Average Error: 61.4 → 2.2
Time: 17.0s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
↓
\[\mathsf{fma}\left(4, \log \left(\sqrt[3]{{\left({f}^{\left(\frac{1}{\pi}\right)}\right)}^{3}}\right) - \frac{\log \left(\frac{4}{\pi}\right)}{\pi}, \left(f \cdot \left(f \cdot \pi\right)\right) \cdot -0.08333333333333333\right) \]
(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
 (fma
  4.0
  (- (log (cbrt (pow (pow f (/ 1.0 PI)) 3.0))) (/ (log (/ 4.0 PI)) PI))
  (* (* f (* f PI)) -0.08333333333333333)))
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 fma(4.0, (log(cbrt(pow(pow(f, (1.0 / ((double) M_PI))), 3.0))) - (log((4.0 / ((double) M_PI))) / ((double) M_PI))), ((f * (f * ((double) M_PI))) * -0.08333333333333333));
}

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)
	return fma(4.0, Float64(log(cbrt(((f ^ Float64(1.0 / pi)) ^ 3.0))) - Float64(log(Float64(4.0 / pi)) / pi)), Float64(Float64(f * Float64(f * pi)) * -0.08333333333333333))
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_] := N[(4.0 * N[(N[Log[N[Power[N[Power[N[Power[f, N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision] - N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] + N[(N[(f * N[(f * Pi), $MachinePrecision]), $MachinePrecision] * -0.08333333333333333), $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)
↓
\mathsf{fma}\left(4, \log \left(\sqrt[3]{{\left({f}^{\left(\frac{1}{\pi}\right)}\right)}^{3}}\right) - \frac{\log \left(\frac{4}{\pi}\right)}{\pi}, \left(f \cdot \left(f \cdot \pi\right)\right) \cdot -0.08333333333333333\right)

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

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

  3. Taylor expanded in f around 0 2.3

    \[\leadsto \log \color{blue}{\left(4 \cdot \frac{1}{f \cdot \pi} + 0.08333333333333333 \cdot \left(f \cdot \pi\right)\right)} \cdot \frac{-4}{\pi} \]

  4. Simplified2.3

    \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{4}{f \cdot \pi}\right)\right)} \cdot \frac{-4}{\pi} \]

  5. Taylor expanded in f around 0 2.3

    \[\leadsto \color{blue}{4 \cdot \frac{\log f}{\pi} - \left(0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right) + 4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi}\right)} \]

  6. Simplified2.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(4, \frac{\log f}{\pi} - \frac{\log \left(\frac{4}{\pi}\right)}{\pi}, \left(f \cdot \left(\pi \cdot f\right)\right) \cdot -0.08333333333333333\right)} \]

  7. Applied egg-rr2.2

    \[\leadsto \mathsf{fma}\left(4, \color{blue}{\log \left({f}^{\left(\frac{1}{\pi}\right)}\right)} - \frac{\log \left(\frac{4}{\pi}\right)}{\pi}, \left(f \cdot \left(\pi \cdot f\right)\right) \cdot -0.08333333333333333\right) \]

  8. Applied egg-rr2.2

    \[\leadsto \mathsf{fma}\left(4, \log \color{blue}{\left(\sqrt[3]{{\left({f}^{\left(\frac{1}{\pi}\right)}\right)}^{3}}\right)} - \frac{\log \left(\frac{4}{\pi}\right)}{\pi}, \left(f \cdot \left(\pi \cdot f\right)\right) \cdot -0.08333333333333333\right) \]

  9. Final simplification2.2

    \[\leadsto \mathsf{fma}\left(4, \log \left(\sqrt[3]{{\left({f}^{\left(\frac{1}{\pi}\right)}\right)}^{3}}\right) - \frac{\log \left(\frac{4}{\pi}\right)}{\pi}, \left(f \cdot \left(f \cdot \pi\right)\right) \cdot -0.08333333333333333\right) \]

Reproduce

herbie shell --seed 2022156 
(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)))))))))