Average Error: 61.4 → 0.7
Time: 22.2s
Precision: binary64
Cost: 26112
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
↓
\[\frac{-\log \tanh \left(\frac{f}{\frac{4}{\pi}}\right)}{\pi \cdot -0.25} \]
(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 (tanh (/ f (/ 4.0 PI))))) (* PI -0.25)))
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(tanh((f / (4.0 / ((double) M_PI))))) / (((double) M_PI) * -0.25);
}

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 -Math.log(Math.tanh((f / (4.0 / Math.PI)))) / (Math.PI * -0.25);
}

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 -math.log(math.tanh((f / (4.0 / math.pi)))) / (math.pi * -0.25)

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(tanh(Float64(f / Float64(4.0 / pi))))) / Float64(pi * -0.25))
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 = -log(tanh((f / (4.0 / pi)))) / (pi * -0.25);
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[Tanh[N[(f / N[(4.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]) / N[(Pi * -0.25), $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)
↓
\frac{-\log \tanh \left(\frac{f}{\frac{4}{\pi}}\right)}{\pi \cdot -0.25}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Applied egg-rr2.0

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(1 \cdot \log \left(\frac{2 \cdot \cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{2 \cdot \sinh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}\right)\right)} \]

  3. Applied egg-rr0.7

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

  4. Final simplification0.7

    \[\leadsto \frac{-\log \tanh \left(\frac{f}{\frac{4}{\pi}}\right)}{\pi \cdot -0.25} \]

Alternatives

Alternative 1
Error2.8
Cost19712
\[\frac{-\log \left(\frac{\pi}{\frac{4}{f}}\right)}{\pi \cdot -0.25} \]
Alternative 2
Error2.8
Cost19648
\[\frac{4}{\frac{\pi}{\log \left(\pi \cdot \left(f \cdot 0.25\right)\right)}} \]

Error

Reproduce

herbie shell --seed 2022294 
(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)))))))))