VandenBroeck and Keller, Equation (20)

Average Error: 61.6 → 1.8

Time: 18.3s

Precision: binary64

Cost: 20224

Math

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

↓

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

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
 (- (/ (log (/ (cosh (* (/ PI 4.0) f)) (sinh (* (/ PI 4.0) f)))) (/ PI 4.0))))

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 -(log(cosh((((double) M_PI) / 4.0) * f) / sinh((((double) M_PI) / 4.0) * f)) / (((double) M_PI) / 4.0));
}

Error

Bits error versus f

Alternatives

Alternative 1
Error	61.6
Cost	106048

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

Alternative 2
Error	61.6
Cost	80448

\[\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\sqrt[3]{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}} \cdot \left(\sqrt[3]{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}} \cdot \sqrt[3]{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)}\right) \cdot \frac{-1}{\frac{\pi}{4}}\]

Alternative 3
Error	2.7
Cost	79872

\[-\sqrt[3]{\log \left(\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right) \cdot \frac{4}{\pi}} \cdot \left(\sqrt[3]{\log \left(\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right) \cdot \frac{4}{\pi}} \cdot \sqrt[3]{\log \left(\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right) \cdot \frac{4}{\pi}}\right)\]

Alternative 4
Error	1.8
Cost	79744

\[-\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \left(\sqrt{\frac{4}{\pi}} \cdot \log \left(\sqrt[3]{\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}} \cdot \left(\sqrt[3]{\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}} \cdot \sqrt[3]{\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}}\right)\right)\right)\]

Alternative 5
Error	2.7
Cost	79424

\[\left(\sqrt[3]{\log \left(\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right)} \cdot \left(\sqrt[3]{\log \left(\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right)} \cdot \sqrt[3]{\log \left(\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right)}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}}\]

Alternative 6
Error	1.9
Cost	73728

\[\log \left(\frac{\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}} \cdot \sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}{2} \cdot \frac{\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right) \cdot \frac{-1}{\frac{\pi}{4}}\]

Alternative 7
Error	61.6
Cost	73728

\[\log \left(\frac{2}{\sqrt[3]{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}} \cdot \sqrt[3]{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}} \cdot \frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sqrt[3]{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}}\right) \cdot \frac{-1}{\frac{\pi}{4}}\]

Alternative 8
Error	61.7
Cost	73472

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

Alternative 9
Error	2.0
Cost	66624

\[\left(\left(2 \cdot \log \left(\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}\right) - \log 2\right) + \log \left(\frac{\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}}\]

Alternative 10
Error	61.6
Cost	66624

\[\left(\left(\log 2 - 2 \cdot \log \left(\sqrt[3]{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\right) + \log \left(\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sqrt[3]{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}}\]

Alternative 11
Error	61.7
Cost	66496

\[\log \left(\frac{2}{\sqrt{e^{\frac{\pi}{4} \cdot f}} + \sqrt{e^{-\frac{\pi}{4} \cdot f}}} \cdot \frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sqrt{e^{\frac{\pi}{4} \cdot f}} - \sqrt{e^{-\frac{\pi}{4} \cdot f}}}\right) \cdot \frac{-1}{\frac{\pi}{4}}\]

Alternative 12
Error	2.1
Cost	66048

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

Alternative 13
Error	2.1
Cost	60096

\[\left(\log \left(\frac{\sqrt{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}{2}\right) + \log \left(\frac{\sqrt{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}}\]

Alternative 14
Error	2.1
Cost	53248

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

Alternative 15
Error	1.9
Cost	53120

\[-\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \left(\sqrt{\frac{4}{\pi}} \cdot \log \left(\frac{1}{\sqrt{\sinh \left(\frac{\pi}{4} \cdot f\right)}} \cdot \frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sqrt{\sinh \left(\frac{\pi}{4} \cdot f\right)}}\right)\right)\]

Alternative 16
Error	2.1
Cost	53056

\[\left(\sqrt{\log \left(\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right)} \cdot \sqrt{\log \left(\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right)}\right) \cdot \frac{-1}{\frac{\pi}{4}}\]

Alternative 17
Error	2.1
Cost	53056

\[\sqrt{\log \left(\cosh \left(\frac{\pi}{4} \cdot f\right) \cdot \frac{1}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right)} \cdot \left(\sqrt{\log \left(\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right)} \cdot \frac{-4}{\pi}\right)\]

Alternative 18
Error	61.6
Cost	46848

\[\sqrt[3]{{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)}^{3}} \cdot \frac{-1}{\frac{\pi}{4}}\]

Alternative 19
Error	61.6
Cost	46784

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

Alternative 20
Error	61.6
Cost	34112

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

Alternative 21
Error	61.6
Cost	33984

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

Alternative 22
Error	2.0
Cost	33664

\[\left(\log 0.5 + \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}}\]

Alternative 23
Error	1.9
Cost	33664

\[\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(2 \cdot \sinh \left(\frac{\pi}{4} \cdot f\right)\right) - \log \left(e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}\right)\right)\]

Alternative 24
Error	1.9
Cost	33536

\[-\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \left(\log \left(\cosh \left(\frac{\pi}{4} \cdot f\right) \cdot \frac{1}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right) \cdot \sqrt{\frac{4}{\pi}}\right)\]

Alternative 25
Error	1.8
Cost	33408

\[-\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \left(\log \left(\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right) \cdot \sqrt{\frac{4}{\pi}}\right)\]

Alternative 26
Error	2.1
Cost	33152

\[\sqrt[3]{{\log \left(\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right)}^{3}} \cdot \frac{-1}{\frac{\pi}{4}}\]

Alternative 27
Error	2.6
Cost	33088

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

Alternative 28
Error	2.1
Cost	33088

\[-\sqrt[3]{{\left(\frac{\log \left(\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right)}{\pi}\right)}^{3} \cdot 64}\]

Alternative 29
Error	2.6
Cost	33024

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

Alternative 30
Error	1.9
Cost	27264

\[\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{2 \cdot \sinh \left(\frac{\pi}{4} \cdot f\right)}\right) \cdot \frac{-1}{\frac{\pi}{4}}\]

Alternative 31
Error	14.9
Cost	26560

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

Alternative 32
Error	2.2
Cost	20608

\[-\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \left(\sqrt{\frac{4}{\pi}} \cdot \log \left(\frac{4}{\pi \cdot f} + \left(\pi \cdot f\right) \cdot 0.08333333333333333\right)\right)\]

Alternative 33
Error	1.9
Cost	20416

\[\log \left(\cosh \left(\frac{\pi}{4} \cdot f\right) \cdot \frac{1}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right) \cdot \frac{-1}{\frac{\pi}{4}}\]

Alternative 34
Error	1.9
Cost	20288

\[\log \left(\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right) \cdot \frac{-1}{\frac{\pi}{4}}\]

Alternative 35
Error	1.9
Cost	20288

\[\log \left(\cosh \left(\frac{\pi}{4} \cdot f\right) \cdot \frac{1}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right) \cdot \frac{-4}{\pi}\]

Alternative 36
Error	2.5
Cost	20224

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

Alternative 37
Error	2.2
Cost	14144

\[4 \cdot \frac{\log f}{\pi} - \left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + 0.08333333333333333 \cdot \left(\pi \cdot \left(f \cdot f\right)\right)\right)\]

Alternative 38
Error	2.5
Cost	13504

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

Alternative 39
Error	2.5
Cost	13376

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

Alternative 40
Error	2.2
Cost	7488

\[\log \left(\frac{4}{\pi \cdot f} + \left(\pi \cdot f\right) \cdot 0.08333333333333333\right) \cdot \frac{-1}{\frac{\pi}{4}}\]

Alternative 41
Error	2.6
Cost	7104

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

Alternative 42
Error	63.0
Cost	64

\[1\]

Alternative 43
Error	60.8
Cost	64

\[0\]

Alternative 44
Error	55.2
Cost	64

\[-1\]

Derivation

1. Initial program 61.6

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

3. Applied sinh-undef_binary64 1.9

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{2 \cdot \sinh \left(\frac{\pi}{4} \cdot f\right)}}\right)\]

4. Applied add-cube-cbrt_binary64 1.9

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

5. Applied times-frac_binary64 1.9

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}} \cdot \sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}{2} \cdot \frac{\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right)}\]
6. Using strategy rm

7. Applied div-inv_binary64 2.0

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

8. Applied associate-*r*_binary64 1.9

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

9. Simplified 1.9

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\color{blue}{\cosh \left(\frac{\pi}{4} \cdot f\right)} \cdot \frac{1}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right)\]
10. Using strategy rm

11. Applied associate-*l/_binary64 1.8

    \[\leadsto -\color{blue}{\frac{1 \cdot \log \left(\cosh \left(\frac{\pi}{4} \cdot f\right) \cdot \frac{1}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right)}{\frac{\pi}{4}}}\]

12. Simplified 1.8

    \[\leadsto -\frac{\color{blue}{\log \left(\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right)}}{\frac{\pi}{4}}\]

13. Simplified 1.8

    \[\leadsto \color{blue}{-\frac{\log \left(\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right)}{\frac{\pi}{4}}}\]

14. Final simplification 1.8

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

Reproduce

herbie shell --seed 2021042 
(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)))))))))