Average Error: 61.6 → 2.3
Time: 14.5s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
↓
\[4 \cdot \log \left({f}^{\left(\frac{1}{\pi}\right)}\right) + \left(\left(\pi \cdot {f}^{2}\right) \cdot -0.08333333333333333 + \frac{\log \left(\frac{4}{\pi}\right)}{\pi} \cdot -4\right) \]
(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
 (+
  (* 4.0 (log (pow f (/ 1.0 PI))))
  (+
   (* (* PI (pow f 2.0)) -0.08333333333333333)
   (* (/ (log (/ 4.0 PI)) PI) -4.0))))
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 (4.0 * log(pow(f, (1.0 / ((double) M_PI))))) + (((((double) M_PI) * pow(f, 2.0)) * -0.08333333333333333) + ((log((4.0 / ((double) M_PI))) / ((double) M_PI)) * -4.0));
}

public static double code(double f) {
	return -((1.0 / (Math.PI / 4.0)) * Math.log(((Math.exp(((Math.PI / 4.0) * f)) + Math.exp(-((Math.PI / 4.0) * f))) / (Math.exp(((Math.PI / 4.0) * f)) - Math.exp(-((Math.PI / 4.0) * f))))));
}
↓
public static double code(double f) {
	return (4.0 * Math.log(Math.pow(f, (1.0 / Math.PI)))) + (((Math.PI * Math.pow(f, 2.0)) * -0.08333333333333333) + ((Math.log((4.0 / Math.PI)) / Math.PI) * -4.0));
}

def code(f):
	return -((1.0 / (math.pi / 4.0)) * math.log(((math.exp(((math.pi / 4.0) * f)) + math.exp(-((math.pi / 4.0) * f))) / (math.exp(((math.pi / 4.0) * f)) - math.exp(-((math.pi / 4.0) * f))))))
↓
def code(f):
	return (4.0 * math.log(math.pow(f, (1.0 / math.pi)))) + (((math.pi * math.pow(f, 2.0)) * -0.08333333333333333) + ((math.log((4.0 / math.pi)) / math.pi) * -4.0))

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(4.0 * log((f ^ Float64(1.0 / pi)))) + Float64(Float64(Float64(pi * (f ^ 2.0)) * -0.08333333333333333) + Float64(Float64(log(Float64(4.0 / pi)) / pi) * -4.0)))
end

function tmp = code(f)
	tmp = -((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))))));
end
↓
function tmp = code(f)
	tmp = (4.0 * log((f ^ (1.0 / pi)))) + (((pi * (f ^ 2.0)) * -0.08333333333333333) + ((log((4.0 / pi)) / pi) * -4.0));
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[(4.0 * N[Log[N[Power[f, N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(Pi * N[Power[f, 2.0], $MachinePrecision]), $MachinePrecision] * -0.08333333333333333), $MachinePrecision] + N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * -4.0), $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)
↓
4 \cdot \log \left({f}^{\left(\frac{1}{\pi}\right)}\right) + \left(\left(\pi \cdot {f}^{2}\right) \cdot -0.08333333333333333 + \frac{\log \left(\frac{4}{\pi}\right)}{\pi} \cdot -4\right)

Error

Bits error versus f

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  3. Taylor expanded in f around 0 2.4

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

  4. Simplified2.4

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

  5. Taylor expanded in f around 0 2.4

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

  6. Applied egg-rr2.3

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

  7. Final simplification2.3

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

Reproduce

herbie shell --seed 2022159 
(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)))))))))