Average Error: 61.7 → 2.2
Time: 14.7s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
↓
\[\log \left(\frac{{\left(e^{\frac{\pi}{4}}\right)}^{f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{\mathsf{fma}\left(0.005208333333333333, {f}^{3} \cdot {\pi}^{3}, \mathsf{fma}\left(0.5, \pi \cdot f, \mathsf{fma}\left(1.6276041666666666 \cdot 10^{-5}, {f}^{5} \cdot {\pi}^{5}, 2.422030009920635 \cdot 10^{-8} \cdot \left({f}^{7} \cdot {\pi}^{7}\right)\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]
(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
   (/
    (+ (pow (exp (/ PI 4.0)) f) (pow (exp -0.25) (* PI f)))
    (fma
     0.005208333333333333
     (* (pow f 3.0) (pow PI 3.0))
     (fma
      0.5
      (* PI f)
      (fma
       1.6276041666666666e-5
       (* (pow f 5.0) (pow PI 5.0))
       (* 2.422030009920635e-8 (* (pow f 7.0) (pow PI 7.0))))))))
  (/ -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(((pow(exp((((double) M_PI) / 4.0)), f) + pow(exp(-0.25), (((double) M_PI) * f))) / fma(0.005208333333333333, (pow(f, 3.0) * pow(((double) M_PI), 3.0)), fma(0.5, (((double) M_PI) * f), fma(1.6276041666666666e-5, (pow(f, 5.0) * pow(((double) M_PI), 5.0)), (2.422030009920635e-8 * (pow(f, 7.0) * pow(((double) M_PI), 7.0)))))))) * (-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(log(Float64(Float64((exp(Float64(pi / 4.0)) ^ f) + (exp(-0.25) ^ Float64(pi * f))) / fma(0.005208333333333333, Float64((f ^ 3.0) * (pi ^ 3.0)), fma(0.5, Float64(pi * f), fma(1.6276041666666666e-5, Float64((f ^ 5.0) * (pi ^ 5.0)), Float64(2.422030009920635e-8 * Float64((f ^ 7.0) * (pi ^ 7.0)))))))) * Float64(-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[Log[N[(N[(N[Power[N[Exp[N[(Pi / 4.0), $MachinePrecision]], $MachinePrecision], f], $MachinePrecision] + N[Power[N[Exp[-0.25], $MachinePrecision], N[(Pi * f), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(0.005208333333333333 * N[(N[Power[f, 3.0], $MachinePrecision] * N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(Pi * f), $MachinePrecision] + N[(1.6276041666666666e-5 * N[(N[Power[f, 5.0], $MachinePrecision] * N[Power[Pi, 5.0], $MachinePrecision]), $MachinePrecision] + N[(2.422030009920635e-8 * N[(N[Power[f, 7.0], $MachinePrecision] * N[Power[Pi, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-4.0 / Pi), $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)
↓
\log \left(\frac{{\left(e^{\frac{\pi}{4}}\right)}^{f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{\mathsf{fma}\left(0.005208333333333333, {f}^{3} \cdot {\pi}^{3}, \mathsf{fma}\left(0.5, \pi \cdot f, \mathsf{fma}\left(1.6276041666666666 \cdot 10^{-5}, {f}^{5} \cdot {\pi}^{5}, 2.422030009920635 \cdot 10^{-8} \cdot \left({f}^{7} \cdot {\pi}^{7}\right)\right)\right)\right)}\right) \cdot \frac{-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 2.2

    \[\leadsto \log \left(\frac{{\left(e^{\frac{\pi}{4}}\right)}^{f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{\color{blue}{0.005208333333333333 \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \left(1.6276041666666666 \cdot 10^{-5} \cdot \left({f}^{5} \cdot {\pi}^{5}\right) + \left(2.422030009920635 \cdot 10^{-8} \cdot \left({f}^{7} \cdot {\pi}^{7}\right) + 0.5 \cdot \left(f \cdot \pi\right)\right)\right)}}\right) \cdot \frac{-4}{\pi} \]

  4. Simplified2.2

    \[\leadsto \log \left(\frac{{\left(e^{\frac{\pi}{4}}\right)}^{f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{\color{blue}{\mathsf{fma}\left(0.005208333333333333, {f}^{3} \cdot {\pi}^{3}, \mathsf{fma}\left(0.5, f \cdot \pi, \mathsf{fma}\left(1.6276041666666666 \cdot 10^{-5}, {f}^{5} \cdot {\pi}^{5}, 2.422030009920635 \cdot 10^{-8} \cdot \left({f}^{7} \cdot {\pi}^{7}\right)\right)\right)\right)}}\right) \cdot \frac{-4}{\pi} \]

  5. Final simplification2.2

    \[\leadsto \log \left(\frac{{\left(e^{\frac{\pi}{4}}\right)}^{f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{\mathsf{fma}\left(0.005208333333333333, {f}^{3} \cdot {\pi}^{3}, \mathsf{fma}\left(0.5, \pi \cdot f, \mathsf{fma}\left(1.6276041666666666 \cdot 10^{-5}, {f}^{5} \cdot {\pi}^{5}, 2.422030009920635 \cdot 10^{-8} \cdot \left({f}^{7} \cdot {\pi}^{7}\right)\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

Reproduce

herbie shell --seed 2022151 
(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)))))))))