Average Error: 61.7 → 2.1
Time: 14.5s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
↓
\[\frac{\log \left(\mathsf{fma}\left({\left(\pi \cdot f\right)}^{3}, -0.00034722222222222224, \mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 2.066798941798942 \cdot 10^{-6}, \frac{\frac{4}{f}}{\pi}\right)\right)\right)\right) \cdot -4}{\pi} \]
(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
      f
      (* PI 0.08333333333333333)
      (fma
       (pow f 5.0)
       (* (pow PI 5.0) 2.066798941798942e-6)
       (/ (/ 4.0 f) PI)))))
   -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(f, (((double) M_PI) * 0.08333333333333333), fma(pow(f, 5.0), (pow(((double) M_PI), 5.0) * 2.066798941798942e-6), ((4.0 / f) / ((double) M_PI)))))) * -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(f, Float64(pi * 0.08333333333333333), fma((f ^ 5.0), Float64((pi ^ 5.0) * 2.066798941798942e-6), Float64(Float64(4.0 / f) / pi))))) * -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[(f * N[(Pi * 0.08333333333333333), $MachinePrecision] + N[(N[Power[f, 5.0], $MachinePrecision] * N[(N[Power[Pi, 5.0], $MachinePrecision] * 2.066798941798942e-6), $MachinePrecision] + N[(N[(4.0 / f), $MachinePrecision] / Pi), $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(f, \pi \cdot 0.08333333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 2.066798941798942 \cdot 10^{-6}, \frac{\frac{4}{f}}{\pi}\right)\right)\right)\right) \cdot -4}{\pi}

Error

Bits error versus f

Derivation

  1. Initial program 61.7

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

  2. Simplified61.7

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

  3. Taylor expanded in f around 0 2.1

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

  4. Simplified2.1

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

  5. Applied egg-rr2.1

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

  6. Final simplification2.1

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

Reproduce

herbie shell --seed 2022162 
(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)))))))))