Average Error: 61.5 → 2.0
Time: 16.2s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
↓
\[\begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ -\frac{\log \left(\frac{2 \cdot \cosh t_0}{2 \cdot \sinh t_0}\right)}{\frac{\pi}{4}} \end{array} \]
(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 (* (/ PI 4.0) f)))
   (- (/ (log (/ (* 2.0 (cosh t_0)) (* 2.0 (sinh t_0)))) (/ 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) {
	double t_0 = (((double) M_PI) / 4.0) * f;
	return -(log(((2.0 * cosh(t_0)) / (2.0 * sinh(t_0)))) / (((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) {
	double t_0 = (Math.PI / 4.0) * f;
	return -(Math.log(((2.0 * Math.cosh(t_0)) / (2.0 * Math.sinh(t_0)))) / (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):
	t_0 = (math.pi / 4.0) * f
	return -(math.log(((2.0 * math.cosh(t_0)) / (2.0 * math.sinh(t_0)))) / (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)
	t_0 = Float64(Float64(pi / 4.0) * f)
	return Float64(-Float64(log(Float64(Float64(2.0 * cosh(t_0)) / Float64(2.0 * sinh(t_0)))) / Float64(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)
	t_0 = (pi / 4.0) * f;
	tmp = -(log(((2.0 * cosh(t_0)) / (2.0 * sinh(t_0)))) / (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_] := Block[{t$95$0 = N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]}, (-N[(N[Log[N[(N[(2.0 * N[Cosh[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[Sinh[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Pi / 4.0), $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 := \frac{\pi}{4} \cdot f\\
-\frac{\log \left(\frac{2 \cdot \cosh t_0}{2 \cdot \sinh t_0}\right)}{\frac{\pi}{4}}
\end{array}

Error

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Your Program's Arguments

Results

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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. Applied egg-rr2.0

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

  3. Final simplification2.0

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

Reproduce

herbie shell --seed 2022203 
(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)))))))))