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

Average Error: 2.8 → 1.2

Time: 5.6s

Precision: binary64

\[x \geq 0.5\]

Math

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

↓

\[\begin{array}{l} t_0 := {\left(e^{x}\right)}^{x}\\ \sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{t_0}{\frac{{x}^{5}}{0.75}} + 1.875 \cdot \frac{t_0}{{x}^{7}}\right) + \left(\frac{t_0}{x} + \frac{{\left({\left(e^{x}\right)}^{\left(\sqrt{x}\right)}\right)}^{\left(\sqrt{x}\right)}}{\frac{{x}^{3}}{0.5}}\right)\right) \end{array} \]

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 (pow (exp x) x)))
   (*
    (sqrt (/ 1.0 PI))
    (+
     (+ (/ t_0 (/ (pow x 5.0) 0.75)) (* 1.875 (/ t_0 (pow x 7.0))))
     (+
      (/ t_0 x)
      (/ (pow (pow (exp x) (sqrt x)) (sqrt x)) (/ (pow x 3.0) 0.5)))))))

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 = pow(exp(x), x);
	return sqrt((1.0 / ((double) M_PI))) * (((t_0 / (pow(x, 5.0) / 0.75)) + (1.875 * (t_0 / pow(x, 7.0)))) + ((t_0 / x) + (pow(pow(exp(x), sqrt(x)), sqrt(x)) / (pow(x, 3.0) / 0.5))));
}

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 = Math.pow(Math.exp(x), x);
	return Math.sqrt((1.0 / Math.PI)) * (((t_0 / (Math.pow(x, 5.0) / 0.75)) + (1.875 * (t_0 / Math.pow(x, 7.0)))) + ((t_0 / x) + (Math.pow(Math.pow(Math.exp(x), Math.sqrt(x)), Math.sqrt(x)) / (Math.pow(x, 3.0) / 0.5))));
}

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 = math.pow(math.exp(x), x)
	return math.sqrt((1.0 / math.pi)) * (((t_0 / (math.pow(x, 5.0) / 0.75)) + (1.875 * (t_0 / math.pow(x, 7.0)))) + ((t_0 / x) + (math.pow(math.pow(math.exp(x), math.sqrt(x)), math.sqrt(x)) / (math.pow(x, 3.0) / 0.5))))

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 = exp(x) ^ x
	return Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(Float64(t_0 / Float64((x ^ 5.0) / 0.75)) + Float64(1.875 * Float64(t_0 / (x ^ 7.0)))) + Float64(Float64(t_0 / x) + Float64(((exp(x) ^ sqrt(x)) ^ sqrt(x)) / Float64((x ^ 3.0) / 0.5)))))
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 = exp(x) ^ x;
	tmp = sqrt((1.0 / pi)) * (((t_0 / ((x ^ 5.0) / 0.75)) + (1.875 * (t_0 / (x ^ 7.0)))) + ((t_0 / x) + (((exp(x) ^ sqrt(x)) ^ sqrt(x)) / ((x ^ 3.0) / 0.5))));
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[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]}, N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(t$95$0 / N[(N[Power[x, 5.0], $MachinePrecision] / 0.75), $MachinePrecision]), $MachinePrecision] + N[(1.875 * N[(t$95$0 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 / x), $MachinePrecision] + N[(N[Power[N[Power[N[Exp[x], $MachinePrecision], N[Sqrt[x], $MachinePrecision]], $MachinePrecision], N[Sqrt[x], $MachinePrecision]], $MachinePrecision] / N[(N[Power[x, 3.0], $MachinePrecision] / 0.5), $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

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

3. Taylor expanded in x around inf 2.6

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

4. Simplified 1.2

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

5. Taylor expanded in x around inf 1.2

   \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{{\left(e^{x}\right)}^{x}}{\frac{{x}^{5}}{0.75}} + 1.875 \cdot \color{blue}{\frac{e^{{x}^{2}}}{{x}^{7}}}\right) + \left(\frac{{\left(e^{x}\right)}^{x}}{x} + \frac{{\left(e^{x}\right)}^{x}}{\frac{{x}^{3}}{0.5}}\right)\right) \]

6. Simplified 1.2

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

7. Applied egg-rr 1.2

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

8. Applied egg-rr 1.2

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

9. Final simplification 1.2

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

Reproduce

herbie shell --seed 2022155 
(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)))))))