Average Error: 61.6 → 2.2
Time: 16.7s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{\log 4}\\ \mathsf{fma}\left(-4, \frac{\mathsf{fma}\left(t_0 \cdot t_0, t_0, -\log \pi\right) - \log f}{\pi}, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \pi \cdot 2, -2 \cdot \frac{\left(\pi \cdot 2\right) \cdot 0.005208333333333333}{0.5}\right), 0\right)}{\pi}, f \cdot f, 0\right)\right) \end{array} \]
(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))))
   (fma
    -4.0
    (/ (- (fma (* t_0 t_0) t_0 (- (log PI))) (log f)) PI)
    (*
     -2.0
     (fma
      (/
       (fma
        (* PI 0.5)
        (fma
         0.0625
         (* PI 2.0)
         (* -2.0 (/ (* (* PI 2.0) 0.005208333333333333) 0.5)))
        0.0)
       PI)
      (* f f)
      0.0)))))
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));
	return fma(-4.0, ((fma((t_0 * t_0), t_0, -log(((double) M_PI))) - log(f)) / ((double) M_PI)), (-2.0 * fma((fma((((double) M_PI) * 0.5), fma(0.0625, (((double) M_PI) * 2.0), (-2.0 * (((((double) M_PI) * 2.0) * 0.005208333333333333) / 0.5))), 0.0) / ((double) M_PI)), (f * f), 0.0)));
}

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))
	return fma(-4.0, Float64(Float64(fma(Float64(t_0 * t_0), t_0, Float64(-log(pi))) - log(f)) / pi), Float64(-2.0 * fma(Float64(fma(Float64(pi * 0.5), fma(0.0625, Float64(pi * 2.0), Float64(-2.0 * Float64(Float64(Float64(pi * 2.0) * 0.005208333333333333) / 0.5))), 0.0) / pi), Float64(f * f), 0.0)))
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]}, N[(-4.0 * N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0 + (-N[Log[Pi], $MachinePrecision])), $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] + N[(-2.0 * N[(N[(N[(N[(Pi * 0.5), $MachinePrecision] * N[(0.0625 * N[(Pi * 2.0), $MachinePrecision] + N[(-2.0 * N[(N[(N[(Pi * 2.0), $MachinePrecision] * 0.005208333333333333), $MachinePrecision] / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision] / Pi), $MachinePrecision] * N[(f * f), $MachinePrecision] + 0.0), $MachinePrecision]), $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}\\
\mathsf{fma}\left(-4, \frac{\mathsf{fma}\left(t_0 \cdot t_0, t_0, -\log \pi\right) - \log f}{\pi}, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \pi \cdot 2, -2 \cdot \frac{\left(\pi \cdot 2\right) \cdot 0.005208333333333333}{0.5}\right), 0\right)}{\pi}, f \cdot f, 0\right)\right)
\end{array}

Error

Derivation

  1. Initial program 61.6

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

  2. Simplified61.6

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

  3. Taylor expanded in f around 0 2.2

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

  4. Simplified2.2

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

  5. Applied egg-rr2.2

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

  6. Final simplification2.2

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

Reproduce

herbie shell --seed 2022209 
(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)))))))))