Average Error: 59.8 → 1.3
Time: 4.8m
Precision: 64
Internal Precision: 1344
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(-(\left(\left(f \cdot f\right) \cdot \pi\right) \cdot \frac{7}{96} + \left(\frac{3}{2} \cdot f\right))_*\right) + \frac{4}{\pi} \cdot \log_* (1 + (e^{\log f - \log \left(\frac{4}{\pi}\right)} - 1)^*) \le -1.4509068521284864 \cdot 10^{+17}:\\ \;\;\;\;\left(-(\left(\left(f \cdot f\right) \cdot \pi\right) \cdot \frac{7}{96} + \left(\frac{3}{2} \cdot f\right))_*\right) + \left(\frac{4}{\pi} \cdot \left(\log f - \log 4\right) + \frac{4}{\pi} \cdot \log \pi\right)\\ \mathbf{if}\;\left(-(\left(\left(f \cdot f\right) \cdot \pi\right) \cdot \frac{7}{96} + \left(\frac{3}{2} \cdot f\right))_*\right) + \frac{4}{\pi} \cdot \log_* (1 + (e^{\log f - \log \left(\frac{4}{\pi}\right)} - 1)^*) \le -52359817.530938186:\\ \;\;\;\;\frac{\log 1}{\pi} \cdot \left(-4\right)\\ \mathbf{if}\;\left(-(\left(\left(f \cdot f\right) \cdot \pi\right) \cdot \frac{7}{96} + \left(\frac{3}{2} \cdot f\right))_*\right) + \frac{4}{\pi} \cdot \log_* (1 + (e^{\log f - \log \left(\frac{4}{\pi}\right)} - 1)^*) \le -24.727664437970862:\\ \;\;\;\;\log \left(\frac{e^{\left(-f\right) \cdot \frac{\pi}{4}} + e^{\frac{\pi}{\frac{4}{f}}}}{(\pi \cdot \left((\left(\frac{1}{192} \cdot f\right) \cdot \left(\left(f \cdot \pi\right) \cdot \left(f \cdot \pi\right)\right) + \left(f \cdot \frac{1}{2}\right))_*\right) + \left(\left(\left(f \cdot \pi\right) \cdot \left(f \cdot \pi\right)\right) \cdot \frac{1}{16}\right))_*}\right) \cdot \left(-\frac{4}{\pi}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\frac{e^{\left(f + f\right) \cdot \frac{\pi}{4}} - {\left(e^{-f}\right)}^{\left(\frac{\pi}{4} + \frac{\pi}{4}\right)}}{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}\right)\\ \end{array}\]

Error

Bits error versus f

Derivation

  1. Split input into 4 regimes

  2. if (+ (- (fma (* (* f f) PI) 7/96 (* 3/2 f))) (* (/ 4 PI) (log1p (expm1 (- (log f) (log (/ 4 PI))))))) < -1.4509068521284864e+17

    1. Initial program 62.0

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

    2. Taylor expanded around 0 0.4

      \[\leadsto -\color{blue}{\left(\left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + \left(\frac{7}{96} \cdot \left({f}^{2} \cdot \pi\right) + \frac{3}{2} \cdot f\right)\right) - 4 \cdot \frac{\log f}{\pi}\right)}\]

    3. Applied simplify0.4

      \[\leadsto \color{blue}{\left(-(\left(\left(f \cdot f\right) \cdot \pi\right) \cdot \frac{7}{96} + \left(\frac{3}{2} \cdot f\right))_*\right) + \frac{4}{\pi} \cdot \left(\log f - \log \left(\frac{4}{\pi}\right)\right)}\]
    4. Using strategy rm

    5. Applied log-div0.4

      \[\leadsto \left(-(\left(\left(f \cdot f\right) \cdot \pi\right) \cdot \frac{7}{96} + \left(\frac{3}{2} \cdot f\right))_*\right) + \frac{4}{\pi} \cdot \left(\log f - \color{blue}{\left(\log 4 - \log \pi\right)}\right)\]

    6. Applied associate--r-0.5

      \[\leadsto \left(-(\left(\left(f \cdot f\right) \cdot \pi\right) \cdot \frac{7}{96} + \left(\frac{3}{2} \cdot f\right))_*\right) + \frac{4}{\pi} \cdot \color{blue}{\left(\left(\log f - \log 4\right) + \log \pi\right)}\]

    7. Applied distribute-lft-in0.4

      \[\leadsto \left(-(\left(\left(f \cdot f\right) \cdot \pi\right) \cdot \frac{7}{96} + \left(\frac{3}{2} \cdot f\right))_*\right) + \color{blue}{\left(\frac{4}{\pi} \cdot \left(\log f - \log 4\right) + \frac{4}{\pi} \cdot \log \pi\right)}\]

    if -1.4509068521284864e+17 < (+ (- (fma (* (* f f) PI) 7/96 (* 3/2 f))) (* (/ 4 PI) (log1p (expm1 (- (log f) (log (/ 4 PI))))))) < -52359817.530938186

    1. Initial program 63.0

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

    2. Taylor expanded around inf 0

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{1}\]

    3. Applied simplify0

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

    if -52359817.530938186 < (+ (- (fma (* (* f f) PI) 7/96 (* 3/2 f))) (* (/ 4 PI) (log1p (expm1 (- (log f) (log (/ 4 PI))))))) < -24.727664437970862

    1. Initial program 39.3

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

    2. Taylor expanded around 0 22.0

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\frac{1}{16} \cdot \left({f}^{2} \cdot {\pi}^{2}\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{2} \cdot \left(f \cdot \pi\right)\right)}}\right)\]

    3. Applied simplify22.0

      \[\leadsto \color{blue}{\log \left(\frac{e^{\left(-f\right) \cdot \frac{\pi}{4}} + e^{\frac{\pi}{\frac{4}{f}}}}{(\pi \cdot \left((\left(\frac{1}{192} \cdot f\right) \cdot \left(\left(f \cdot \pi\right) \cdot \left(f \cdot \pi\right)\right) + \left(f \cdot \frac{1}{2}\right))_*\right) + \left(\left(\left(f \cdot \pi\right) \cdot \left(f \cdot \pi\right)\right) \cdot \frac{1}{16}\right))_*}\right) \cdot \left(-\frac{4}{\pi}\right)}\]

    if -24.727664437970862 < (+ (- (fma (* (* f f) PI) 7/96 (* 3/2 f))) (* (/ 4 PI) (log1p (expm1 (- (log f) (log (/ 4 PI)))))))

    1. Initial program 8.9

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

    3. Applied flip--9.0

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

    4. Applied simplify8.4

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\frac{\color{blue}{e^{\left(f + f\right) \cdot \frac{\pi}{4}} - {\left(e^{-f}\right)}^{\left(\frac{\pi}{4} + \frac{\pi}{4}\right)}}}{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}\right)\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 4.8m)Debug logProfile

herbie shell --seed 2020178 +o rules:numerics
(FPCore (f)
  :name "VandenBroeck and Keller, Equation (20)"
  (- (* (/ 1 (/ PI 4)) (log (/ (+ (exp (* (/ PI 4) f)) (exp (- (* (/ PI 4) f)))) (- (exp (* (/ PI 4) f)) (exp (- (* (/ PI 4) f)))))))))