VandenBroeck and Keller, Equation (20)

Average Error: 61.7 → 0.8

Time: 24.6s

Precision: binary64

Cost: 20801

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 1.9408592242812768:\\ \;\;\;\;-\frac{\log \left(\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right)}{\frac{\pi}{4}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

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
 (if (<= (* (/ PI 4.0) f) 1.9408592242812768)
   (- (/ (log (/ (cosh (* (/ PI 4.0) f)) (sinh (* (/ PI 4.0) f)))) (/ PI 4.0)))
   0.0))

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) {
	double tmp;
	if (((((double) M_PI) / 4.0) * f) <= 1.9408592242812768) {
		tmp = -(log(cosh((((double) M_PI) / 4.0) * f) / sinh((((double) M_PI) / 4.0) * f)) / (((double) M_PI) / 4.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Error

Bits error versus f

Alternatives

Alternative 1
Error	1.9
Cost	80256

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

Alternative 2
Error	2.8
Cost	79872

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

Alternative 3
Error	61.7
Cost	73856

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

Alternative 4
Error	2.2
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 5
Error	2.5
Cost	66048

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

Alternative 6
Error	61.7
Cost	60416

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

Alternative 7
Error	1.9
Cost	60224

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

Alternative 8
Error	2.2
Cost	59264

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

Alternative 9
Error	61.7
Cost	53696

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

Alternative 10
Error	2.0
Cost	53312

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

Alternative 11
Error	2.2
Cost	53248

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

Alternative 12
Error	2.7
Cost	52864

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

Alternative 13
Error	2.6
Cost	52864

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

Alternative 14
Error	2.4
Cost	48128

\[\left(\left(4 \cdot \frac{\log f}{\pi} + \left(\pi \cdot \left(f \cdot f\right)\right) \cdot 0.041666666666666664\right) - \left(0.125 \cdot \frac{f \cdot f}{\frac{2}{\pi}} + 4 \cdot \frac{\log \left(4 \cdot \frac{\sqrt{2}}{\pi}\right)}{\pi}\right)\right) - \frac{4}{\pi} \cdot \log \left(\frac{\sqrt{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}{2}\right)\]

Alternative 15
Error	2.0
Cost	47232

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

Alternative 16
Error	2.6
Cost	46720

\[\log \left(\frac{\sqrt{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right) \cdot \frac{-4}{\pi} + \log \left(0.5 \cdot \sqrt{2}\right) \cdot \frac{-4}{\pi}\]

Alternative 17
Error	2.0
Cost	46656

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

Alternative 18
Error	2.6
Cost	40320

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

Alternative 19
Error	8.7
Cost	39744

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

Alternative 20
Error	61.7
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	61.7
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	33792

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

Alternative 23
Error	2.1
Cost	33792

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

Alternative 24
Error	1.9
Cost	33408

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

Alternative 25
Error	2.0
Cost	33344

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

Alternative 26
Error	1.9
Cost	33280

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

Alternative 27
Error	2.0
Cost	33280

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

Alternative 28
Error	2.6
Cost	33216

\[\frac{\log \left(\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right) \cdot -2}{\sqrt{\frac{\pi}{4}} \cdot \sqrt{\pi}}\]

Alternative 29
Error	2.2
Cost	33088

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

Alternative 30
Error	2.7
Cost	33024

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

Alternative 31
Error	2.0
Cost	27264

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

Alternative 32
Error	2.0
Cost	27136

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

Alternative 33
Error	15.5
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 34
Error	2.3
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 35
Error	2.0
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 36
Error	2.6
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	1.9
Cost	20224

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

Alternative 38
Error	2.3
Cost	14336

\[\log \left(\left(\frac{4}{\pi \cdot f} + \left(\pi \cdot f\right) \cdot 0.08333333333333333\right) - {\left(\pi \cdot f\right)}^{3} \cdot 0.00034722222222222224\right) \cdot \frac{-1}{\frac{\pi}{4}}\]

Alternative 39
Error	2.3
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 40
Error	2.5
Cost	13376

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

Alternative 41
Error	2.3
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 42
Error	2.6
Cost	7104

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

Alternative 43
Error	63.0
Cost	64

\[1\]

Alternative 44
Error	60.8
Cost	64

\[0\]

Alternative 45
Error	55.2
Cost	64

\[-1\]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 PI.f64 4) f) < 1.9408592242812768

   1. Initial program 61.8

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

   3. Applied sinh-undef_binary64 0.5

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

   5. Applied associate-*l/_binary64 0.4

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

   6. Simplified 0.4

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

   7. Simplified 0.4

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

   if 1.9408592242812768 < (*.f64 (/.f64 PI.f64 4) f)

   1. Initial program 15.9

      \[0\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 1.9408592242812768:\\ \;\;\;\;-\frac{\log \left(\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right)}{\frac{\pi}{4}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2021022 
(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)))))))))