Average Error: 61.5 → 2.6
Time: 15.1s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
↓
\[\frac{\log \left(\mathsf{fma}\left(f \cdot \pi, 0.08333333333333333, \frac{4}{f \cdot \pi}\right)\right)}{\frac{\pi}{-4}} \]
(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
 (/ (log (fma (* f PI) 0.08333333333333333 (/ 4.0 (* f PI)))) (/ PI -4.0)))
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) {
	return log(fma((f * ((double) M_PI)), 0.08333333333333333, (4.0 / (f * ((double) M_PI))))) / (((double) M_PI) / -4.0);
}

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

Error

Bits error versus f

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

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

  3. Taylor expanded in f around 0 2.7

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

  4. Simplified2.7

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

  5. Applied egg-rr2.6

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

  6. Final simplification2.6

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

Reproduce

herbie shell --seed 2022134 
(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)))))))))