Average Error: 61.5 → 2.3
Time: 14.1s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
↓
\[\frac{\log \left(\frac{{\left(e^{\pi \cdot 0.25}\right)}^{f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{\mathsf{fma}\left(\pi, f \cdot 0.5, \mathsf{fma}\left({\pi}^{3}, 0.005208333333333333 \cdot {f}^{3}, {\pi}^{5} \cdot \left(1.6276041666666666 \cdot 10^{-5} \cdot {f}^{5}\right)\right)\right)}\right) \cdot -4}{\pi} \]
(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
    (/
     (+ (pow (exp (* PI 0.25)) f) (pow (exp -0.25) (* PI f)))
     (fma
      PI
      (* f 0.5)
      (fma
       (pow PI 3.0)
       (* 0.005208333333333333 (pow f 3.0))
       (* (pow PI 5.0) (* 1.6276041666666666e-5 (pow f 5.0)))))))
   -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) * 0.25)), f) + pow(exp(-0.25), (((double) M_PI) * f))) / fma(((double) M_PI), (f * 0.5), fma(pow(((double) M_PI), 3.0), (0.005208333333333333 * pow(f, 3.0)), (pow(((double) M_PI), 5.0) * (1.6276041666666666e-5 * pow(f, 5.0))))))) * -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(Float64(log(Float64(Float64((exp(Float64(pi * 0.25)) ^ f) + (exp(-0.25) ^ Float64(pi * f))) / fma(pi, Float64(f * 0.5), fma((pi ^ 3.0), Float64(0.005208333333333333 * (f ^ 3.0)), Float64((pi ^ 5.0) * Float64(1.6276041666666666e-5 * (f ^ 5.0))))))) * -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[(N[Power[N[Exp[N[(Pi * 0.25), $MachinePrecision]], $MachinePrecision], f], $MachinePrecision] + N[Power[N[Exp[-0.25], $MachinePrecision], N[(Pi * f), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(Pi * N[(f * 0.5), $MachinePrecision] + N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(0.005208333333333333 * N[Power[f, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[Pi, 5.0], $MachinePrecision] * N[(1.6276041666666666e-5 * N[Power[f, 5.0], $MachinePrecision]), $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(\frac{{\left(e^{\pi \cdot 0.25}\right)}^{f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{\mathsf{fma}\left(\pi, f \cdot 0.5, \mathsf{fma}\left({\pi}^{3}, 0.005208333333333333 \cdot {f}^{3}, {\pi}^{5} \cdot \left(1.6276041666666666 \cdot 10^{-5} \cdot {f}^{5}\right)\right)\right)}\right) \cdot -4}{\pi}

Error

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

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

  3. Taylor expanded in f around 0 2.4

    \[\leadsto \log \left(\frac{{\left(e^{\frac{\pi}{4}}\right)}^{f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{\color{blue}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}}\right) \cdot \frac{-4}{\pi} \]

  4. Simplified2.4

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

  5. Applied egg-rr2.3

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

  6. Final simplification2.3

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

Reproduce

herbie shell --seed 2022202 
(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)))))))))