VandenBroeck and Keller, Equation (20)

Average Error: 61.4 → 2.4

Time: 14.6s

Precision: binary64

Math

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

↓

\[\log \left(\left(\frac{\pi}{\frac{\pi}{-0.5}} + \left(0.125 \cdot \frac{{\pi}^{3} \cdot \left(f \cdot 0.0625\right)}{{\pi}^{2} \cdot 0.25} + \left(0.5 \cdot \left(f \cdot \frac{{\pi}^{2}}{\frac{\pi}{0.125}}\right) + \left(0.001953125 \cdot \frac{f \cdot {\pi}^{4}}{0.125 \cdot {\pi}^{3}} + \left(\frac{2}{f \cdot \left(\pi \cdot 0.5\right)} + \left(0.03125 \cdot \frac{f \cdot {\pi}^{2}}{\pi \cdot 0.5} + \left(0.8333333333333334 \cdot \frac{f \cdot \left({\pi}^{3} \cdot -0.015625\right)}{{\pi}^{2} \cdot 0.25} + \left(0.5 \cdot \left(\frac{f \cdot {\pi}^{4}}{0.125 \cdot {\pi}^{3}} \cdot 0.00390625\right) + \left(\frac{\pi \cdot 0.5}{\pi} + \frac{0.0625 \cdot {\pi}^{2}}{{\pi}^{2} \cdot 0.25}\right)\right)\right)\right)\right)\right)\right)\right)\right) - \left(0.03125 \cdot \frac{{\pi}^{3} \cdot \left(f \cdot -0.25\right)}{{\pi}^{2} \cdot 0.25} + \left(0.013020833333333334 \cdot \frac{{\pi}^{3} \cdot f}{{\pi}^{2} \cdot 0.25} + 0.0625 \cdot \left(\frac{{\pi}^{2}}{{\pi}^{2} \cdot 0.25} + 0.0625 \cdot \frac{f \cdot {\left({\pi}^{\left(\sqrt[3]{4} \cdot \sqrt[3]{4}\right)}\right)}^{\left(\sqrt[3]{4}\right)}}{0.125 \cdot {\pi}^{3}}\right)\right)\right)\right) \cdot \frac{-4}{\pi}\]

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
   (-
    (+
     (/ PI (/ PI -0.5))
     (+
      (* 0.125 (/ (* (pow PI 3.0) (* f 0.0625)) (* (pow PI 2.0) 0.25)))
      (+
       (* 0.5 (* f (/ (pow PI 2.0) (/ PI 0.125))))
       (+
        (* 0.001953125 (/ (* f (pow PI 4.0)) (* 0.125 (pow PI 3.0))))
        (+
         (/ 2.0 (* f (* PI 0.5)))
         (+
          (* 0.03125 (/ (* f (pow PI 2.0)) (* PI 0.5)))
          (+
           (*
            0.8333333333333334
            (/ (* f (* (pow PI 3.0) -0.015625)) (* (pow PI 2.0) 0.25)))
           (+
            (*
             0.5
             (* (/ (* f (pow PI 4.0)) (* 0.125 (pow PI 3.0))) 0.00390625))
            (+
             (/ (* PI 0.5) PI)
             (/ (* 0.0625 (pow PI 2.0)) (* (pow PI 2.0) 0.25)))))))))))
    (+
     (* 0.03125 (/ (* (pow PI 3.0) (* f -0.25)) (* (pow PI 2.0) 0.25)))
     (+
      (* 0.013020833333333334 (/ (* (pow PI 3.0) f) (* (pow PI 2.0) 0.25)))
      (*
       0.0625
       (+
        (/ (pow PI 2.0) (* (pow PI 2.0) 0.25))
        (*
         0.0625
         (/
          (* f (pow (pow PI (* (cbrt 4.0) (cbrt 4.0))) (cbrt 4.0)))
          (* 0.125 (pow PI 3.0))))))))))
  (/ -4.0 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 log(((((double) M_PI) / (((double) M_PI) / -0.5)) + ((0.125 * ((pow(((double) M_PI), 3.0) * (f * 0.0625)) / (pow(((double) M_PI), 2.0) * 0.25))) + ((0.5 * (f * (pow(((double) M_PI), 2.0) / (((double) M_PI) / 0.125)))) + ((0.001953125 * ((f * pow(((double) M_PI), 4.0)) / (0.125 * pow(((double) M_PI), 3.0)))) + ((2.0 / (f * (((double) M_PI) * 0.5))) + ((0.03125 * ((f * pow(((double) M_PI), 2.0)) / (((double) M_PI) * 0.5))) + ((0.8333333333333334 * ((f * (pow(((double) M_PI), 3.0) * -0.015625)) / (pow(((double) M_PI), 2.0) * 0.25))) + ((0.5 * (((f * pow(((double) M_PI), 4.0)) / (0.125 * pow(((double) M_PI), 3.0))) * 0.00390625)) + (((((double) M_PI) * 0.5) / ((double) M_PI)) + ((0.0625 * pow(((double) M_PI), 2.0)) / (pow(((double) M_PI), 2.0) * 0.25))))))))))) - ((0.03125 * ((pow(((double) M_PI), 3.0) * (f * -0.25)) / (pow(((double) M_PI), 2.0) * 0.25))) + ((0.013020833333333334 * ((pow(((double) M_PI), 3.0) * f) / (pow(((double) M_PI), 2.0) * 0.25))) + (0.0625 * ((pow(((double) M_PI), 2.0) / (pow(((double) M_PI), 2.0) * 0.25)) + (0.0625 * ((f * pow(pow(((double) M_PI), (cbrt(4.0) * cbrt(4.0))), cbrt(4.0))) / (0.125 * pow(((double) M_PI), 3.0))))))))) * (-4.0 / ((double) M_PI));
}

Error

Bits error versus f

Derivation

1. Initial program 61.4

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

2. Simplified 61.4

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

3. Taylor expanded around 0 2.4

   \[\leadsto \log \color{blue}{\left(\left(\frac{\log \left(e^{-0.25}\right) \cdot \pi}{0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)} + \left(0.125 \cdot \frac{{\pi}^{3} \cdot \left({\log \left(e^{-0.25}\right)}^{2} \cdot f\right)}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{2}} + \left(0.5 \cdot \frac{{\pi}^{2} \cdot \left({\log \left(e^{-0.25}\right)}^{2} \cdot f\right)}{0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)} + \left(0.001953125 \cdot \frac{{\pi}^{4} \cdot f}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{3}} + \left(2 \cdot \frac{1}{f \cdot \left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)} + \left(0.03125 \cdot \frac{{\pi}^{2} \cdot f}{0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)} + \left(0.8333333333333334 \cdot \frac{{\pi}^{3} \cdot \left({\log \left(e^{-0.25}\right)}^{3} \cdot f\right)}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{2}} + \left(0.5 \cdot \frac{{\pi}^{4} \cdot \left({\log \left(e^{-0.25}\right)}^{4} \cdot f\right)}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{3}} + \left(0.25 \cdot \frac{\pi}{0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)} + \frac{{\pi}^{2} \cdot {\log \left(e^{-0.25}\right)}^{2}}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{2}}\right)\right)\right)\right)\right)\right)\right)\right)\right) - \left(0.03125 \cdot \frac{{\pi}^{3} \cdot \left(\log \left(e^{-0.25}\right) \cdot f\right)}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{2}} + \left(0.0625 \cdot \frac{{\pi}^{4} \cdot \left({\log \left(e^{-0.25}\right)}^{2} \cdot f\right)}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{3}} + \left(0.013020833333333334 \cdot \frac{{\pi}^{3} \cdot f}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{2}} + 0.0625 \cdot \frac{{\pi}^{2}}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{2}}\right)\right)\right)\right)} \cdot \frac{-4}{\pi}\]

4. Simplified 2.4

   \[\leadsto \log \color{blue}{\left(\left(\frac{\pi}{\frac{\pi}{-0.5}} + \left(0.125 \cdot \frac{{\pi}^{3} \cdot \left(f \cdot 0.0625\right)}{{\pi}^{2} \cdot 0.25} + \left(0.5 \cdot \left(\frac{{\pi}^{2}}{\frac{\pi}{0.125}} \cdot f\right) + \left(0.001953125 \cdot \frac{f \cdot {\pi}^{4}}{{\pi}^{3} \cdot 0.125} + \left(\frac{2}{f \cdot \left(\pi \cdot 0.5\right)} + \left(0.03125 \cdot \frac{f \cdot {\pi}^{2}}{\pi \cdot 0.5} + \left(0.8333333333333334 \cdot \frac{f \cdot \left({\pi}^{3} \cdot -0.015625\right)}{{\pi}^{2} \cdot 0.25} + \left(0.5 \cdot \left(\frac{f \cdot {\pi}^{4}}{{\pi}^{3} \cdot 0.125} \cdot 0.00390625\right) + \left(\frac{\pi \cdot 0.5}{\pi} + \frac{{\pi}^{2} \cdot 0.0625}{{\pi}^{2} \cdot 0.25}\right)\right)\right)\right)\right)\right)\right)\right)\right) - \left(0.03125 \cdot \frac{{\pi}^{3} \cdot \left(f \cdot -0.25\right)}{{\pi}^{2} \cdot 0.25} + \left(0.013020833333333334 \cdot \frac{f \cdot {\pi}^{3}}{{\pi}^{2} \cdot 0.25} + 0.0625 \cdot \left(\frac{{\pi}^{2}}{{\pi}^{2} \cdot 0.25} + \frac{f \cdot {\pi}^{4}}{{\pi}^{3} \cdot 0.125} \cdot 0.0625\right)\right)\right)\right)} \cdot \frac{-4}{\pi}\]
5. Using strategy rm

6. Applied add-cube-cbrt_binary64 2.4

   \[\leadsto \log \left(\left(\frac{\pi}{\frac{\pi}{-0.5}} + \left(0.125 \cdot \frac{{\pi}^{3} \cdot \left(f \cdot 0.0625\right)}{{\pi}^{2} \cdot 0.25} + \left(0.5 \cdot \left(\frac{{\pi}^{2}}{\frac{\pi}{0.125}} \cdot f\right) + \left(0.001953125 \cdot \frac{f \cdot {\pi}^{4}}{{\pi}^{3} \cdot 0.125} + \left(\frac{2}{f \cdot \left(\pi \cdot 0.5\right)} + \left(0.03125 \cdot \frac{f \cdot {\pi}^{2}}{\pi \cdot 0.5} + \left(0.8333333333333334 \cdot \frac{f \cdot \left({\pi}^{3} \cdot -0.015625\right)}{{\pi}^{2} \cdot 0.25} + \left(0.5 \cdot \left(\frac{f \cdot {\pi}^{4}}{{\pi}^{3} \cdot 0.125} \cdot 0.00390625\right) + \left(\frac{\pi \cdot 0.5}{\pi} + \frac{{\pi}^{2} \cdot 0.0625}{{\pi}^{2} \cdot 0.25}\right)\right)\right)\right)\right)\right)\right)\right)\right) - \left(0.03125 \cdot \frac{{\pi}^{3} \cdot \left(f \cdot -0.25\right)}{{\pi}^{2} \cdot 0.25} + \left(0.013020833333333334 \cdot \frac{f \cdot {\pi}^{3}}{{\pi}^{2} \cdot 0.25} + 0.0625 \cdot \left(\frac{{\pi}^{2}}{{\pi}^{2} \cdot 0.25} + \frac{f \cdot {\pi}^{\color{blue}{\left(\left(\sqrt[3]{4} \cdot \sqrt[3]{4}\right) \cdot \sqrt[3]{4}\right)}}}{{\pi}^{3} \cdot 0.125} \cdot 0.0625\right)\right)\right)\right) \cdot \frac{-4}{\pi}\]

7. Applied pow-unpow_binary64 2.4

   \[\leadsto \log \left(\left(\frac{\pi}{\frac{\pi}{-0.5}} + \left(0.125 \cdot \frac{{\pi}^{3} \cdot \left(f \cdot 0.0625\right)}{{\pi}^{2} \cdot 0.25} + \left(0.5 \cdot \left(\frac{{\pi}^{2}}{\frac{\pi}{0.125}} \cdot f\right) + \left(0.001953125 \cdot \frac{f \cdot {\pi}^{4}}{{\pi}^{3} \cdot 0.125} + \left(\frac{2}{f \cdot \left(\pi \cdot 0.5\right)} + \left(0.03125 \cdot \frac{f \cdot {\pi}^{2}}{\pi \cdot 0.5} + \left(0.8333333333333334 \cdot \frac{f \cdot \left({\pi}^{3} \cdot -0.015625\right)}{{\pi}^{2} \cdot 0.25} + \left(0.5 \cdot \left(\frac{f \cdot {\pi}^{4}}{{\pi}^{3} \cdot 0.125} \cdot 0.00390625\right) + \left(\frac{\pi \cdot 0.5}{\pi} + \frac{{\pi}^{2} \cdot 0.0625}{{\pi}^{2} \cdot 0.25}\right)\right)\right)\right)\right)\right)\right)\right)\right) - \left(0.03125 \cdot \frac{{\pi}^{3} \cdot \left(f \cdot -0.25\right)}{{\pi}^{2} \cdot 0.25} + \left(0.013020833333333334 \cdot \frac{f \cdot {\pi}^{3}}{{\pi}^{2} \cdot 0.25} + 0.0625 \cdot \left(\frac{{\pi}^{2}}{{\pi}^{2} \cdot 0.25} + \frac{f \cdot \color{blue}{{\left({\pi}^{\left(\sqrt[3]{4} \cdot \sqrt[3]{4}\right)}\right)}^{\left(\sqrt[3]{4}\right)}}}{{\pi}^{3} \cdot 0.125} \cdot 0.0625\right)\right)\right)\right) \cdot \frac{-4}{\pi}\]

8. Final simplification 2.4

   \[\leadsto \log \left(\left(\frac{\pi}{\frac{\pi}{-0.5}} + \left(0.125 \cdot \frac{{\pi}^{3} \cdot \left(f \cdot 0.0625\right)}{{\pi}^{2} \cdot 0.25} + \left(0.5 \cdot \left(f \cdot \frac{{\pi}^{2}}{\frac{\pi}{0.125}}\right) + \left(0.001953125 \cdot \frac{f \cdot {\pi}^{4}}{0.125 \cdot {\pi}^{3}} + \left(\frac{2}{f \cdot \left(\pi \cdot 0.5\right)} + \left(0.03125 \cdot \frac{f \cdot {\pi}^{2}}{\pi \cdot 0.5} + \left(0.8333333333333334 \cdot \frac{f \cdot \left({\pi}^{3} \cdot -0.015625\right)}{{\pi}^{2} \cdot 0.25} + \left(0.5 \cdot \left(\frac{f \cdot {\pi}^{4}}{0.125 \cdot {\pi}^{3}} \cdot 0.00390625\right) + \left(\frac{\pi \cdot 0.5}{\pi} + \frac{0.0625 \cdot {\pi}^{2}}{{\pi}^{2} \cdot 0.25}\right)\right)\right)\right)\right)\right)\right)\right)\right) - \left(0.03125 \cdot \frac{{\pi}^{3} \cdot \left(f \cdot -0.25\right)}{{\pi}^{2} \cdot 0.25} + \left(0.013020833333333334 \cdot \frac{{\pi}^{3} \cdot f}{{\pi}^{2} \cdot 0.25} + 0.0625 \cdot \left(\frac{{\pi}^{2}}{{\pi}^{2} \cdot 0.25} + 0.0625 \cdot \frac{f \cdot {\left({\pi}^{\left(\sqrt[3]{4} \cdot \sqrt[3]{4}\right)}\right)}^{\left(\sqrt[3]{4}\right)}}{0.125 \cdot {\pi}^{3}}\right)\right)\right)\right) \cdot \frac{-4}{\pi}\]

Reproduce

herbie shell --seed 2020281 
(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)))))))))