Average Error: 61.6 → 2.0
Time: 17.3s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
↓
\[\frac{\log \left(\mathsf{fma}\left({\left(\pi \cdot f\right)}^{3}, -0.00034722222222222224, \mathsf{fma}\left({\left(\pi \cdot f\right)}^{5}, 2.066798941798942 \cdot 10^{-6}, \mathsf{fma}\left(\pi \cdot f, 0.08333333333333333, \frac{4}{\pi \cdot f}\right)\right)\right)\right) \cdot -4}{\pi} \]
(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
     (pow (* PI f) 3.0)
     -0.00034722222222222224
     (fma
      (pow (* PI f) 5.0)
      2.066798941798942e-6
      (fma (* PI f) 0.08333333333333333 (/ 4.0 (* PI 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) {
	return (log(fma(pow((((double) M_PI) * f), 3.0), -0.00034722222222222224, fma(pow((((double) M_PI) * f), 5.0), 2.066798941798942e-6, fma((((double) M_PI) * f), 0.08333333333333333, (4.0 / (((double) M_PI) * 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)
	return Float64(Float64(log(fma((Float64(pi * f) ^ 3.0), -0.00034722222222222224, fma((Float64(pi * f) ^ 5.0), 2.066798941798942e-6, fma(Float64(pi * f), 0.08333333333333333, Float64(4.0 / Float64(pi * f)))))) * -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_] := N[(N[(N[Log[N[(N[Power[N[(Pi * f), $MachinePrecision], 3.0], $MachinePrecision] * -0.00034722222222222224 + N[(N[Power[N[(Pi * f), $MachinePrecision], 5.0], $MachinePrecision] * 2.066798941798942e-6 + N[(N[(Pi * f), $MachinePrecision] * 0.08333333333333333 + N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -4.0), $MachinePrecision] / Pi), $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({\left(\pi \cdot f\right)}^{3}, -0.00034722222222222224, \mathsf{fma}\left({\left(\pi \cdot f\right)}^{5}, 2.066798941798942 \cdot 10^{-6}, \mathsf{fma}\left(\pi \cdot f, 0.08333333333333333, \frac{4}{\pi \cdot f}\right)\right)\right)\right) \cdot -4}{\pi}

Error

Bits error versus f

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

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

  3. Taylor expanded in f around 0 2.1

    \[\leadsto \log \color{blue}{\left(\left(2.066798941798942 \cdot 10^{-6} \cdot \left({f}^{5} \cdot {\pi}^{5}\right) + \left(4 \cdot \frac{1}{f \cdot \pi} + 0.08333333333333333 \cdot \left(f \cdot \pi\right)\right)\right) - 0.00034722222222222224 \cdot \left({f}^{3} \cdot {\pi}^{3}\right)\right)} \cdot \frac{-4}{\pi} \]

  4. Simplified2.1

    \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left({f}^{3} \cdot {\pi}^{3}, -0.00034722222222222224, \mathsf{fma}\left({f}^{5} \cdot {\pi}^{5}, 2.066798941798942 \cdot 10^{-6}, \mathsf{fma}\left(f \cdot \pi, 0.08333333333333333, \frac{4}{f \cdot \pi}\right)\right)\right)\right)} \cdot \frac{-4}{\pi} \]

  5. Applied egg-rr2.0

    \[\leadsto \color{blue}{\frac{\log \left(\mathsf{fma}\left({\left(\pi \cdot f\right)}^{3}, -0.00034722222222222224, \mathsf{fma}\left({\left(\pi \cdot f\right)}^{5}, 2.066798941798942 \cdot 10^{-6}, \mathsf{fma}\left(\pi \cdot f, 0.08333333333333333, \frac{4}{\pi \cdot f}\right)\right)\right)\right) \cdot -4}{\pi}} \]

  6. Final simplification2.0

    \[\leadsto \frac{\log \left(\mathsf{fma}\left({\left(\pi \cdot f\right)}^{3}, -0.00034722222222222224, \mathsf{fma}\left({\left(\pi \cdot f\right)}^{5}, 2.066798941798942 \cdot 10^{-6}, \mathsf{fma}\left(\pi \cdot f, 0.08333333333333333, \frac{4}{\pi \cdot f}\right)\right)\right)\right) \cdot -4}{\pi} \]

Reproduce

herbie shell --seed 2022140 
(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)))))))))