VandenBroeck and Keller, Equation (20)

Average Error: 61.3 → 2.4

Time: 22.7s

Precision: binary64

Cost: 150592

Math

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

↓

\[\begin{array}{l} t_0 := 8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\\ t_1 := {\pi}^{3} \cdot -0.0026041666666666665\\ t_2 := {\pi}^{7} \cdot -1.2110150049603175 \cdot 10^{-8}\\ \log \left(\frac{e^{\frac{\pi \cdot f}{4}} + e^{\left(\pi \cdot f\right) \cdot -0.25}}{{f}^{5} \cdot \left(t_0 + t_0\right) - \left(f \cdot \left(\pi \cdot -0.25 + \pi \cdot -0.25\right) + \left({f}^{7} \cdot \left(t_2 + t_2\right) + {f}^{3} \cdot \left(t_1 + t_1\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \end{array} \]

FPCore

(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 (* 8.138020833333333e-6 (pow PI 5.0)))
        (t_1 (* (pow PI 3.0) -0.0026041666666666665))
        (t_2 (* (pow PI 7.0) -1.2110150049603175e-8)))
   (*
    (log
     (/
      (+ (exp (/ (* PI f) 4.0)) (exp (* (* PI f) -0.25)))
      (-
       (* (pow f 5.0) (+ t_0 t_0))
       (+
        (* f (+ (* PI -0.25) (* PI -0.25)))
        (+ (* (pow f 7.0) (+ t_2 t_2)) (* (pow f 3.0) (+ t_1 t_1)))))))
    (/ -4.0 PI))))

C

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 = 8.138020833333333e-6 * pow(((double) M_PI), 5.0);
	double t_1 = pow(((double) M_PI), 3.0) * -0.0026041666666666665;
	double t_2 = pow(((double) M_PI), 7.0) * -1.2110150049603175e-8;
	return log(((exp(((((double) M_PI) * f) / 4.0)) + exp(((((double) M_PI) * f) * -0.25))) / ((pow(f, 5.0) * (t_0 + t_0)) - ((f * ((((double) M_PI) * -0.25) + (((double) M_PI) * -0.25))) + ((pow(f, 7.0) * (t_2 + t_2)) + (pow(f, 3.0) * (t_1 + t_1))))))) * (-4.0 / ((double) M_PI));
}

Java

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 = 8.138020833333333e-6 * Math.pow(Math.PI, 5.0);
	double t_1 = Math.pow(Math.PI, 3.0) * -0.0026041666666666665;
	double t_2 = Math.pow(Math.PI, 7.0) * -1.2110150049603175e-8;
	return Math.log(((Math.exp(((Math.PI * f) / 4.0)) + Math.exp(((Math.PI * f) * -0.25))) / ((Math.pow(f, 5.0) * (t_0 + t_0)) - ((f * ((Math.PI * -0.25) + (Math.PI * -0.25))) + ((Math.pow(f, 7.0) * (t_2 + t_2)) + (Math.pow(f, 3.0) * (t_1 + t_1))))))) * (-4.0 / Math.PI);
}

Python

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 = 8.138020833333333e-6 * math.pow(math.pi, 5.0)
	t_1 = math.pow(math.pi, 3.0) * -0.0026041666666666665
	t_2 = math.pow(math.pi, 7.0) * -1.2110150049603175e-8
	return math.log(((math.exp(((math.pi * f) / 4.0)) + math.exp(((math.pi * f) * -0.25))) / ((math.pow(f, 5.0) * (t_0 + t_0)) - ((f * ((math.pi * -0.25) + (math.pi * -0.25))) + ((math.pow(f, 7.0) * (t_2 + t_2)) + (math.pow(f, 3.0) * (t_1 + t_1))))))) * (-4.0 / math.pi)

Julia

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(8.138020833333333e-6 * (pi ^ 5.0))
	t_1 = Float64((pi ^ 3.0) * -0.0026041666666666665)
	t_2 = Float64((pi ^ 7.0) * -1.2110150049603175e-8)
	return Float64(log(Float64(Float64(exp(Float64(Float64(pi * f) / 4.0)) + exp(Float64(Float64(pi * f) * -0.25))) / Float64(Float64((f ^ 5.0) * Float64(t_0 + t_0)) - Float64(Float64(f * Float64(Float64(pi * -0.25) + Float64(pi * -0.25))) + Float64(Float64((f ^ 7.0) * Float64(t_2 + t_2)) + Float64((f ^ 3.0) * Float64(t_1 + t_1))))))) * Float64(-4.0 / pi))
end

MATLAB

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 = 8.138020833333333e-6 * (pi ^ 5.0);
	t_1 = (pi ^ 3.0) * -0.0026041666666666665;
	t_2 = (pi ^ 7.0) * -1.2110150049603175e-8;
	tmp = log(((exp(((pi * f) / 4.0)) + exp(((pi * f) * -0.25))) / (((f ^ 5.0) * (t_0 + t_0)) - ((f * ((pi * -0.25) + (pi * -0.25))) + (((f ^ 7.0) * (t_2 + t_2)) + ((f ^ 3.0) * (t_1 + t_1))))))) * (-4.0 / pi);
end

Wolfram

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[(8.138020833333333e-6 * N[Power[Pi, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[Pi, 3.0], $MachinePrecision] * -0.0026041666666666665), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[Pi, 7.0], $MachinePrecision] * -1.2110150049603175e-8), $MachinePrecision]}, N[(N[Log[N[(N[(N[Exp[N[(N[(Pi * f), $MachinePrecision] / 4.0), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(N[(Pi * f), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[f, 5.0], $MachinePrecision] * N[(t$95$0 + t$95$0), $MachinePrecision]), $MachinePrecision] - N[(N[(f * N[(N[(Pi * -0.25), $MachinePrecision] + N[(Pi * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[f, 7.0], $MachinePrecision] * N[(t$95$2 + t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[Power[f, 3.0], $MachinePrecision] * N[(t$95$1 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 61.3

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

2. Simplified 61.3

   \[\leadsto \color{blue}{\log \left(\frac{e^{\frac{\pi \cdot f}{4}} + e^{-0.25 \cdot \left(\pi \cdot f\right)}}{e^{\frac{\pi \cdot f}{4}} - e^{-0.25 \cdot \left(\pi \cdot f\right)}}\right) \cdot \frac{-4}{\pi}} \]
   Proof
   (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (/.f64 (*.f64 (PI.f64) f) 4)) (exp.f64 (*.f64 -1/4 (*.f64 (PI.f64) f)))) (-.f64 (exp.f64 (/.f64 (*.f64 (PI.f64) f) 4)) (exp.f64 (*.f64 -1/4 (*.f64 (PI.f64) f)))))) (/.f64 -4 (PI.f64))): 0 points increase in error, 0 points decrease in error
   (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (PI.f64) 4) f))) (exp.f64 (*.f64 -1/4 (*.f64 (PI.f64) f)))) (-.f64 (exp.f64 (/.f64 (*.f64 (PI.f64) f) 4)) (exp.f64 (*.f64 -1/4 (*.f64 (PI.f64) f)))))) (/.f64 -4 (PI.f64))): 0 points increase in error, 0 points decrease in error
   (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (*.f64 (Rewrite<= metadata-eval (/.f64 -1 4)) (*.f64 (PI.f64) f)))) (-.f64 (exp.f64 (/.f64 (*.f64 (PI.f64) f) 4)) (exp.f64 (*.f64 -1/4 (*.f64 (PI.f64) f)))))) (/.f64 -4 (PI.f64))): 0 points increase in error, 0 points decrease in error
   (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (Rewrite<= associate-/r/_binary64 (/.f64 -1 (/.f64 4 (*.f64 (PI.f64) f)))))) (-.f64 (exp.f64 (/.f64 (*.f64 (PI.f64) f) 4)) (exp.f64 (*.f64 -1/4 (*.f64 (PI.f64) f)))))) (/.f64 -4 (PI.f64))): 0 points increase in error, 0 points decrease in error
   (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 -1 (*.f64 (PI.f64) f)) 4)))) (-.f64 (exp.f64 (/.f64 (*.f64 (PI.f64) f) 4)) (exp.f64 (*.f64 -1/4 (*.f64 (PI.f64) f)))))) (/.f64 -4 (PI.f64))): 0 points increase in error, 0 points decrease in error
   (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (*.f64 (PI.f64) f) 4))))) (-.f64 (exp.f64 (/.f64 (*.f64 (PI.f64) f) 4)) (exp.f64 (*.f64 -1/4 (*.f64 (PI.f64) f)))))) (/.f64 -4 (PI.f64))): 0 points increase in error, 0 points decrease in error
   (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (*.f64 -1 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (PI.f64) 4) f))))) (-.f64 (exp.f64 (/.f64 (*.f64 (PI.f64) f) 4)) (exp.f64 (*.f64 -1/4 (*.f64 (PI.f64) f)))))) (/.f64 -4 (PI.f64))): 0 points increase in error, 0 points decrease in error
   (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f))))) (-.f64 (exp.f64 (/.f64 (*.f64 (PI.f64) f) 4)) (exp.f64 (*.f64 -1/4 (*.f64 (PI.f64) f)))))) (/.f64 -4 (PI.f64))): 0 points increase in error, 0 points decrease in error
   (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (exp.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (PI.f64) 4) f))) (exp.f64 (*.f64 -1/4 (*.f64 (PI.f64) f)))))) (/.f64 -4 (PI.f64))): 0 points increase in error, 0 points decrease in error
   (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (*.f64 (Rewrite<= metadata-eval (/.f64 -1 4)) (*.f64 (PI.f64) f)))))) (/.f64 -4 (PI.f64))): 0 points increase in error, 0 points decrease in error
   (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (Rewrite<= associate-/r/_binary64 (/.f64 -1 (/.f64 4 (*.f64 (PI.f64) f)))))))) (/.f64 -4 (PI.f64))): 0 points increase in error, 0 points decrease in error
   (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 -1 (*.f64 (PI.f64) f)) 4)))))) (/.f64 -4 (PI.f64))): 0 points increase in error, 0 points decrease in error
   (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (*.f64 (PI.f64) f) 4))))))) (/.f64 -4 (PI.f64))): 0 points increase in error, 0 points decrease in error
   (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (*.f64 -1 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (PI.f64) 4) f))))))) (/.f64 -4 (PI.f64))): 0 points increase in error, 0 points decrease in error
   (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f))))))) (/.f64 -4 (PI.f64))): 0 points increase in error, 0 points decrease in error
   (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))))) (/.f64 (Rewrite<= metadata-eval (neg.f64 4)) (PI.f64))): 0 points increase in error, 0 points decrease in error
   (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))))) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 4 (PI.f64))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))))) (neg.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 1 4)) (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))))) (neg.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (PI.f64)) 4)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))))) (neg.f64 (Rewrite<= associate-/r/_binary64 (/.f64 1 (/.f64 (PI.f64) 4))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (/.f64 1 (/.f64 (PI.f64) 4))) (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 (/.f64 1 (/.f64 (PI.f64) 4)) (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f))))))))): 0 points increase in error, 0 points decrease in error

3. Taylor expanded in f around 0 2.4

   \[\leadsto \log \left(\frac{e^{\frac{\pi \cdot f}{4}} + e^{-0.25 \cdot \left(\pi \cdot f\right)}}{\color{blue}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}}\right) \cdot \frac{-4}{\pi} \]

4. Final simplification 2.4

   \[\leadsto \log \left(\frac{e^{\frac{\pi \cdot f}{4}} + e^{\left(\pi \cdot f\right) \cdot -0.25}}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} + 8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) - \left(f \cdot \left(\pi \cdot -0.25 + \pi \cdot -0.25\right) + \left({f}^{7} \cdot \left({\pi}^{7} \cdot -1.2110150049603175 \cdot 10^{-8} + {\pi}^{7} \cdot -1.2110150049603175 \cdot 10^{-8}\right) + {f}^{3} \cdot \left({\pi}^{3} \cdot -0.0026041666666666665 + {\pi}^{3} \cdot -0.0026041666666666665\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

Alternatives

Alternative 1
Error	2.4
Cost	117888

\[\begin{array}{l} t_0 := {\pi}^{3} \cdot -0.0026041666666666665\\ t_1 := 8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\\ \frac{-4}{\pi} \cdot \log \left(\frac{e^{\frac{\pi \cdot f}{4}} + e^{\left(\pi \cdot f\right) \cdot -0.25}}{{f}^{5} \cdot \left(t_1 + t_1\right) + \left(f \cdot \left(\pi \cdot 0.25 + \pi \cdot 0.25\right) - {f}^{3} \cdot \left(t_0 + t_0\right)\right)}\right) \end{array} \]

Alternative 2
Error	2.6
Cost	85184

\[\begin{array}{l} t_0 := {\pi}^{3} \cdot -0.0026041666666666665\\ \frac{-4}{\pi} \cdot \log \left(\frac{e^{\frac{\pi \cdot f}{4}} + e^{\left(\pi \cdot f\right) \cdot -0.25}}{f \cdot \left(\pi \cdot 0.25 + \pi \cdot 0.25\right) - {f}^{3} \cdot \left(t_0 + t_0\right)}\right) \end{array} \]

Alternative 3
Error	2.9
Cost	32576

\[\frac{\log 2 - \log \left(\left(\pi \cdot f\right) \cdot 0.5\right)}{\frac{\pi}{\sqrt[3]{-64}}} \]

Alternative 4
Error	2.8
Cost	32576

\[-4 \cdot \frac{-3 \cdot \log \left(\sqrt[3]{\frac{\pi}{4}}\right) - \log f}{\pi} \]

Alternative 5
Error	2.8
Cost	19648

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

Alternative 6
Error	54.7
Cost	19392

\[\sqrt[3]{\frac{-64}{{\pi}^{-1}}} \]

Alternative 7
Error	55.2
Cost	12928

\[\sqrt[3]{\frac{-4}{\pi}} \]

Alternative 8
Error	63.0
Cost	6656

\[\frac{\frac{-1}{\pi}}{-0.25} \]

Reproduce

herbie shell --seed 2022296 
(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)))))))))