(0.75 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(t$95$0 * N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(0.5 / x), $MachinePrecision] + N[(1.875 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Error

Bits error versus x

Derivation

1. Initial program 2.8

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

2. Simplified 1.3

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

3. Taylor expanded in x around inf 2.7

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

4. Simplified 1.2

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

5. Applied egg-rr 1.2

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

6. Applied egg-rr 1.2

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

7. Applied egg-rr 1.2

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

8. Final simplification 1.2

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

Reproduce

herbie shell --seed 2022178 
(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)))))))