Average Error: 61.4 → 2.6
Time: 56.9s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
↓
\[\begin{array}{l} t_0 := f \cdot {\pi}^{3}\\ t_1 := f \cdot {\pi}^{4}\\ t_2 := {\left(\pi \cdot 0.5\right)}^{4}\\ t_3 := f \cdot \left({\pi}^{3} \cdot -0.015625\right)\\ t_4 := \left(f \cdot f\right) \cdot {\pi}^{5}\\ t_5 := \frac{\pi}{\pi \cdot 0.5}\\ t_6 := {\left(\pi \cdot 0.5\right)}^{3}\\ t_7 := {\left(\pi \cdot 0.5\right)}^{2}\\ t_8 := f \cdot \left({\pi}^{4} \cdot 0.00390625\right)\\ t_9 := \frac{{\pi}^{2}}{t_7}\\ \log \left(\mathsf{fma}\left(0.16666666666666666, \frac{f \cdot t_3}{\pi \cdot 0.5}, \mathsf{fma}\left(0.001953125, \frac{t_1}{t_6}, \mathsf{fma}\left(0.5833333333333334, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{5} \cdot -0.0009765625\right)}{t_6}, \mathsf{fma}\left(0.125, \frac{f \cdot \left({\pi}^{3} \cdot 0.0625\right)}{t_7}, \mathsf{fma}\left(0.5, \frac{t_8}{t_6}, \mathsf{fma}\left(0.0029296875, \frac{\left(f \cdot f\right) \cdot \left(0.0625 \cdot {\left({\pi}^{\left(\sqrt[3]{6} \cdot \sqrt[3]{6}\right)}\right)}^{\left(\sqrt[3]{6}\right)}\right)}{t_2}, \mathsf{fma}\left(0.5, \frac{f \cdot \left(0.0625 \cdot {\pi}^{2}\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.0026041666666666665, \frac{f \cdot t_0}{\pi \cdot 0.5}, \mathsf{fma}\left(0.0005696614583333334, \frac{t_4}{t_6}, \mathsf{fma}\left(0.0009765625, \frac{t_4 \cdot -0.25}{t_6}, \mathsf{fma}\left(0.041666666666666664, \frac{\left(f \cdot f\right) \cdot \left(-0.015625 \cdot {\left({\left({\pi}^{\left(\sqrt[3]{4} \cdot \sqrt[3]{4}\right)}\right)}^{\left(\sqrt[3]{2}\right)}\right)}^{\left(\sqrt[3]{2}\right)}\right)}{t_7}, \mathsf{fma}\left(0.25, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{6} \cdot 0.000244140625\right)}{t_2}, \mathsf{fma}\left(0.0625, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{5} \cdot 0.00390625\right)}{t_6}, \mathsf{fma}\left(0.5, \frac{f \cdot t_8}{t_7}, \mathsf{fma}\left(0.25, t_5, \frac{2}{f \cdot \left(\pi \cdot 0.5\right)} + \mathsf{fma}\left(0.0625, t_9, \mathsf{fma}\left(0.03125, \frac{f \cdot {\pi}^{2}}{\pi \cdot 0.5}, \mathsf{fma}\left(0.8333333333333334, \frac{t_3}{t_7}, -0.25 \cdot t_5\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \mathsf{fma}\left(0.03125, \frac{f \cdot \left({\pi}^{3} \cdot -0.25\right)}{t_7}, \mathsf{fma}\left(0.0625, \frac{f \cdot \left({\pi}^{4} \cdot 0.0625\right)}{t_6}, \mathsf{fma}\left(0.013020833333333334, \frac{t_0}{t_7}, \mathsf{fma}\left(0.052083333333333336, \frac{\left(f \cdot f\right) \cdot \left(-0.015625 \cdot {\pi}^{5}\right)}{t_6}, \mathsf{fma}\left(0.0625, t_9, \mathsf{fma}\left(0.0026041666666666665, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{4} \cdot -0.25\right)}{t_7}, \mathsf{fma}\left(0.046875, \frac{\left(f \cdot f\right) \cdot \left(0.00390625 \cdot {\pi}^{6}\right)}{t_2}, \mathsf{fma}\left(0.013020833333333334, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{5} \cdot 0.0625\right)}{t_6}, \mathsf{fma}\left(0.001953125, \frac{f \cdot t_1}{t_7}, 6.103515625 \cdot 10^{-5} \cdot \frac{\left(f \cdot f\right) \cdot {\pi}^{6}}{t_2}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot \frac{-4}{\pi} \end{array} \]
-\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}
t_0 := f \cdot {\pi}^{3}\\
t_1 := f \cdot {\pi}^{4}\\
t_2 := {\left(\pi \cdot 0.5\right)}^{4}\\
t_3 := f \cdot \left({\pi}^{3} \cdot -0.015625\right)\\
t_4 := \left(f \cdot f\right) \cdot {\pi}^{5}\\
t_5 := \frac{\pi}{\pi \cdot 0.5}\\
t_6 := {\left(\pi \cdot 0.5\right)}^{3}\\
t_7 := {\left(\pi \cdot 0.5\right)}^{2}\\
t_8 := f \cdot \left({\pi}^{4} \cdot 0.00390625\right)\\
t_9 := \frac{{\pi}^{2}}{t_7}\\
\log \left(\mathsf{fma}\left(0.16666666666666666, \frac{f \cdot t_3}{\pi \cdot 0.5}, \mathsf{fma}\left(0.001953125, \frac{t_1}{t_6}, \mathsf{fma}\left(0.5833333333333334, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{5} \cdot -0.0009765625\right)}{t_6}, \mathsf{fma}\left(0.125, \frac{f \cdot \left({\pi}^{3} \cdot 0.0625\right)}{t_7}, \mathsf{fma}\left(0.5, \frac{t_8}{t_6}, \mathsf{fma}\left(0.0029296875, \frac{\left(f \cdot f\right) \cdot \left(0.0625 \cdot {\left({\pi}^{\left(\sqrt[3]{6} \cdot \sqrt[3]{6}\right)}\right)}^{\left(\sqrt[3]{6}\right)}\right)}{t_2}, \mathsf{fma}\left(0.5, \frac{f \cdot \left(0.0625 \cdot {\pi}^{2}\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.0026041666666666665, \frac{f \cdot t_0}{\pi \cdot 0.5}, \mathsf{fma}\left(0.0005696614583333334, \frac{t_4}{t_6}, \mathsf{fma}\left(0.0009765625, \frac{t_4 \cdot -0.25}{t_6}, \mathsf{fma}\left(0.041666666666666664, \frac{\left(f \cdot f\right) \cdot \left(-0.015625 \cdot {\left({\left({\pi}^{\left(\sqrt[3]{4} \cdot \sqrt[3]{4}\right)}\right)}^{\left(\sqrt[3]{2}\right)}\right)}^{\left(\sqrt[3]{2}\right)}\right)}{t_7}, \mathsf{fma}\left(0.25, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{6} \cdot 0.000244140625\right)}{t_2}, \mathsf{fma}\left(0.0625, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{5} \cdot 0.00390625\right)}{t_6}, \mathsf{fma}\left(0.5, \frac{f \cdot t_8}{t_7}, \mathsf{fma}\left(0.25, t_5, \frac{2}{f \cdot \left(\pi \cdot 0.5\right)} + \mathsf{fma}\left(0.0625, t_9, \mathsf{fma}\left(0.03125, \frac{f \cdot {\pi}^{2}}{\pi \cdot 0.5}, \mathsf{fma}\left(0.8333333333333334, \frac{t_3}{t_7}, -0.25 \cdot t_5\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \mathsf{fma}\left(0.03125, \frac{f \cdot \left({\pi}^{3} \cdot -0.25\right)}{t_7}, \mathsf{fma}\left(0.0625, \frac{f \cdot \left({\pi}^{4} \cdot 0.0625\right)}{t_6}, \mathsf{fma}\left(0.013020833333333334, \frac{t_0}{t_7}, \mathsf{fma}\left(0.052083333333333336, \frac{\left(f \cdot f\right) \cdot \left(-0.015625 \cdot {\pi}^{5}\right)}{t_6}, \mathsf{fma}\left(0.0625, t_9, \mathsf{fma}\left(0.0026041666666666665, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{4} \cdot -0.25\right)}{t_7}, \mathsf{fma}\left(0.046875, \frac{\left(f \cdot f\right) \cdot \left(0.00390625 \cdot {\pi}^{6}\right)}{t_2}, \mathsf{fma}\left(0.013020833333333334, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{5} \cdot 0.0625\right)}{t_6}, \mathsf{fma}\left(0.001953125, \frac{f \cdot t_1}{t_7}, 6.103515625 \cdot 10^{-5} \cdot \frac{\left(f \cdot f\right) \cdot {\pi}^{6}}{t_2}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot \frac{-4}{\pi}
\end{array}

(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
 (let* ((t_0 (* f (pow PI 3.0)))
        (t_1 (* f (pow PI 4.0)))
        (t_2 (pow (* PI 0.5) 4.0))
        (t_3 (* f (* (pow PI 3.0) -0.015625)))
        (t_4 (* (* f f) (pow PI 5.0)))
        (t_5 (/ PI (* PI 0.5)))
        (t_6 (pow (* PI 0.5) 3.0))
        (t_7 (pow (* PI 0.5) 2.0))
        (t_8 (* f (* (pow PI 4.0) 0.00390625)))
        (t_9 (/ (pow PI 2.0) t_7)))
   (*
    (log
     (-
      (fma
       0.16666666666666666
       (/ (* f t_3) (* PI 0.5))
       (fma
        0.001953125
        (/ t_1 t_6)
        (fma
         0.5833333333333334
         (/ (* (* f f) (* (pow PI 5.0) -0.0009765625)) t_6)
         (fma
          0.125
          (/ (* f (* (pow PI 3.0) 0.0625)) t_7)
          (fma
           0.5
           (/ t_8 t_6)
           (fma
            0.0029296875
            (/
             (*
              (* f f)
              (* 0.0625 (pow (pow PI (* (cbrt 6.0) (cbrt 6.0))) (cbrt 6.0))))
             t_2)
            (fma
             0.5
             (/ (* f (* 0.0625 (pow PI 2.0))) (* PI 0.5))
             (fma
              0.0026041666666666665
              (/ (* f t_0) (* PI 0.5))
              (fma
               0.0005696614583333334
               (/ t_4 t_6)
               (fma
                0.0009765625
                (/ (* t_4 -0.25) t_6)
                (fma
                 0.041666666666666664
                 (/
                  (*
                   (* f f)
                   (*
                    -0.015625
                    (pow
                     (pow (pow PI (* (cbrt 4.0) (cbrt 4.0))) (cbrt 2.0))
                     (cbrt 2.0))))
                  t_7)
                 (fma
                  0.25
                  (/ (* (* f f) (* (pow PI 6.0) 0.000244140625)) t_2)
                  (fma
                   0.0625
                   (/ (* (* f f) (* (pow PI 5.0) 0.00390625)) t_6)
                   (fma
                    0.5
                    (/ (* f t_8) t_7)
                    (fma
                     0.25
                     t_5
                     (+
                      (/ 2.0 (* f (* PI 0.5)))
                      (fma
                       0.0625
                       t_9
                       (fma
                        0.03125
                        (/ (* f (pow PI 2.0)) (* PI 0.5))
                        (fma
                         0.8333333333333334
                         (/ t_3 t_7)
                         (* -0.25 t_5))))))))))))))))))))
      (fma
       0.03125
       (/ (* f (* (pow PI 3.0) -0.25)) t_7)
       (fma
        0.0625
        (/ (* f (* (pow PI 4.0) 0.0625)) t_6)
        (fma
         0.013020833333333334
         (/ t_0 t_7)
         (fma
          0.052083333333333336
          (/ (* (* f f) (* -0.015625 (pow PI 5.0))) t_6)
          (fma
           0.0625
           t_9
           (fma
            0.0026041666666666665
            (/ (* (* f f) (* (pow PI 4.0) -0.25)) t_7)
            (fma
             0.046875
             (/ (* (* f f) (* 0.00390625 (pow PI 6.0))) t_2)
             (fma
              0.013020833333333334
              (/ (* (* f f) (* (pow PI 5.0) 0.0625)) t_6)
              (fma
               0.001953125
               (/ (* f t_1) t_7)
               (* 6.103515625e-5 (/ (* (* f f) (pow PI 6.0)) t_2)))))))))))))
    (/ -4.0 PI))))
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 t_0 = f * pow(((double) M_PI), 3.0);
	double t_1 = f * pow(((double) M_PI), 4.0);
	double t_2 = pow((((double) M_PI) * 0.5), 4.0);
	double t_3 = f * (pow(((double) M_PI), 3.0) * -0.015625);
	double t_4 = (f * f) * pow(((double) M_PI), 5.0);
	double t_5 = ((double) M_PI) / (((double) M_PI) * 0.5);
	double t_6 = pow((((double) M_PI) * 0.5), 3.0);
	double t_7 = pow((((double) M_PI) * 0.5), 2.0);
	double t_8 = f * (pow(((double) M_PI), 4.0) * 0.00390625);
	double t_9 = pow(((double) M_PI), 2.0) / t_7;
	return log(fma(0.16666666666666666, ((f * t_3) / (((double) M_PI) * 0.5)), fma(0.001953125, (t_1 / t_6), fma(0.5833333333333334, (((f * f) * (pow(((double) M_PI), 5.0) * -0.0009765625)) / t_6), fma(0.125, ((f * (pow(((double) M_PI), 3.0) * 0.0625)) / t_7), fma(0.5, (t_8 / t_6), fma(0.0029296875, (((f * f) * (0.0625 * pow(pow(((double) M_PI), (cbrt(6.0) * cbrt(6.0))), cbrt(6.0)))) / t_2), fma(0.5, ((f * (0.0625 * pow(((double) M_PI), 2.0))) / (((double) M_PI) * 0.5)), fma(0.0026041666666666665, ((f * t_0) / (((double) M_PI) * 0.5)), fma(0.0005696614583333334, (t_4 / t_6), fma(0.0009765625, ((t_4 * -0.25) / t_6), fma(0.041666666666666664, (((f * f) * (-0.015625 * pow(pow(pow(((double) M_PI), (cbrt(4.0) * cbrt(4.0))), cbrt(2.0)), cbrt(2.0)))) / t_7), fma(0.25, (((f * f) * (pow(((double) M_PI), 6.0) * 0.000244140625)) / t_2), fma(0.0625, (((f * f) * (pow(((double) M_PI), 5.0) * 0.00390625)) / t_6), fma(0.5, ((f * t_8) / t_7), fma(0.25, t_5, ((2.0 / (f * (((double) M_PI) * 0.5))) + fma(0.0625, t_9, fma(0.03125, ((f * pow(((double) M_PI), 2.0)) / (((double) M_PI) * 0.5)), fma(0.8333333333333334, (t_3 / t_7), (-0.25 * t_5)))))))))))))))))))) - fma(0.03125, ((f * (pow(((double) M_PI), 3.0) * -0.25)) / t_7), fma(0.0625, ((f * (pow(((double) M_PI), 4.0) * 0.0625)) / t_6), fma(0.013020833333333334, (t_0 / t_7), fma(0.052083333333333336, (((f * f) * (-0.015625 * pow(((double) M_PI), 5.0))) / t_6), fma(0.0625, t_9, fma(0.0026041666666666665, (((f * f) * (pow(((double) M_PI), 4.0) * -0.25)) / t_7), fma(0.046875, (((f * f) * (0.00390625 * pow(((double) M_PI), 6.0))) / t_2), fma(0.013020833333333334, (((f * f) * (pow(((double) M_PI), 5.0) * 0.0625)) / t_6), fma(0.001953125, ((f * t_1) / t_7), (6.103515625e-5 * (((f * f) * pow(((double) M_PI), 6.0)) / t_2)))))))))))) * (-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. Simplified61.4

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

  3. Taylor expanded in f around 0 2.6

    \[\leadsto \log \color{blue}{\left(\left(0.16666666666666666 \cdot \frac{{f}^{2} \cdot \left({\pi}^{3} \cdot {\log \left(e^{-0.25}\right)}^{3}\right)}{0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)} + \left(0.001953125 \cdot \frac{f \cdot {\pi}^{4}}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{3}} + \left(0.5833333333333334 \cdot \frac{{f}^{2} \cdot \left({\pi}^{5} \cdot {\log \left(e^{-0.25}\right)}^{5}\right)}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{3}} + \left(0.125 \cdot \frac{f \cdot \left({\pi}^{3} \cdot {\log \left(e^{-0.25}\right)}^{2}\right)}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{2}} + \left(0.5 \cdot \frac{f \cdot \left({\pi}^{4} \cdot {\log \left(e^{-0.25}\right)}^{4}\right)}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{3}} + \left(0.0029296875 \cdot \frac{{f}^{2} \cdot \left({\pi}^{6} \cdot {\log \left(e^{-0.25}\right)}^{2}\right)}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{4}} + \left(0.5 \cdot \frac{f \cdot \left({\pi}^{2} \cdot {\log \left(e^{-0.25}\right)}^{2}\right)}{0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)} + \left(0.0026041666666666665 \cdot \frac{{f}^{2} \cdot {\pi}^{3}}{0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)} + \left(0.0005696614583333334 \cdot \frac{{f}^{2} \cdot {\pi}^{5}}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{3}} + \left(0.0009765625 \cdot \frac{{f}^{2} \cdot \left({\pi}^{5} \cdot \log \left(e^{-0.25}\right)\right)}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{3}} + \left(0.041666666666666664 \cdot \frac{{f}^{2} \cdot \left({\pi}^{4} \cdot {\log \left(e^{-0.25}\right)}^{3}\right)}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{2}} + \left(0.25 \cdot \frac{{f}^{2} \cdot \left({\pi}^{6} \cdot {\log \left(e^{-0.25}\right)}^{6}\right)}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{4}} + \left(0.0625 \cdot \frac{{f}^{2} \cdot \left({\pi}^{5} \cdot {\log \left(e^{-0.25}\right)}^{4}\right)}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{3}} + \left(0.5 \cdot \frac{{f}^{2} \cdot \left({\pi}^{4} \cdot {\log \left(e^{-0.25}\right)}^{4}\right)}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{2}} + \left(0.25 \cdot \frac{\pi}{0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)} + \left(2 \cdot \frac{1}{f \cdot \left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)} + \left(\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}} + \left(0.03125 \cdot \frac{f \cdot {\pi}^{2}}{0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)} + \left(0.8333333333333334 \cdot \frac{f \cdot \left({\pi}^{3} \cdot {\log \left(e^{-0.25}\right)}^{3}\right)}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{2}} + \frac{\pi \cdot \log \left(e^{-0.25}\right)}{0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \left(0.03125 \cdot \frac{f \cdot \left({\pi}^{3} \cdot \log \left(e^{-0.25}\right)\right)}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{2}} + \left(0.0625 \cdot \frac{f \cdot \left({\pi}^{4} \cdot {\log \left(e^{-0.25}\right)}^{2}\right)}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{3}} + \left(0.013020833333333334 \cdot \frac{f \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{2}} + \left(0.052083333333333336 \cdot \frac{{f}^{2} \cdot \left({\pi}^{5} \cdot {\log \left(e^{-0.25}\right)}^{3}\right)}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{3}} + \left(0.0625 \cdot \frac{{\pi}^{2}}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{2}} + \left(0.0026041666666666665 \cdot \frac{{f}^{2} \cdot \left({\pi}^{4} \cdot \log \left(e^{-0.25}\right)\right)}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{2}} + \left(0.046875 \cdot \frac{{f}^{2} \cdot \left({\pi}^{6} \cdot {\log \left(e^{-0.25}\right)}^{4}\right)}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{4}} + \left(0.013020833333333334 \cdot \frac{{f}^{2} \cdot \left({\pi}^{5} \cdot {\log \left(e^{-0.25}\right)}^{2}\right)}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{3}} + \left(0.001953125 \cdot \frac{{f}^{2} \cdot {\pi}^{4}}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{2}} + 6.103515625 \cdot 10^{-5} \cdot \frac{{f}^{2} \cdot {\pi}^{6}}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{4}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)} \cdot \frac{-4}{\pi} \]

  4. Simplified2.6

    \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(0.16666666666666666, \frac{f \cdot \left(f \cdot \left({\pi}^{3} \cdot -0.015625\right)\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.001953125, \frac{f \cdot {\pi}^{4}}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.5833333333333334, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{5} \cdot -0.0009765625\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.125, \frac{f \cdot \left(0.0625 \cdot {\pi}^{3}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left({\pi}^{4} \cdot 0.00390625\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0029296875, \frac{\left(f \cdot f\right) \cdot \left(0.0625 \cdot {\pi}^{6}\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left({\pi}^{2} \cdot 0.0625\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.0026041666666666665, \frac{f \cdot \left(f \cdot {\pi}^{3}\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.0005696614583333334, \frac{\left(f \cdot f\right) \cdot {\pi}^{5}}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0009765625, \frac{-0.25 \cdot \left(\left(f \cdot f\right) \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.041666666666666664, \frac{\left(f \cdot f\right) \cdot \left(-0.015625 \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.25, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{6} \cdot 0.000244140625\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.0625, \frac{\left(f \cdot f\right) \cdot \left(0.00390625 \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left(f \cdot \left({\pi}^{4} \cdot 0.00390625\right)\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.25, \frac{\pi}{\pi \cdot 0.5}, \frac{2}{f \cdot \left(\pi \cdot 0.5\right)} + \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.03125, \frac{f \cdot {\pi}^{2}}{\pi \cdot 0.5}, \mathsf{fma}\left(0.8333333333333334, \frac{f \cdot \left({\pi}^{3} \cdot -0.015625\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \frac{\pi}{\pi \cdot 0.5} \cdot -0.25\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \mathsf{fma}\left(0.03125, \frac{f \cdot \left(-0.25 \cdot {\pi}^{3}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.0625, \frac{f \cdot \left(0.0625 \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.013020833333333334, \frac{f \cdot {\pi}^{3}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.052083333333333336, \frac{\left(f \cdot f\right) \cdot \left(-0.015625 \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.0026041666666666665, \frac{\left(f \cdot f\right) \cdot \left(-0.25 \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.046875, \frac{\left(f \cdot f\right) \cdot \left(0.00390625 \cdot {\pi}^{6}\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.013020833333333334, \frac{\left(f \cdot f\right) \cdot \left(0.0625 \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.001953125, \frac{f \cdot \left(f \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, 6.103515625 \cdot 10^{-5} \cdot \frac{\left(f \cdot f\right) \cdot {\pi}^{6}}{{\left(\pi \cdot 0.5\right)}^{4}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)} \cdot \frac{-4}{\pi} \]

  5. Applied add-cube-cbrt_binary642.6

    \[\leadsto \log \left(\mathsf{fma}\left(0.16666666666666666, \frac{f \cdot \left(f \cdot \left({\pi}^{3} \cdot -0.015625\right)\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.001953125, \frac{f \cdot {\pi}^{4}}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.5833333333333334, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{5} \cdot -0.0009765625\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.125, \frac{f \cdot \left(0.0625 \cdot {\pi}^{3}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left({\pi}^{4} \cdot 0.00390625\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0029296875, \frac{\left(f \cdot f\right) \cdot \left(0.0625 \cdot {\pi}^{\color{blue}{\left(\left(\sqrt[3]{6} \cdot \sqrt[3]{6}\right) \cdot \sqrt[3]{6}\right)}}\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left({\pi}^{2} \cdot 0.0625\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.0026041666666666665, \frac{f \cdot \left(f \cdot {\pi}^{3}\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.0005696614583333334, \frac{\left(f \cdot f\right) \cdot {\pi}^{5}}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0009765625, \frac{-0.25 \cdot \left(\left(f \cdot f\right) \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.041666666666666664, \frac{\left(f \cdot f\right) \cdot \left(-0.015625 \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.25, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{6} \cdot 0.000244140625\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.0625, \frac{\left(f \cdot f\right) \cdot \left(0.00390625 \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left(f \cdot \left({\pi}^{4} \cdot 0.00390625\right)\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.25, \frac{\pi}{\pi \cdot 0.5}, \frac{2}{f \cdot \left(\pi \cdot 0.5\right)} + \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.03125, \frac{f \cdot {\pi}^{2}}{\pi \cdot 0.5}, \mathsf{fma}\left(0.8333333333333334, \frac{f \cdot \left({\pi}^{3} \cdot -0.015625\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \frac{\pi}{\pi \cdot 0.5} \cdot -0.25\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \mathsf{fma}\left(0.03125, \frac{f \cdot \left(-0.25 \cdot {\pi}^{3}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.0625, \frac{f \cdot \left(0.0625 \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.013020833333333334, \frac{f \cdot {\pi}^{3}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.052083333333333336, \frac{\left(f \cdot f\right) \cdot \left(-0.015625 \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.0026041666666666665, \frac{\left(f \cdot f\right) \cdot \left(-0.25 \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.046875, \frac{\left(f \cdot f\right) \cdot \left(0.00390625 \cdot {\pi}^{6}\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.013020833333333334, \frac{\left(f \cdot f\right) \cdot \left(0.0625 \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.001953125, \frac{f \cdot \left(f \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, 6.103515625 \cdot 10^{-5} \cdot \frac{\left(f \cdot f\right) \cdot {\pi}^{6}}{{\left(\pi \cdot 0.5\right)}^{4}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot \frac{-4}{\pi} \]

  6. Applied pow-unpow_binary642.6

    \[\leadsto \log \left(\mathsf{fma}\left(0.16666666666666666, \frac{f \cdot \left(f \cdot \left({\pi}^{3} \cdot -0.015625\right)\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.001953125, \frac{f \cdot {\pi}^{4}}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.5833333333333334, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{5} \cdot -0.0009765625\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.125, \frac{f \cdot \left(0.0625 \cdot {\pi}^{3}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left({\pi}^{4} \cdot 0.00390625\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0029296875, \frac{\left(f \cdot f\right) \cdot \left(0.0625 \cdot \color{blue}{{\left({\pi}^{\left(\sqrt[3]{6} \cdot \sqrt[3]{6}\right)}\right)}^{\left(\sqrt[3]{6}\right)}}\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left({\pi}^{2} \cdot 0.0625\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.0026041666666666665, \frac{f \cdot \left(f \cdot {\pi}^{3}\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.0005696614583333334, \frac{\left(f \cdot f\right) \cdot {\pi}^{5}}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0009765625, \frac{-0.25 \cdot \left(\left(f \cdot f\right) \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.041666666666666664, \frac{\left(f \cdot f\right) \cdot \left(-0.015625 \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.25, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{6} \cdot 0.000244140625\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.0625, \frac{\left(f \cdot f\right) \cdot \left(0.00390625 \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left(f \cdot \left({\pi}^{4} \cdot 0.00390625\right)\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.25, \frac{\pi}{\pi \cdot 0.5}, \frac{2}{f \cdot \left(\pi \cdot 0.5\right)} + \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.03125, \frac{f \cdot {\pi}^{2}}{\pi \cdot 0.5}, \mathsf{fma}\left(0.8333333333333334, \frac{f \cdot \left({\pi}^{3} \cdot -0.015625\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \frac{\pi}{\pi \cdot 0.5} \cdot -0.25\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \mathsf{fma}\left(0.03125, \frac{f \cdot \left(-0.25 \cdot {\pi}^{3}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.0625, \frac{f \cdot \left(0.0625 \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.013020833333333334, \frac{f \cdot {\pi}^{3}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.052083333333333336, \frac{\left(f \cdot f\right) \cdot \left(-0.015625 \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.0026041666666666665, \frac{\left(f \cdot f\right) \cdot \left(-0.25 \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.046875, \frac{\left(f \cdot f\right) \cdot \left(0.00390625 \cdot {\pi}^{6}\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.013020833333333334, \frac{\left(f \cdot f\right) \cdot \left(0.0625 \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.001953125, \frac{f \cdot \left(f \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, 6.103515625 \cdot 10^{-5} \cdot \frac{\left(f \cdot f\right) \cdot {\pi}^{6}}{{\left(\pi \cdot 0.5\right)}^{4}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot \frac{-4}{\pi} \]

  7. Applied add-cube-cbrt_binary642.6

    \[\leadsto \log \left(\mathsf{fma}\left(0.16666666666666666, \frac{f \cdot \left(f \cdot \left({\pi}^{3} \cdot -0.015625\right)\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.001953125, \frac{f \cdot {\pi}^{4}}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.5833333333333334, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{5} \cdot -0.0009765625\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.125, \frac{f \cdot \left(0.0625 \cdot {\pi}^{3}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left({\pi}^{4} \cdot 0.00390625\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0029296875, \frac{\left(f \cdot f\right) \cdot \left(0.0625 \cdot {\left({\pi}^{\left(\sqrt[3]{6} \cdot \sqrt[3]{6}\right)}\right)}^{\left(\sqrt[3]{6}\right)}\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left({\pi}^{2} \cdot 0.0625\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.0026041666666666665, \frac{f \cdot \left(f \cdot {\pi}^{3}\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.0005696614583333334, \frac{\left(f \cdot f\right) \cdot {\pi}^{5}}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0009765625, \frac{-0.25 \cdot \left(\left(f \cdot f\right) \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.041666666666666664, \frac{\left(f \cdot f\right) \cdot \left(-0.015625 \cdot {\pi}^{\color{blue}{\left(\left(\sqrt[3]{4} \cdot \sqrt[3]{4}\right) \cdot \sqrt[3]{4}\right)}}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.25, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{6} \cdot 0.000244140625\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.0625, \frac{\left(f \cdot f\right) \cdot \left(0.00390625 \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left(f \cdot \left({\pi}^{4} \cdot 0.00390625\right)\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.25, \frac{\pi}{\pi \cdot 0.5}, \frac{2}{f \cdot \left(\pi \cdot 0.5\right)} + \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.03125, \frac{f \cdot {\pi}^{2}}{\pi \cdot 0.5}, \mathsf{fma}\left(0.8333333333333334, \frac{f \cdot \left({\pi}^{3} \cdot -0.015625\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \frac{\pi}{\pi \cdot 0.5} \cdot -0.25\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \mathsf{fma}\left(0.03125, \frac{f \cdot \left(-0.25 \cdot {\pi}^{3}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.0625, \frac{f \cdot \left(0.0625 \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.013020833333333334, \frac{f \cdot {\pi}^{3}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.052083333333333336, \frac{\left(f \cdot f\right) \cdot \left(-0.015625 \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.0026041666666666665, \frac{\left(f \cdot f\right) \cdot \left(-0.25 \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.046875, \frac{\left(f \cdot f\right) \cdot \left(0.00390625 \cdot {\pi}^{6}\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.013020833333333334, \frac{\left(f \cdot f\right) \cdot \left(0.0625 \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.001953125, \frac{f \cdot \left(f \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, 6.103515625 \cdot 10^{-5} \cdot \frac{\left(f \cdot f\right) \cdot {\pi}^{6}}{{\left(\pi \cdot 0.5\right)}^{4}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot \frac{-4}{\pi} \]

  8. Applied pow-unpow_binary642.6

    \[\leadsto \log \left(\mathsf{fma}\left(0.16666666666666666, \frac{f \cdot \left(f \cdot \left({\pi}^{3} \cdot -0.015625\right)\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.001953125, \frac{f \cdot {\pi}^{4}}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.5833333333333334, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{5} \cdot -0.0009765625\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.125, \frac{f \cdot \left(0.0625 \cdot {\pi}^{3}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left({\pi}^{4} \cdot 0.00390625\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0029296875, \frac{\left(f \cdot f\right) \cdot \left(0.0625 \cdot {\left({\pi}^{\left(\sqrt[3]{6} \cdot \sqrt[3]{6}\right)}\right)}^{\left(\sqrt[3]{6}\right)}\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left({\pi}^{2} \cdot 0.0625\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.0026041666666666665, \frac{f \cdot \left(f \cdot {\pi}^{3}\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.0005696614583333334, \frac{\left(f \cdot f\right) \cdot {\pi}^{5}}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0009765625, \frac{-0.25 \cdot \left(\left(f \cdot f\right) \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.041666666666666664, \frac{\left(f \cdot f\right) \cdot \left(-0.015625 \cdot \color{blue}{{\left({\pi}^{\left(\sqrt[3]{4} \cdot \sqrt[3]{4}\right)}\right)}^{\left(\sqrt[3]{4}\right)}}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.25, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{6} \cdot 0.000244140625\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.0625, \frac{\left(f \cdot f\right) \cdot \left(0.00390625 \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left(f \cdot \left({\pi}^{4} \cdot 0.00390625\right)\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.25, \frac{\pi}{\pi \cdot 0.5}, \frac{2}{f \cdot \left(\pi \cdot 0.5\right)} + \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.03125, \frac{f \cdot {\pi}^{2}}{\pi \cdot 0.5}, \mathsf{fma}\left(0.8333333333333334, \frac{f \cdot \left({\pi}^{3} \cdot -0.015625\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \frac{\pi}{\pi \cdot 0.5} \cdot -0.25\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \mathsf{fma}\left(0.03125, \frac{f \cdot \left(-0.25 \cdot {\pi}^{3}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.0625, \frac{f \cdot \left(0.0625 \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.013020833333333334, \frac{f \cdot {\pi}^{3}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.052083333333333336, \frac{\left(f \cdot f\right) \cdot \left(-0.015625 \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.0026041666666666665, \frac{\left(f \cdot f\right) \cdot \left(-0.25 \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.046875, \frac{\left(f \cdot f\right) \cdot \left(0.00390625 \cdot {\pi}^{6}\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.013020833333333334, \frac{\left(f \cdot f\right) \cdot \left(0.0625 \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.001953125, \frac{f \cdot \left(f \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, 6.103515625 \cdot 10^{-5} \cdot \frac{\left(f \cdot f\right) \cdot {\pi}^{6}}{{\left(\pi \cdot 0.5\right)}^{4}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot \frac{-4}{\pi} \]

  9. Applied add-sqr-sqrt_binary642.6

    \[\leadsto \log \left(\mathsf{fma}\left(0.16666666666666666, \frac{f \cdot \left(f \cdot \left({\pi}^{3} \cdot -0.015625\right)\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.001953125, \frac{f \cdot {\pi}^{4}}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.5833333333333334, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{5} \cdot -0.0009765625\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.125, \frac{f \cdot \left(0.0625 \cdot {\pi}^{3}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left({\pi}^{4} \cdot 0.00390625\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0029296875, \frac{\left(f \cdot f\right) \cdot \left(0.0625 \cdot {\left({\pi}^{\left(\sqrt[3]{6} \cdot \sqrt[3]{6}\right)}\right)}^{\left(\sqrt[3]{6}\right)}\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left({\pi}^{2} \cdot 0.0625\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.0026041666666666665, \frac{f \cdot \left(f \cdot {\pi}^{3}\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.0005696614583333334, \frac{\left(f \cdot f\right) \cdot {\pi}^{5}}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0009765625, \frac{-0.25 \cdot \left(\left(f \cdot f\right) \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.041666666666666664, \frac{\left(f \cdot f\right) \cdot \left(-0.015625 \cdot {\left({\pi}^{\left(\sqrt[3]{4} \cdot \sqrt[3]{4}\right)}\right)}^{\left(\sqrt[3]{\color{blue}{\sqrt{4} \cdot \sqrt{4}}}\right)}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.25, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{6} \cdot 0.000244140625\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.0625, \frac{\left(f \cdot f\right) \cdot \left(0.00390625 \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left(f \cdot \left({\pi}^{4} \cdot 0.00390625\right)\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.25, \frac{\pi}{\pi \cdot 0.5}, \frac{2}{f \cdot \left(\pi \cdot 0.5\right)} + \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.03125, \frac{f \cdot {\pi}^{2}}{\pi \cdot 0.5}, \mathsf{fma}\left(0.8333333333333334, \frac{f \cdot \left({\pi}^{3} \cdot -0.015625\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \frac{\pi}{\pi \cdot 0.5} \cdot -0.25\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \mathsf{fma}\left(0.03125, \frac{f \cdot \left(-0.25 \cdot {\pi}^{3}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.0625, \frac{f \cdot \left(0.0625 \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.013020833333333334, \frac{f \cdot {\pi}^{3}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.052083333333333336, \frac{\left(f \cdot f\right) \cdot \left(-0.015625 \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.0026041666666666665, \frac{\left(f \cdot f\right) \cdot \left(-0.25 \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.046875, \frac{\left(f \cdot f\right) \cdot \left(0.00390625 \cdot {\pi}^{6}\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.013020833333333334, \frac{\left(f \cdot f\right) \cdot \left(0.0625 \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.001953125, \frac{f \cdot \left(f \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, 6.103515625 \cdot 10^{-5} \cdot \frac{\left(f \cdot f\right) \cdot {\pi}^{6}}{{\left(\pi \cdot 0.5\right)}^{4}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot \frac{-4}{\pi} \]

  10. Applied cbrt-prod_binary642.6

    \[\leadsto \log \left(\mathsf{fma}\left(0.16666666666666666, \frac{f \cdot \left(f \cdot \left({\pi}^{3} \cdot -0.015625\right)\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.001953125, \frac{f \cdot {\pi}^{4}}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.5833333333333334, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{5} \cdot -0.0009765625\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.125, \frac{f \cdot \left(0.0625 \cdot {\pi}^{3}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left({\pi}^{4} \cdot 0.00390625\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0029296875, \frac{\left(f \cdot f\right) \cdot \left(0.0625 \cdot {\left({\pi}^{\left(\sqrt[3]{6} \cdot \sqrt[3]{6}\right)}\right)}^{\left(\sqrt[3]{6}\right)}\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left({\pi}^{2} \cdot 0.0625\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.0026041666666666665, \frac{f \cdot \left(f \cdot {\pi}^{3}\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.0005696614583333334, \frac{\left(f \cdot f\right) \cdot {\pi}^{5}}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0009765625, \frac{-0.25 \cdot \left(\left(f \cdot f\right) \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.041666666666666664, \frac{\left(f \cdot f\right) \cdot \left(-0.015625 \cdot {\left({\pi}^{\left(\sqrt[3]{4} \cdot \sqrt[3]{4}\right)}\right)}^{\color{blue}{\left(\sqrt[3]{\sqrt{4}} \cdot \sqrt[3]{\sqrt{4}}\right)}}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.25, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{6} \cdot 0.000244140625\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.0625, \frac{\left(f \cdot f\right) \cdot \left(0.00390625 \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left(f \cdot \left({\pi}^{4} \cdot 0.00390625\right)\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.25, \frac{\pi}{\pi \cdot 0.5}, \frac{2}{f \cdot \left(\pi \cdot 0.5\right)} + \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.03125, \frac{f \cdot {\pi}^{2}}{\pi \cdot 0.5}, \mathsf{fma}\left(0.8333333333333334, \frac{f \cdot \left({\pi}^{3} \cdot -0.015625\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \frac{\pi}{\pi \cdot 0.5} \cdot -0.25\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \mathsf{fma}\left(0.03125, \frac{f \cdot \left(-0.25 \cdot {\pi}^{3}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.0625, \frac{f \cdot \left(0.0625 \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.013020833333333334, \frac{f \cdot {\pi}^{3}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.052083333333333336, \frac{\left(f \cdot f\right) \cdot \left(-0.015625 \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.0026041666666666665, \frac{\left(f \cdot f\right) \cdot \left(-0.25 \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.046875, \frac{\left(f \cdot f\right) \cdot \left(0.00390625 \cdot {\pi}^{6}\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.013020833333333334, \frac{\left(f \cdot f\right) \cdot \left(0.0625 \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.001953125, \frac{f \cdot \left(f \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, 6.103515625 \cdot 10^{-5} \cdot \frac{\left(f \cdot f\right) \cdot {\pi}^{6}}{{\left(\pi \cdot 0.5\right)}^{4}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot \frac{-4}{\pi} \]

  11. Applied pow-unpow_binary642.6

    \[\leadsto \log \left(\mathsf{fma}\left(0.16666666666666666, \frac{f \cdot \left(f \cdot \left({\pi}^{3} \cdot -0.015625\right)\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.001953125, \frac{f \cdot {\pi}^{4}}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.5833333333333334, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{5} \cdot -0.0009765625\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.125, \frac{f \cdot \left(0.0625 \cdot {\pi}^{3}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left({\pi}^{4} \cdot 0.00390625\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0029296875, \frac{\left(f \cdot f\right) \cdot \left(0.0625 \cdot {\left({\pi}^{\left(\sqrt[3]{6} \cdot \sqrt[3]{6}\right)}\right)}^{\left(\sqrt[3]{6}\right)}\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left({\pi}^{2} \cdot 0.0625\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.0026041666666666665, \frac{f \cdot \left(f \cdot {\pi}^{3}\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.0005696614583333334, \frac{\left(f \cdot f\right) \cdot {\pi}^{5}}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0009765625, \frac{-0.25 \cdot \left(\left(f \cdot f\right) \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.041666666666666664, \frac{\left(f \cdot f\right) \cdot \left(-0.015625 \cdot \color{blue}{{\left({\left({\pi}^{\left(\sqrt[3]{4} \cdot \sqrt[3]{4}\right)}\right)}^{\left(\sqrt[3]{\sqrt{4}}\right)}\right)}^{\left(\sqrt[3]{\sqrt{4}}\right)}}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.25, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{6} \cdot 0.000244140625\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.0625, \frac{\left(f \cdot f\right) \cdot \left(0.00390625 \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left(f \cdot \left({\pi}^{4} \cdot 0.00390625\right)\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.25, \frac{\pi}{\pi \cdot 0.5}, \frac{2}{f \cdot \left(\pi \cdot 0.5\right)} + \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.03125, \frac{f \cdot {\pi}^{2}}{\pi \cdot 0.5}, \mathsf{fma}\left(0.8333333333333334, \frac{f \cdot \left({\pi}^{3} \cdot -0.015625\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \frac{\pi}{\pi \cdot 0.5} \cdot -0.25\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \mathsf{fma}\left(0.03125, \frac{f \cdot \left(-0.25 \cdot {\pi}^{3}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.0625, \frac{f \cdot \left(0.0625 \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.013020833333333334, \frac{f \cdot {\pi}^{3}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.052083333333333336, \frac{\left(f \cdot f\right) \cdot \left(-0.015625 \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.0026041666666666665, \frac{\left(f \cdot f\right) \cdot \left(-0.25 \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.046875, \frac{\left(f \cdot f\right) \cdot \left(0.00390625 \cdot {\pi}^{6}\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.013020833333333334, \frac{\left(f \cdot f\right) \cdot \left(0.0625 \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.001953125, \frac{f \cdot \left(f \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, 6.103515625 \cdot 10^{-5} \cdot \frac{\left(f \cdot f\right) \cdot {\pi}^{6}}{{\left(\pi \cdot 0.5\right)}^{4}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot \frac{-4}{\pi} \]

  12. Simplified2.6

    \[\leadsto \log \left(\mathsf{fma}\left(0.16666666666666666, \frac{f \cdot \left(f \cdot \left({\pi}^{3} \cdot -0.015625\right)\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.001953125, \frac{f \cdot {\pi}^{4}}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.5833333333333334, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{5} \cdot -0.0009765625\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.125, \frac{f \cdot \left(0.0625 \cdot {\pi}^{3}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left({\pi}^{4} \cdot 0.00390625\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0029296875, \frac{\left(f \cdot f\right) \cdot \left(0.0625 \cdot {\left({\pi}^{\left(\sqrt[3]{6} \cdot \sqrt[3]{6}\right)}\right)}^{\left(\sqrt[3]{6}\right)}\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left({\pi}^{2} \cdot 0.0625\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.0026041666666666665, \frac{f \cdot \left(f \cdot {\pi}^{3}\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.0005696614583333334, \frac{\left(f \cdot f\right) \cdot {\pi}^{5}}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0009765625, \frac{-0.25 \cdot \left(\left(f \cdot f\right) \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.041666666666666664, \frac{\left(f \cdot f\right) \cdot \left(-0.015625 \cdot {\color{blue}{\left({\left({\pi}^{\left(\sqrt[3]{4} \cdot \sqrt[3]{4}\right)}\right)}^{\left(\sqrt[3]{2}\right)}\right)}}^{\left(\sqrt[3]{\sqrt{4}}\right)}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.25, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{6} \cdot 0.000244140625\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.0625, \frac{\left(f \cdot f\right) \cdot \left(0.00390625 \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left(f \cdot \left({\pi}^{4} \cdot 0.00390625\right)\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.25, \frac{\pi}{\pi \cdot 0.5}, \frac{2}{f \cdot \left(\pi \cdot 0.5\right)} + \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.03125, \frac{f \cdot {\pi}^{2}}{\pi \cdot 0.5}, \mathsf{fma}\left(0.8333333333333334, \frac{f \cdot \left({\pi}^{3} \cdot -0.015625\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \frac{\pi}{\pi \cdot 0.5} \cdot -0.25\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \mathsf{fma}\left(0.03125, \frac{f \cdot \left(-0.25 \cdot {\pi}^{3}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.0625, \frac{f \cdot \left(0.0625 \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.013020833333333334, \frac{f \cdot {\pi}^{3}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.052083333333333336, \frac{\left(f \cdot f\right) \cdot \left(-0.015625 \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.0026041666666666665, \frac{\left(f \cdot f\right) \cdot \left(-0.25 \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.046875, \frac{\left(f \cdot f\right) \cdot \left(0.00390625 \cdot {\pi}^{6}\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.013020833333333334, \frac{\left(f \cdot f\right) \cdot \left(0.0625 \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.001953125, \frac{f \cdot \left(f \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, 6.103515625 \cdot 10^{-5} \cdot \frac{\left(f \cdot f\right) \cdot {\pi}^{6}}{{\left(\pi \cdot 0.5\right)}^{4}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot \frac{-4}{\pi} \]

  13. Final simplification2.6

    \[\leadsto \log \left(\mathsf{fma}\left(0.16666666666666666, \frac{f \cdot \left(f \cdot \left({\pi}^{3} \cdot -0.015625\right)\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.001953125, \frac{f \cdot {\pi}^{4}}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.5833333333333334, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{5} \cdot -0.0009765625\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.125, \frac{f \cdot \left({\pi}^{3} \cdot 0.0625\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left({\pi}^{4} \cdot 0.00390625\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0029296875, \frac{\left(f \cdot f\right) \cdot \left(0.0625 \cdot {\left({\pi}^{\left(\sqrt[3]{6} \cdot \sqrt[3]{6}\right)}\right)}^{\left(\sqrt[3]{6}\right)}\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left(0.0625 \cdot {\pi}^{2}\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.0026041666666666665, \frac{f \cdot \left(f \cdot {\pi}^{3}\right)}{\pi \cdot 0.5}, \mathsf{fma}\left(0.0005696614583333334, \frac{\left(f \cdot f\right) \cdot {\pi}^{5}}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0009765625, \frac{\left(\left(f \cdot f\right) \cdot {\pi}^{5}\right) \cdot -0.25}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.041666666666666664, \frac{\left(f \cdot f\right) \cdot \left(-0.015625 \cdot {\left({\left({\pi}^{\left(\sqrt[3]{4} \cdot \sqrt[3]{4}\right)}\right)}^{\left(\sqrt[3]{2}\right)}\right)}^{\left(\sqrt[3]{2}\right)}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.25, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{6} \cdot 0.000244140625\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.0625, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{5} \cdot 0.00390625\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.5, \frac{f \cdot \left(f \cdot \left({\pi}^{4} \cdot 0.00390625\right)\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.25, \frac{\pi}{\pi \cdot 0.5}, \frac{2}{f \cdot \left(\pi \cdot 0.5\right)} + \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.03125, \frac{f \cdot {\pi}^{2}}{\pi \cdot 0.5}, \mathsf{fma}\left(0.8333333333333334, \frac{f \cdot \left({\pi}^{3} \cdot -0.015625\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, -0.25 \cdot \frac{\pi}{\pi \cdot 0.5}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \mathsf{fma}\left(0.03125, \frac{f \cdot \left({\pi}^{3} \cdot -0.25\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.0625, \frac{f \cdot \left({\pi}^{4} \cdot 0.0625\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.013020833333333334, \frac{f \cdot {\pi}^{3}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.052083333333333336, \frac{\left(f \cdot f\right) \cdot \left(-0.015625 \cdot {\pi}^{5}\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.0026041666666666665, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{4} \cdot -0.25\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \mathsf{fma}\left(0.046875, \frac{\left(f \cdot f\right) \cdot \left(0.00390625 \cdot {\pi}^{6}\right)}{{\left(\pi \cdot 0.5\right)}^{4}}, \mathsf{fma}\left(0.013020833333333334, \frac{\left(f \cdot f\right) \cdot \left({\pi}^{5} \cdot 0.0625\right)}{{\left(\pi \cdot 0.5\right)}^{3}}, \mathsf{fma}\left(0.001953125, \frac{f \cdot \left(f \cdot {\pi}^{4}\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, 6.103515625 \cdot 10^{-5} \cdot \frac{\left(f \cdot f\right) \cdot {\pi}^{6}}{{\left(\pi \cdot 0.5\right)}^{4}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot \frac{-4}{\pi} \]

Reproduce

herbie shell --seed 2021220 
(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)))))))))