Average Error: 61.5 → 2.1
Time: 22.6s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{\log 4}\\ t_1 := \sqrt[3]{\log \pi}\\ t_2 := -t_1\\ t_3 := t_1 \cdot t_1\\ \left(\mathsf{fma}\left(f \cdot f, 0.5 \cdot \mathsf{fma}\left(0.5 \cdot \pi, \mathsf{fma}\left(0.0625, \pi \cdot 2, \frac{0.005208333333333333}{0.5 \cdot \frac{0.5}{\pi}} \cdot -2\right), 0\right), \mathsf{fma}\left(t_0 \cdot t_0, t_0, t_1 \cdot \left(t_1 \cdot t_2\right)\right) + \mathsf{fma}\left(t_2, t_3, t_1 \cdot t_3\right)\right) - \log f\right) \cdot \frac{-4}{\pi} \end{array} \]
(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 (cbrt (log 4.0)))
        (t_1 (cbrt (log PI)))
        (t_2 (- t_1))
        (t_3 (* t_1 t_1)))
   (*
    (-
     (fma
      (* f f)
      (*
       0.5
       (fma
        (* 0.5 PI)
        (fma
         0.0625
         (* PI 2.0)
         (* (/ 0.005208333333333333 (* 0.5 (/ 0.5 PI))) -2.0))
        0.0))
      (+ (fma (* t_0 t_0) t_0 (* t_1 (* t_1 t_2))) (fma t_2 t_3 (* t_1 t_3))))
     (log f))
    (/ -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 = cbrt(log(4.0));
	double t_1 = cbrt(log(((double) M_PI)));
	double t_2 = -t_1;
	double t_3 = t_1 * t_1;
	return (fma((f * f), (0.5 * fma((0.5 * ((double) M_PI)), fma(0.0625, (((double) M_PI) * 2.0), ((0.005208333333333333 / (0.5 * (0.5 / ((double) M_PI)))) * -2.0)), 0.0)), (fma((t_0 * t_0), t_0, (t_1 * (t_1 * t_2))) + fma(t_2, t_3, (t_1 * t_3)))) - log(f)) * (-4.0 / ((double) M_PI));
}

function code(f)
	return Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(exp(Float64(Float64(pi / 4.0) * f)) + exp(Float64(-Float64(Float64(pi / 4.0) * f)))) / Float64(exp(Float64(Float64(pi / 4.0) * f)) - exp(Float64(-Float64(Float64(pi / 4.0) * f))))))))
end
↓
function code(f)
	t_0 = cbrt(log(4.0))
	t_1 = cbrt(log(pi))
	t_2 = Float64(-t_1)
	t_3 = Float64(t_1 * t_1)
	return Float64(Float64(fma(Float64(f * f), Float64(0.5 * fma(Float64(0.5 * pi), fma(0.0625, Float64(pi * 2.0), Float64(Float64(0.005208333333333333 / Float64(0.5 * Float64(0.5 / pi))) * -2.0)), 0.0)), Float64(fma(Float64(t_0 * t_0), t_0, Float64(t_1 * Float64(t_1 * t_2))) + fma(t_2, t_3, Float64(t_1 * t_3)))) - log(f)) * Float64(-4.0 / pi))
end

code[f_] := (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]], $MachinePrecision] + N[Exp[(-N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] / N[(N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]], $MachinePrecision] - N[Exp[(-N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])
↓
code[f_] := Block[{t$95$0 = N[Power[N[Log[4.0], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Log[Pi], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = (-t$95$1)}, Block[{t$95$3 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(N[(N[(f * f), $MachinePrecision] * N[(0.5 * N[(N[(0.5 * Pi), $MachinePrecision] * N[(0.0625 * N[(Pi * 2.0), $MachinePrecision] + N[(N[(0.005208333333333333 / N[(0.5 * N[(0.5 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0 + N[(t$95$1 * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * t$95$3 + N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision]]]]]

-\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 := \sqrt[3]{\log 4}\\
t_1 := \sqrt[3]{\log \pi}\\
t_2 := -t_1\\
t_3 := t_1 \cdot t_1\\
\left(\mathsf{fma}\left(f \cdot f, 0.5 \cdot \mathsf{fma}\left(0.5 \cdot \pi, \mathsf{fma}\left(0.0625, \pi \cdot 2, \frac{0.005208333333333333}{0.5 \cdot \frac{0.5}{\pi}} \cdot -2\right), 0\right), \mathsf{fma}\left(t_0 \cdot t_0, t_0, t_1 \cdot \left(t_1 \cdot t_2\right)\right) + \mathsf{fma}\left(t_2, t_3, t_1 \cdot t_3\right)\right) - \log f\right) \cdot \frac{-4}{\pi}
\end{array}

Error

Derivation

  1. Initial program 61.5

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

  2. Simplified61.5

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

  3. Taylor expanded in f around 0 2.1

    \[\leadsto \color{blue}{\left(0.5 \cdot \left(\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right)\right) \cdot {f}^{2}\right) + \left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + \left(0.5 \cdot \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)\right) + -1 \cdot \log f\right)\right)\right)} \cdot \frac{-4}{\pi} \]

  4. Simplified2.1

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(f \cdot f, 0.5 \cdot \mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, 2 \cdot \pi, \frac{0.005208333333333333}{0.5 \cdot \frac{0.5}{\pi}} \cdot -2\right), 0\right), \log \left(\frac{4}{\pi}\right)\right) - \log f\right)} \cdot \frac{-4}{\pi} \]

  5. Applied egg-rr2.1

    \[\leadsto \left(\mathsf{fma}\left(f \cdot f, 0.5 \cdot \mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, 2 \cdot \pi, \frac{0.005208333333333333}{0.5 \cdot \frac{0.5}{\pi}} \cdot -2\right), 0\right), \color{blue}{\mathsf{fma}\left(\sqrt[3]{\log 4} \cdot \sqrt[3]{\log 4}, \sqrt[3]{\log 4}, -\sqrt[3]{\log \pi} \cdot \left(\sqrt[3]{\log \pi} \cdot \sqrt[3]{\log \pi}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\log \pi}, \sqrt[3]{\log \pi} \cdot \sqrt[3]{\log \pi}, \sqrt[3]{\log \pi} \cdot \left(\sqrt[3]{\log \pi} \cdot \sqrt[3]{\log \pi}\right)\right)}\right) - \log f\right) \cdot \frac{-4}{\pi} \]

  6. Final simplification2.1

    \[\leadsto \left(\mathsf{fma}\left(f \cdot f, 0.5 \cdot \mathsf{fma}\left(0.5 \cdot \pi, \mathsf{fma}\left(0.0625, \pi \cdot 2, \frac{0.005208333333333333}{0.5 \cdot \frac{0.5}{\pi}} \cdot -2\right), 0\right), \mathsf{fma}\left(\sqrt[3]{\log 4} \cdot \sqrt[3]{\log 4}, \sqrt[3]{\log 4}, \sqrt[3]{\log \pi} \cdot \left(\sqrt[3]{\log \pi} \cdot \left(-\sqrt[3]{\log \pi}\right)\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\log \pi}, \sqrt[3]{\log \pi} \cdot \sqrt[3]{\log \pi}, \sqrt[3]{\log \pi} \cdot \left(\sqrt[3]{\log \pi} \cdot \sqrt[3]{\log \pi}\right)\right)\right) - \log f\right) \cdot \frac{-4}{\pi} \]

Reproduce

herbie shell --seed 2022210 
(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)))))))))