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

Average Error: 2.8 → 1.2

Time: 34.6s

Precision: binary64

\[x \geq 0.5\]

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\sqrt{\pi}}\\ \frac{1}{t_0} \cdot \left(\frac{\frac{{\left(e^{x}\right)}^{x}}{t_0}}{\left|x\right|} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right)\right)\right)\right) \end{array} \]

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 (sqrt (sqrt PI))))
   (*
    (/ 1.0 t_0)
    (*
     (/ (/ (pow (exp x) x) t_0) (fabs x))
     (+
      1.0
      (+
       (/ 1.875 (pow x 6.0))
       (log1p (expm1 (/ (+ 0.5 (/ 0.75 (* x x))) (* 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 = sqrt(sqrt(((double) M_PI)));
	return (1.0 / t_0) * (((pow(exp(x), x) / t_0) / fabs(x)) * (1.0 + ((1.875 / pow(x, 6.0)) + log1p(expm1(((0.5 + (0.75 / (x * x))) / (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 = Math.sqrt(Math.sqrt(Math.PI));
	return (1.0 / t_0) * (((Math.pow(Math.exp(x), x) / t_0) / Math.abs(x)) * (1.0 + ((1.875 / Math.pow(x, 6.0)) + Math.log1p(Math.expm1(((0.5 + (0.75 / (x * x))) / (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 = math.sqrt(math.sqrt(math.pi))
	return (1.0 / t_0) * (((math.pow(math.exp(x), x) / t_0) / math.fabs(x)) * (1.0 + ((1.875 / math.pow(x, 6.0)) + math.log1p(math.expm1(((0.5 + (0.75 / (x * x))) / (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 = sqrt(sqrt(pi))
	return Float64(Float64(1.0 / t_0) * Float64(Float64(Float64((exp(x) ^ x) / t_0) / abs(x)) * Float64(1.0 + Float64(Float64(1.875 / (x ^ 6.0)) + log1p(expm1(Float64(Float64(0.5 + Float64(0.75 / Float64(x * x))) / Float64(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[Sqrt[N[Sqrt[Pi], $MachinePrecision]], $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / t$95$0), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[Log[1 + N[(Exp[N[(N[(0.5 + N[(0.75 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $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 2.7

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

3. Applied add-log-exp_binary64 2.7

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

4. Applied exp-to-pow_binary64 1.3

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

5. Applied *-un-lft-identity_binary64 1.3

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

6. Applied add-sqr-sqrt_binary64 1.2

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

7. Applied *-un-lft-identity_binary64 1.2

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

8. Applied times-frac_binary64 1.2

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

9. Applied times-frac_binary64 1.2

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

10. Applied associate-*l*_binary64 1.2

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

11. Applied log1p-expm1-u_binary64 1.2

    \[\leadsto \frac{\frac{1}{\sqrt{\sqrt{\pi}}}}{1} \cdot \left(\frac{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\sqrt{\pi}}}}{\left|x\right|} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right)}\right)\right)\right) \]

12. Final simplification 1.2

    \[\leadsto \frac{1}{\sqrt{\sqrt{\pi}}} \cdot \left(\frac{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\sqrt{\pi}}}}{\left|x\right|} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right)\right)\right)\right) \]

Reproduce

herbie shell --seed 2022129 
(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)))))))