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

Average Error: 2.8 → 1.2

Time: 22.4s

Precision: binary64

\[x \geq 0.5\]

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ {\left(e^{x}\right)}^{x} \cdot \left(\mathsf{fma}\left(0.5, \frac{t_0}{{\left(\left|x\right|\right)}^{3}}, \mathsf{fma}\left(1.875, \frac{t_0}{\left|x\right| \cdot {x}^{6}}, 0.75 \cdot \frac{t_0}{\left|x\right| \cdot {x}^{4}}\right)\right) + \frac{t_0}{\left|x\right|}\right) \end{array} \]

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 (/ 1.0 PI))))
   (*
    (pow (exp x) x)
    (+
     (fma
      0.5
      (/ t_0 (pow (fabs x) 3.0))
      (fma
       1.875
       (/ t_0 (* (fabs x) (pow x 6.0)))
       (* 0.75 (/ t_0 (* (fabs x) (pow x 4.0))))))
     (/ t_0 (fabs 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((1.0 / ((double) M_PI)));
	return pow(exp(x), x) * (fma(0.5, (t_0 / pow(fabs(x), 3.0)), fma(1.875, (t_0 / (fabs(x) * pow(x, 6.0))), (0.75 * (t_0 / (fabs(x) * pow(x, 4.0)))))) + (t_0 / fabs(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(Float64(1.0 / pi))
	return Float64((exp(x) ^ x) * Float64(fma(0.5, Float64(t_0 / (abs(x) ^ 3.0)), fma(1.875, Float64(t_0 / Float64(abs(x) * (x ^ 6.0))), Float64(0.75 * Float64(t_0 / Float64(abs(x) * (x ^ 4.0)))))) + Float64(t_0 / abs(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[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] * N[(N[(0.5 * N[(t$95$0 / N[Power[N[Abs[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + N[(1.875 * N[(t$95$0 / N[(N[Abs[x], $MachinePrecision] * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.75 * N[(t$95$0 / N[(N[Abs[x], $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 / N[Abs[x], $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 div-inv_binary64 1.2

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

7. Applied times-frac_binary64 1.2

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

8. Applied associate-*l*_binary64 1.2

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

9. Applied add-sqr-sqrt_binary64 1.2

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

10. Applied associate-/l*_binary64 1.2

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

11. Simplified 1.2

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

12. Taylor expanded in x around 0 1.2

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{1} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \frac{1}{\left|x\right|} + \left(0.5 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \frac{1}{\left|x\right| \cdot {x}^{2}}\right) + \left(1.875 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \frac{1}{\left|x\right| \cdot {x}^{6}}\right) + 0.75 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \frac{1}{\left|x\right| \cdot {x}^{4}}\right)\right)\right)\right)} \]

13. Simplified 1.2

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{1} \cdot \color{blue}{\left(\mathsf{fma}\left(0.5, \frac{\sqrt{\frac{1}{\pi}}}{{\left(\left|x\right|\right)}^{3}}, \mathsf{fma}\left(1.875, \frac{\sqrt{\frac{1}{\pi}}}{\left|x\right| \cdot {x}^{6}}, 0.75 \cdot \frac{\sqrt{\frac{1}{\pi}}}{\left|x\right| \cdot {x}^{4}}\right)\right) + \frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|}\right)} \]

14. Final simplification 1.2

    \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\mathsf{fma}\left(0.5, \frac{\sqrt{\frac{1}{\pi}}}{{\left(\left|x\right|\right)}^{3}}, \mathsf{fma}\left(1.875, \frac{\sqrt{\frac{1}{\pi}}}{\left|x\right| \cdot {x}^{6}}, 0.75 \cdot \frac{\sqrt{\frac{1}{\pi}}}{\left|x\right| \cdot {x}^{4}}\right)\right) + \frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|}\right) \]

Reproduce

herbie shell --seed 2022133 
(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)))))))