VandenBroeck and Keller, Equation (20)

Average Error: 61.2 → 2.2

Time: 19.9s

Precision: binary64

Cost: 236864

Math

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

↓

\[\begin{array}{l} t_0 := 0.25 \cdot \pi + 0.25 \cdot \pi\\ t_1 := \frac{\pi}{t_0}\\ t_2 := -0.25 \cdot t_1 + 0.25 \cdot t_1\\ t_3 := {t_0}^{2}\\ t_4 := {\pi}^{3} \cdot -0.0026041666666666665\\ \left(0.5 \cdot \left(\left(-0.25 \cdot \left(t_3 \cdot {t_2}^{2}\right) + t_0 \cdot \left(0.0625 \cdot \frac{{\pi}^{2}}{t_0} + 2 \cdot \frac{t_4 + t_4}{t_3}\right)\right) \cdot {f}^{2}\right) + \left(\left(0.5 \cdot \left(t_0 \cdot \left(t_2 \cdot f\right)\right) - \log f\right) - \log \left(\frac{\pi}{4}\right)\right)\right) \cdot \frac{-4}{\pi} \end{array} \]

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 (+ (* 0.25 PI) (* 0.25 PI)))
        (t_1 (/ PI t_0))
        (t_2 (+ (* -0.25 t_1) (* 0.25 t_1)))
        (t_3 (pow t_0 2.0))
        (t_4 (* (pow PI 3.0) -0.0026041666666666665)))
   (*
    (+
     (*
      0.5
      (*
       (+
        (* -0.25 (* t_3 (pow t_2 2.0)))
        (*
         t_0
         (+ (* 0.0625 (/ (pow PI 2.0) t_0)) (* 2.0 (/ (+ t_4 t_4) t_3)))))
       (pow f 2.0)))
     (- (- (* 0.5 (* t_0 (* t_2 f))) (log f)) (log (/ PI 4.0))))
    (/ -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 = (0.25 * ((double) M_PI)) + (0.25 * ((double) M_PI));
	double t_1 = ((double) M_PI) / t_0;
	double t_2 = (-0.25 * t_1) + (0.25 * t_1);
	double t_3 = pow(t_0, 2.0);
	double t_4 = pow(((double) M_PI), 3.0) * -0.0026041666666666665;
	return ((0.5 * (((-0.25 * (t_3 * pow(t_2, 2.0))) + (t_0 * ((0.0625 * (pow(((double) M_PI), 2.0) / t_0)) + (2.0 * ((t_4 + t_4) / t_3))))) * pow(f, 2.0))) + (((0.5 * (t_0 * (t_2 * f))) - log(f)) - log((((double) M_PI) / 4.0)))) * (-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 = (0.25 * Math.PI) + (0.25 * Math.PI);
	double t_1 = Math.PI / t_0;
	double t_2 = (-0.25 * t_1) + (0.25 * t_1);
	double t_3 = Math.pow(t_0, 2.0);
	double t_4 = Math.pow(Math.PI, 3.0) * -0.0026041666666666665;
	return ((0.5 * (((-0.25 * (t_3 * Math.pow(t_2, 2.0))) + (t_0 * ((0.0625 * (Math.pow(Math.PI, 2.0) / t_0)) + (2.0 * ((t_4 + t_4) / t_3))))) * Math.pow(f, 2.0))) + (((0.5 * (t_0 * (t_2 * f))) - Math.log(f)) - Math.log((Math.PI / 4.0)))) * (-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 = (0.25 * math.pi) + (0.25 * math.pi)
	t_1 = math.pi / t_0
	t_2 = (-0.25 * t_1) + (0.25 * t_1)
	t_3 = math.pow(t_0, 2.0)
	t_4 = math.pow(math.pi, 3.0) * -0.0026041666666666665
	return ((0.5 * (((-0.25 * (t_3 * math.pow(t_2, 2.0))) + (t_0 * ((0.0625 * (math.pow(math.pi, 2.0) / t_0)) + (2.0 * ((t_4 + t_4) / t_3))))) * math.pow(f, 2.0))) + (((0.5 * (t_0 * (t_2 * f))) - math.log(f)) - math.log((math.pi / 4.0)))) * (-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(Float64(0.25 * pi) + Float64(0.25 * pi))
	t_1 = Float64(pi / t_0)
	t_2 = Float64(Float64(-0.25 * t_1) + Float64(0.25 * t_1))
	t_3 = t_0 ^ 2.0
	t_4 = Float64((pi ^ 3.0) * -0.0026041666666666665)
	return Float64(Float64(Float64(0.5 * Float64(Float64(Float64(-0.25 * Float64(t_3 * (t_2 ^ 2.0))) + Float64(t_0 * Float64(Float64(0.0625 * Float64((pi ^ 2.0) / t_0)) + Float64(2.0 * Float64(Float64(t_4 + t_4) / t_3))))) * (f ^ 2.0))) + Float64(Float64(Float64(0.5 * Float64(t_0 * Float64(t_2 * f))) - log(f)) - log(Float64(pi / 4.0)))) * 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 = (0.25 * pi) + (0.25 * pi);
	t_1 = pi / t_0;
	t_2 = (-0.25 * t_1) + (0.25 * t_1);
	t_3 = t_0 ^ 2.0;
	t_4 = (pi ^ 3.0) * -0.0026041666666666665;
	tmp = ((0.5 * (((-0.25 * (t_3 * (t_2 ^ 2.0))) + (t_0 * ((0.0625 * ((pi ^ 2.0) / t_0)) + (2.0 * ((t_4 + t_4) / t_3))))) * (f ^ 2.0))) + (((0.5 * (t_0 * (t_2 * f))) - log(f)) - log((pi / 4.0)))) * (-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[(N[(0.25 * Pi), $MachinePrecision] + N[(0.25 * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-0.25 * t$95$1), $MachinePrecision] + N[(0.25 * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$0, 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[Pi, 3.0], $MachinePrecision] * -0.0026041666666666665), $MachinePrecision]}, N[(N[(N[(0.5 * N[(N[(N[(-0.25 * N[(t$95$3 * N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(N[(0.0625 * N[(N[Power[Pi, 2.0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(t$95$4 + t$95$4), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[f, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.5 * N[(t$95$0 * N[(t$95$2 * f), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] - N[Log[N[(Pi / 4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 61.2

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

2. Simplified 61.2

   \[\leadsto \color{blue}{\log \left(\frac{e^{\frac{\pi \cdot f}{4}} + e^{-0.25 \cdot \left(\pi \cdot f\right)}}{e^{\frac{\pi \cdot f}{4}} - e^{-0.25 \cdot \left(\pi \cdot f\right)}}\right) \cdot \frac{-4}{\pi}} \]
   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 2.2

   \[\leadsto \color{blue}{\left(0.5 \cdot \left(\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right)\right) \cdot {f}^{2}\right) + \left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + \left(0.5 \cdot \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)\right) + -1 \cdot \log f\right)\right)\right)} \cdot \frac{-4}{\pi} \]

4. Applied egg-rr 2.2

   \[\leadsto \left(0.5 \cdot \left(\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right)\right) \cdot {f}^{2}\right) + \left(\color{blue}{\left(0 + \left(-\log \left(\frac{\pi}{4}\right)\right)\right)} + \left(0.5 \cdot \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)\right) + -1 \cdot \log f\right)\right)\right) \cdot \frac{-4}{\pi} \]

5. Simplified 2.2

   \[\leadsto \left(0.5 \cdot \left(\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right)\right) \cdot {f}^{2}\right) + \left(\color{blue}{\left(-\log \left(\frac{\pi}{4}\right)\right)} + \left(0.5 \cdot \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)\right) + -1 \cdot \log f\right)\right)\right) \cdot \frac{-4}{\pi} \]
   Proof
   (neg.f64 (log.f64 (/.f64 (PI.f64) 4))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-lft-identity_binary64 (+.f64 0 (neg.f64 (log.f64 (/.f64 (PI.f64) 4))))): 0 points increase in error, 0 points decrease in error

6. Final simplification 2.2

   \[\leadsto \left(0.5 \cdot \left(\left(-0.25 \cdot \left({\left(0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi + 0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi + 0.25 \cdot \pi}\right)}^{2}\right) + \left(0.25 \cdot \pi + 0.25 \cdot \pi\right) \cdot \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi + 0.25 \cdot \pi} + 2 \cdot \frac{{\pi}^{3} \cdot -0.0026041666666666665 + {\pi}^{3} \cdot -0.0026041666666666665}{{\left(0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2}}\right)\right) \cdot {f}^{2}\right) + \left(\left(0.5 \cdot \left(\left(0.25 \cdot \pi + 0.25 \cdot \pi\right) \cdot \left(\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi + 0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi + 0.25 \cdot \pi}\right) \cdot f\right)\right) - \log f\right) - \log \left(\frac{\pi}{4}\right)\right)\right) \cdot \frac{-4}{\pi} \]

Alternatives

Alternative 1
Error	2.2
Cost	52608

\[\frac{-4}{\pi} \cdot \log \left(\frac{\frac{4}{\pi}}{f} + f \cdot \left(0.0625 \cdot \frac{\pi}{0.5} + \frac{{\pi}^{3} \cdot -0.010416666666666666}{0.25 \cdot {\pi}^{2}}\right)\right) \]

Alternative 2
Error	2.5
Cost	26048

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

Alternative 3
Error	2.6
Cost	19648

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

Alternative 4
Error	60.5
Cost	13120

\[\frac{-16}{{\pi}^{2} \cdot f} \]

Alternative 5
Error	60.9
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022338 
(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)))))))))