Average Error: 61.4 → 2.5
Time: 16.7s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{\log 4}\\ t_1 := \sqrt[3]{\log \pi}\\ t_2 := t_1 \cdot t_1\\ t_3 := t_1 \cdot t_2\\ 4 \cdot \log \left({f}^{\left(\frac{1}{\pi}\right)}\right) - \left(0.08333333333333333 \cdot \left(\pi \cdot {f}^{2}\right) + 4 \cdot \frac{\mathsf{fma}\left(t_0 \cdot t_0, t_0, -t_3\right) + \mathsf{fma}\left(-t_1, t_2, t_3\right)}{\pi}\right) \end{array} \]
(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)))
        (t_1 (cbrt (log PI)))
        (t_2 (* t_1 t_1))
        (t_3 (* t_1 t_2)))
   (-
    (* 4.0 (log (pow f (/ 1.0 PI))))
    (+
     (* 0.08333333333333333 (* PI (pow f 2.0)))
     (* 4.0 (/ (+ (fma (* t_0 t_0) t_0 (- t_3)) (fma (- t_1) t_2 t_3)) 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) {
	double t_0 = cbrt(log(4.0));
	double t_1 = cbrt(log(((double) M_PI)));
	double t_2 = t_1 * t_1;
	double t_3 = t_1 * t_2;
	return (4.0 * log(pow(f, (1.0 / ((double) M_PI))))) - ((0.08333333333333333 * (((double) M_PI) * pow(f, 2.0))) + (4.0 * ((fma((t_0 * t_0), t_0, -t_3) + fma(-t_1, t_2, t_3)) / ((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)
	t_0 = cbrt(log(4.0))
	t_1 = cbrt(log(pi))
	t_2 = Float64(t_1 * t_1)
	t_3 = Float64(t_1 * t_2)
	return Float64(Float64(4.0 * log((f ^ Float64(1.0 / pi)))) - Float64(Float64(0.08333333333333333 * Float64(pi * (f ^ 2.0))) + Float64(4.0 * Float64(Float64(fma(Float64(t_0 * t_0), t_0, Float64(-t_3)) + fma(Float64(-t_1), t_2, t_3)) / 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_] := Block[{t$95$0 = N[Power[N[Log[4.0], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Log[Pi], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * t$95$2), $MachinePrecision]}, N[(N[(4.0 * N[Log[N[Power[f, N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(0.08333333333333333 * N[(Pi * N[Power[f, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0 + (-t$95$3)), $MachinePrecision] + N[((-t$95$1) * t$95$2 + t$95$3), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $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}\\
t_1 := \sqrt[3]{\log \pi}\\
t_2 := t_1 \cdot t_1\\
t_3 := t_1 \cdot t_2\\
4 \cdot \log \left({f}^{\left(\frac{1}{\pi}\right)}\right) - \left(0.08333333333333333 \cdot \left(\pi \cdot {f}^{2}\right) + 4 \cdot \frac{\mathsf{fma}\left(t_0 \cdot t_0, t_0, -t_3\right) + \mathsf{fma}\left(-t_1, t_2, t_3\right)}{\pi}\right)
\end{array}

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

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

  3. Taylor expanded in f around 0 2.6

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

  4. Simplified2.6

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

  5. Taylor expanded in f around 0 2.6

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

  6. Applied egg-rr2.5

    \[\leadsto 4 \cdot \color{blue}{\log \left({f}^{\left(\frac{1}{\pi}\right)}\right)} - \left(0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right) + 4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi}\right) \]

  7. Applied egg-rr2.5

    \[\leadsto 4 \cdot \log \left({f}^{\left(\frac{1}{\pi}\right)}\right) - \left(0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right) + 4 \cdot \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\log 4} \cdot \sqrt[3]{\log 4}, \sqrt[3]{\log 4}, -\sqrt[3]{\log \pi} \cdot \left(\sqrt[3]{\log \pi} \cdot \sqrt[3]{\log \pi}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\log \pi}, \sqrt[3]{\log \pi} \cdot \sqrt[3]{\log \pi}, \sqrt[3]{\log \pi} \cdot \left(\sqrt[3]{\log \pi} \cdot \sqrt[3]{\log \pi}\right)\right)}}{\pi}\right) \]

  8. Final simplification2.5

    \[\leadsto 4 \cdot \log \left({f}^{\left(\frac{1}{\pi}\right)}\right) - \left(0.08333333333333333 \cdot \left(\pi \cdot {f}^{2}\right) + 4 \cdot \frac{\mathsf{fma}\left(\sqrt[3]{\log 4} \cdot \sqrt[3]{\log 4}, \sqrt[3]{\log 4}, -\sqrt[3]{\log \pi} \cdot \left(\sqrt[3]{\log \pi} \cdot \sqrt[3]{\log \pi}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\log \pi}, \sqrt[3]{\log \pi} \cdot \sqrt[3]{\log \pi}, \sqrt[3]{\log \pi} \cdot \left(\sqrt[3]{\log \pi} \cdot \sqrt[3]{\log \pi}\right)\right)}{\pi}\right) \]

Reproduce

herbie shell --seed 2022131 
(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)))))))))