VandenBroeck and Keller, Equation (20)

Average Error: 61.5 → 2.5

Time: 19.6s

Precision: binary64

Cost: 59008

Math

\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]

↓

\[\mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{{\pi}^{2} \cdot 0.041666666666666664}}, -4 \cdot \frac{\log \left(\frac{\frac{4}{f}}{\pi}\right)}{\pi}\right) \]

FPCore

(FPCore (f)
 :precision binary64
 (-
  (*
   (/ 1.0 (/ PI 4.0))
   (log
    (/
     (+ (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f))))
     (- (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f)))))))))

↓

(FPCore (f)
 :precision binary64
 (fma
  -2.0
  (+
   (/ (* f (* PI 0.0)) PI)
   (/ (* f f) (/ PI (* (pow PI 2.0) 0.041666666666666664))))
  (* -4.0 (/ (log (/ (/ 4.0 f) PI)) PI))))

C

double code(double f) {
	return -((1.0 / (((double) M_PI) / 4.0)) * log(((exp(((((double) M_PI) / 4.0) * f)) + exp(-((((double) M_PI) / 4.0) * f))) / (exp(((((double) M_PI) / 4.0) * f)) - exp(-((((double) M_PI) / 4.0) * f))))));
}

↓

double code(double f) {
	return fma(-2.0, (((f * (((double) M_PI) * 0.0)) / ((double) M_PI)) + ((f * f) / (((double) M_PI) / (pow(((double) M_PI), 2.0) * 0.041666666666666664)))), (-4.0 * (log(((4.0 / f) / ((double) M_PI))) / ((double) M_PI))));
}

Julia

function code(f)
	return Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(exp(Float64(Float64(pi / 4.0) * f)) + exp(Float64(-Float64(Float64(pi / 4.0) * f)))) / Float64(exp(Float64(Float64(pi / 4.0) * f)) - exp(Float64(-Float64(Float64(pi / 4.0) * f))))))))
end

↓

function code(f)
	return fma(-2.0, Float64(Float64(Float64(f * Float64(pi * 0.0)) / pi) + Float64(Float64(f * f) / Float64(pi / Float64((pi ^ 2.0) * 0.041666666666666664)))), Float64(-4.0 * Float64(log(Float64(Float64(4.0 / f) / pi)) / pi)))
end

Wolfram

code[f_] := (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]], $MachinePrecision] + N[Exp[(-N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] / N[(N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]], $MachinePrecision] - N[Exp[(-N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])

↓

code[f_] := N[(-2.0 * N[(N[(N[(f * N[(Pi * 0.0), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] + N[(N[(f * f), $MachinePrecision] / N[(Pi / N[(N[Power[Pi, 2.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(N[Log[N[(N[(4.0 / f), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.5

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]

2. Simplified 61.5

   \[\leadsto \color{blue}{\log \left(\frac{{\left(e^{f}\right)}^{\left(\frac{\pi}{4}\right)} + {\left(e^{-0.25 \cdot \pi}\right)}^{f}}{{\left(e^{f}\right)}^{\left(\frac{\pi}{4}\right)} - {\left(e^{-0.25 \cdot \pi}\right)}^{f}}\right) \cdot \frac{-4}{\pi}} \]
   Proof

   [Start] 61.5	\[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
   *-commutative [=>] 61.5	\[ -\color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \frac{1}{\frac{\pi}{4}}} \]
   distribute-rgt-neg-in [=>] 61.5	\[ \color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \left(-\frac{1}{\frac{\pi}{4}}\right)} \]

3. Taylor expanded in f around 0 2.4

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right)\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right)} \]

4. Simplified 2.4

   \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3}}{\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}} \cdot -2\right), {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right)}}, -4 \cdot \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}\right)} \]
   Proof

   [Start] 2.4

   \[ -4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right)\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]

5. Applied egg-rr 2.4

   \[\leadsto \mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{\color{blue}{0 + \pi \cdot \left(0.5 \cdot \mathsf{fma}\left(0.0625, 2 \cdot \pi, \frac{{\pi}^{3}}{{\pi}^{2} \cdot 48} \cdot -2\right)\right)}}}, -4 \cdot \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}\right) \]

6. Simplified 2.4

   \[\leadsto \mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{\color{blue}{\pi \cdot \left(\pi \cdot 0.0625 + \pi \cdot -0.020833333333333332\right)}}}, -4 \cdot \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}\right) \]
   Proof

   [Start] 2.4	\[ \mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{0 + \pi \cdot \left(0.5 \cdot \mathsf{fma}\left(0.0625, 2 \cdot \pi, \frac{{\pi}^{3}}{{\pi}^{2} \cdot 48} \cdot -2\right)\right)}}, -4 \cdot \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}\right) \]
   +-lft-identity [=>] 2.4	\[ \mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{\color{blue}{\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(0.0625, 2 \cdot \pi, \frac{{\pi}^{3}}{{\pi}^{2} \cdot 48} \cdot -2\right)\right)}}}, -4 \cdot \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}\right) \]
   associate-*r* [=>] 2.4	\[ \mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{\color{blue}{\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(0.0625, 2 \cdot \pi, \frac{{\pi}^{3}}{{\pi}^{2} \cdot 48} \cdot -2\right)}}}, -4 \cdot \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}\right) \]
   fma-udef [=>] 2.4	\[ \mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(0.0625 \cdot \left(2 \cdot \pi\right) + \frac{{\pi}^{3}}{{\pi}^{2} \cdot 48} \cdot -2\right)}}}, -4 \cdot \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}\right) \]
   +-commutative [=>] 2.4	\[ \mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(\frac{{\pi}^{3}}{{\pi}^{2} \cdot 48} \cdot -2 + 0.0625 \cdot \left(2 \cdot \pi\right)\right)}}}, -4 \cdot \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}\right) \]
   associate-*r* [=>] 2.4	\[ \mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{\left(\pi \cdot 0.5\right) \cdot \left(\frac{{\pi}^{3}}{{\pi}^{2} \cdot 48} \cdot -2 + \color{blue}{\left(0.0625 \cdot 2\right) \cdot \pi}\right)}}, -4 \cdot \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}\right) \]
   metadata-eval [=>] 2.4	\[ \mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{\left(\pi \cdot 0.5\right) \cdot \left(\frac{{\pi}^{3}}{{\pi}^{2} \cdot 48} \cdot -2 + \color{blue}{0.125} \cdot \pi\right)}}, -4 \cdot \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}\right) \]
   metadata-eval [<=] 2.4	\[ \mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{\left(\pi \cdot 0.5\right) \cdot \left(\frac{{\pi}^{3}}{{\pi}^{2} \cdot 48} \cdot -2 + \color{blue}{\frac{0.0625}{0.5}} \cdot \pi\right)}}, -4 \cdot \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}\right) \]
   associate-/r/ [<=] 2.4	\[ \mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{\left(\pi \cdot 0.5\right) \cdot \left(\frac{{\pi}^{3}}{{\pi}^{2} \cdot 48} \cdot -2 + \color{blue}{\frac{0.0625}{\frac{0.5}{\pi}}}\right)}}, -4 \cdot \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}\right) \]
   associate-/l* [<=] 2.4	\[ \mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{\left(\pi \cdot 0.5\right) \cdot \left(\frac{{\pi}^{3}}{{\pi}^{2} \cdot 48} \cdot -2 + \color{blue}{\frac{0.0625 \cdot \pi}{0.5}}\right)}}, -4 \cdot \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}\right) \]
   +-commutative [<=] 2.4	\[ \mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(\frac{0.0625 \cdot \pi}{0.5} + \frac{{\pi}^{3}}{{\pi}^{2} \cdot 48} \cdot -2\right)}}}, -4 \cdot \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}\right) \]
   associate-*l* [=>] 2.4	\[ \mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{\color{blue}{\pi \cdot \left(0.5 \cdot \left(\frac{0.0625 \cdot \pi}{0.5} + \frac{{\pi}^{3}}{{\pi}^{2} \cdot 48} \cdot -2\right)\right)}}}, -4 \cdot \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}\right) \]

7. Applied egg-rr 2.4

   \[\leadsto \mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{\color{blue}{0.0625 \cdot {\pi}^{2} + -0.020833333333333332 \cdot {\pi}^{2}}}}, -4 \cdot \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}\right) \]

8. Simplified 2.4

   \[\leadsto \mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{\color{blue}{{\pi}^{2} \cdot 0.041666666666666664}}}, -4 \cdot \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}\right) \]
   Proof

   [Start] 2.4	\[ \mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{0.0625 \cdot {\pi}^{2} + -0.020833333333333332 \cdot {\pi}^{2}}}, -4 \cdot \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}\right) \]
   distribute-rgt-out [=>] 2.4	\[ \mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{\color{blue}{{\pi}^{2} \cdot \left(0.0625 + -0.020833333333333332\right)}}}, -4 \cdot \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}\right) \]
   metadata-eval [=>] 2.4	\[ \mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{{\pi}^{2} \cdot \color{blue}{0.041666666666666664}}}, -4 \cdot \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}\right) \]

9. Taylor expanded in f around 0 2.4

   \[\leadsto \mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{{\pi}^{2} \cdot 0.041666666666666664}}, -4 \cdot \color{blue}{\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}}\right) \]

10. Simplified 2.5

    \[\leadsto \mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{{\pi}^{2} \cdot 0.041666666666666664}}, -4 \cdot \color{blue}{\frac{\log \left(\frac{\frac{4}{f}}{\pi}\right)}{\pi}}\right) \]
    Proof

    [Start] 2.4	\[ \mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{{\pi}^{2} \cdot 0.041666666666666664}}, -4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}\right) \]
    log-div [=>] 2.4	\[ \mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{{\pi}^{2} \cdot 0.041666666666666664}}, -4 \cdot \frac{\color{blue}{\left(\log 4 - \log \pi\right)} - \log f}{\pi}\right) \]
    associate--l- [=>] 2.4	\[ \mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{{\pi}^{2} \cdot 0.041666666666666664}}, -4 \cdot \frac{\color{blue}{\log 4 - \left(\log \pi + \log f\right)}}{\pi}\right) \]
    log-prod [<=] 2.4	\[ \mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{{\pi}^{2} \cdot 0.041666666666666664}}, -4 \cdot \frac{\log 4 - \color{blue}{\log \left(\pi \cdot f\right)}}{\pi}\right) \]
    *-commutative [<=] 2.4	\[ \mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{{\pi}^{2} \cdot 0.041666666666666664}}, -4 \cdot \frac{\log 4 - \log \color{blue}{\left(f \cdot \pi\right)}}{\pi}\right) \]
    log-div [<=] 2.4	\[ \mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{{\pi}^{2} \cdot 0.041666666666666664}}, -4 \cdot \frac{\color{blue}{\log \left(\frac{4}{f \cdot \pi}\right)}}{\pi}\right) \]
    associate-/r* [=>] 2.5	\[ \mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{{\pi}^{2} \cdot 0.041666666666666664}}, -4 \cdot \frac{\log \color{blue}{\left(\frac{\frac{4}{f}}{\pi}\right)}}{\pi}\right) \]

11. Final simplification 2.5

    \[\leadsto \mathsf{fma}\left(-2, \frac{f \cdot \left(\pi \cdot 0\right)}{\pi} + \frac{f \cdot f}{\frac{\pi}{{\pi}^{2} \cdot 0.041666666666666664}}, -4 \cdot \frac{\log \left(\frac{\frac{4}{f}}{\pi}\right)}{\pi}\right) \]

Alternatives

Alternative 1
Error	2.7
Cost	19648

\[-4 \cdot \frac{\log \left(\frac{4}{f \cdot \pi}\right)}{\pi} \]

Alternative 2
Error	55.3
Cost	6528

\[\frac{-2}{\pi} \]

Reproduce

herbie shell --seed 2023045 
(FPCore (f)
  :name "VandenBroeck and Keller, Equation (20)"
  :precision binary64
  (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f)))) (- (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f)))))))))