VandenBroeck and Keller, Equation (20)

Average Error: 61.4 → 1.0

Time: 26.8s

Precision: binary64

Cost: 203652

Math

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

↓

\[\begin{array}{l} t_0 := \pi \cdot 0.25 + \pi \cdot 0.25\\ t_1 := \frac{\pi}{t_0}\\ t_2 := -0.25 \cdot t_1 + 0.25 \cdot t_1\\ \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 100:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({t_0}^{2} \cdot {t_2}^{2}\right) + \frac{\pi \cdot \left({\pi}^{4} \cdot 0.041666666666666664\right)}{{\pi}^{3}}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{t_0 \cdot \left(f \cdot t_2\right)}{\pi}\right)\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \left(4 \cdot \log \left(\pi \cdot f\right)\right)\\ \end{array} \]

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 (+ (* PI 0.25) (* PI 0.25)))
        (t_1 (/ PI t_0))
        (t_2 (+ (* -0.25 t_1) (* 0.25 t_1))))
   (if (<= (* (/ PI 4.0) f) 100.0)
     (+
      (* -4.0 (/ (- (log (/ 4.0 PI)) (log f)) PI))
      (+
       (*
        -2.0
        (/
         (*
          (+
           (* -0.25 (* (pow t_0 2.0) (pow t_2 2.0)))
           (/ (* PI (* (pow PI 4.0) 0.041666666666666664)) (pow PI 3.0)))
          (pow f 2.0))
         PI))
       (* -2.0 (/ (* t_0 (* f t_2)) PI))))
     (* 0.0 (* 4.0 (log (* PI f)))))))

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 = (((double) M_PI) * 0.25) + (((double) M_PI) * 0.25);
	double t_1 = ((double) M_PI) / t_0;
	double t_2 = (-0.25 * t_1) + (0.25 * t_1);
	double tmp;
	if (((((double) M_PI) / 4.0) * f) <= 100.0) {
		tmp = (-4.0 * ((log((4.0 / ((double) M_PI))) - log(f)) / ((double) M_PI))) + ((-2.0 * ((((-0.25 * (pow(t_0, 2.0) * pow(t_2, 2.0))) + ((((double) M_PI) * (pow(((double) M_PI), 4.0) * 0.041666666666666664)) / pow(((double) M_PI), 3.0))) * pow(f, 2.0)) / ((double) M_PI))) + (-2.0 * ((t_0 * (f * t_2)) / ((double) M_PI))));
	} else {
		tmp = 0.0 * (4.0 * log((((double) M_PI) * f)));
	}
	return tmp;
}

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 = (Math.PI * 0.25) + (Math.PI * 0.25);
	double t_1 = Math.PI / t_0;
	double t_2 = (-0.25 * t_1) + (0.25 * t_1);
	double tmp;
	if (((Math.PI / 4.0) * f) <= 100.0) {
		tmp = (-4.0 * ((Math.log((4.0 / Math.PI)) - Math.log(f)) / Math.PI)) + ((-2.0 * ((((-0.25 * (Math.pow(t_0, 2.0) * Math.pow(t_2, 2.0))) + ((Math.PI * (Math.pow(Math.PI, 4.0) * 0.041666666666666664)) / Math.pow(Math.PI, 3.0))) * Math.pow(f, 2.0)) / Math.PI)) + (-2.0 * ((t_0 * (f * t_2)) / Math.PI)));
	} else {
		tmp = 0.0 * (4.0 * Math.log((Math.PI * f)));
	}
	return tmp;
}

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 = (math.pi * 0.25) + (math.pi * 0.25)
	t_1 = math.pi / t_0
	t_2 = (-0.25 * t_1) + (0.25 * t_1)
	tmp = 0
	if ((math.pi / 4.0) * f) <= 100.0:
		tmp = (-4.0 * ((math.log((4.0 / math.pi)) - math.log(f)) / math.pi)) + ((-2.0 * ((((-0.25 * (math.pow(t_0, 2.0) * math.pow(t_2, 2.0))) + ((math.pi * (math.pow(math.pi, 4.0) * 0.041666666666666664)) / math.pow(math.pi, 3.0))) * math.pow(f, 2.0)) / math.pi)) + (-2.0 * ((t_0 * (f * t_2)) / math.pi)))
	else:
		tmp = 0.0 * (4.0 * math.log((math.pi * f)))
	return tmp

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(pi * 0.25) + Float64(pi * 0.25))
	t_1 = Float64(pi / t_0)
	t_2 = Float64(Float64(-0.25 * t_1) + Float64(0.25 * t_1))
	tmp = 0.0
	if (Float64(Float64(pi / 4.0) * f) <= 100.0)
		tmp = Float64(Float64(-4.0 * Float64(Float64(log(Float64(4.0 / pi)) - log(f)) / pi)) + Float64(Float64(-2.0 * Float64(Float64(Float64(Float64(-0.25 * Float64((t_0 ^ 2.0) * (t_2 ^ 2.0))) + Float64(Float64(pi * Float64((pi ^ 4.0) * 0.041666666666666664)) / (pi ^ 3.0))) * (f ^ 2.0)) / pi)) + Float64(-2.0 * Float64(Float64(t_0 * Float64(f * t_2)) / pi))));
	else
		tmp = Float64(0.0 * Float64(4.0 * log(Float64(pi * f))));
	end
	return tmp
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_2 = code(f)
	t_0 = (pi * 0.25) + (pi * 0.25);
	t_1 = pi / t_0;
	t_2 = (-0.25 * t_1) + (0.25 * t_1);
	tmp = 0.0;
	if (((pi / 4.0) * f) <= 100.0)
		tmp = (-4.0 * ((log((4.0 / pi)) - log(f)) / pi)) + ((-2.0 * ((((-0.25 * ((t_0 ^ 2.0) * (t_2 ^ 2.0))) + ((pi * ((pi ^ 4.0) * 0.041666666666666664)) / (pi ^ 3.0))) * (f ^ 2.0)) / pi)) + (-2.0 * ((t_0 * (f * t_2)) / pi)));
	else
		tmp = 0.0 * (4.0 * log((pi * f)));
	end
	tmp_2 = tmp;
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[(Pi * 0.25), $MachinePrecision] + N[(Pi * 0.25), $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]}, If[LessEqual[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision], 100.0], N[(N[(-4.0 * N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(N[(N[(N[(-0.25 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(Pi * N[(N[Power[Pi, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] / N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[f, 2.0], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(t$95$0 * N[(f * t$95$2), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 * N[(4.0 * N[Log[N[(Pi * f), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (PI.f64) 4) f) < 100

   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. Simplified 61.5

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

      [Start] 61.5	\[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
      *-commutative [=>] 61.5	\[ -\color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \frac{1}{\frac{\pi}{4}}} \]
      distribute-rgt-neg-in [=>] 61.5	\[ \color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \left(-\frac{1}{\frac{\pi}{4}}\right)} \]

   3. Taylor expanded in f around 0 1.0

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

   4. Applied egg-rr 1.0

      \[\leadsto -4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \color{blue}{\frac{\left(0.0625 \cdot {\pi}^{2}\right) \cdot \left({\pi}^{2} \cdot 0.25\right) - \left(\pi \cdot 0.5\right) \cdot \left(2 \cdot \left({\pi}^{3} \cdot 0.005208333333333333\right)\right)}{\left(\pi \cdot 0.5\right) \cdot \left({\pi}^{2} \cdot 0.25\right)}}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]

   5. Simplified 1.0

      \[\leadsto -4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \color{blue}{\frac{0.0625 \cdot \left({\pi}^{4} \cdot 0.25\right) - \pi \cdot \left({\pi}^{3} \cdot 0.005208333333333333\right)}{0.25 \cdot \left(0.5 \cdot {\pi}^{3}\right)}}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]
      Proof

      [Start] 1.0	\[ -4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \frac{\left(0.0625 \cdot {\pi}^{2}\right) \cdot \left({\pi}^{2} \cdot 0.25\right) - \left(\pi \cdot 0.5\right) \cdot \left(2 \cdot \left({\pi}^{3} \cdot 0.005208333333333333\right)\right)}{\left(\pi \cdot 0.5\right) \cdot \left({\pi}^{2} \cdot 0.25\right)}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]
      associate-*l* [=>] 1.0	\[ -4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \frac{\left(0.0625 \cdot {\pi}^{2}\right) \cdot \left({\pi}^{2} \cdot 0.25\right) - \color{blue}{\pi \cdot \left(0.5 \cdot \left(2 \cdot \left({\pi}^{3} \cdot 0.005208333333333333\right)\right)\right)}}{\left(\pi \cdot 0.5\right) \cdot \left({\pi}^{2} \cdot 0.25\right)}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]
      associate-*r* [=>] 1.0	\[ -4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \frac{\left(0.0625 \cdot {\pi}^{2}\right) \cdot \left({\pi}^{2} \cdot 0.25\right) - \pi \cdot \color{blue}{\left(\left(0.5 \cdot 2\right) \cdot \left({\pi}^{3} \cdot 0.005208333333333333\right)\right)}}{\left(\pi \cdot 0.5\right) \cdot \left({\pi}^{2} \cdot 0.25\right)}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]
      metadata-eval [=>] 1.0	\[ -4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \frac{\left(0.0625 \cdot {\pi}^{2}\right) \cdot \left({\pi}^{2} \cdot 0.25\right) - \pi \cdot \left(\color{blue}{1} \cdot \left({\pi}^{3} \cdot 0.005208333333333333\right)\right)}{\left(\pi \cdot 0.5\right) \cdot \left({\pi}^{2} \cdot 0.25\right)}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]
      associate-*l* [<=] 1.0	\[ -4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \frac{\left(0.0625 \cdot {\pi}^{2}\right) \cdot \left({\pi}^{2} \cdot 0.25\right) - \pi \cdot \color{blue}{\left(\left(1 \cdot {\pi}^{3}\right) \cdot 0.005208333333333333\right)}}{\left(\pi \cdot 0.5\right) \cdot \left({\pi}^{2} \cdot 0.25\right)}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]
      *-lft-identity [=>] 1.0	\[ -4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \frac{\left(0.0625 \cdot {\pi}^{2}\right) \cdot \left({\pi}^{2} \cdot 0.25\right) - \pi \cdot \left(\color{blue}{{\pi}^{3}} \cdot 0.005208333333333333\right)}{\left(\pi \cdot 0.5\right) \cdot \left({\pi}^{2} \cdot 0.25\right)}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]
      associate-*l* [=>] 1.0	\[ -4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \frac{\color{blue}{0.0625 \cdot \left({\pi}^{2} \cdot \left({\pi}^{2} \cdot 0.25\right)\right)} - \pi \cdot \left({\pi}^{3} \cdot 0.005208333333333333\right)}{\left(\pi \cdot 0.5\right) \cdot \left({\pi}^{2} \cdot 0.25\right)}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]
      associate-*r* [=>] 1.0	\[ -4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \frac{0.0625 \cdot \color{blue}{\left(\left({\pi}^{2} \cdot {\pi}^{2}\right) \cdot 0.25\right)} - \pi \cdot \left({\pi}^{3} \cdot 0.005208333333333333\right)}{\left(\pi \cdot 0.5\right) \cdot \left({\pi}^{2} \cdot 0.25\right)}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]
      pow-sqr [=>] 1.0	\[ -4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \frac{0.0625 \cdot \left(\color{blue}{{\pi}^{\left(2 \cdot 2\right)}} \cdot 0.25\right) - \pi \cdot \left({\pi}^{3} \cdot 0.005208333333333333\right)}{\left(\pi \cdot 0.5\right) \cdot \left({\pi}^{2} \cdot 0.25\right)}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]
      metadata-eval [=>] 1.0	\[ -4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \frac{0.0625 \cdot \left({\pi}^{\color{blue}{4}} \cdot 0.25\right) - \pi \cdot \left({\pi}^{3} \cdot 0.005208333333333333\right)}{\left(\pi \cdot 0.5\right) \cdot \left({\pi}^{2} \cdot 0.25\right)}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]
      associate-*r* [=>] 1.0	\[ -4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \frac{0.0625 \cdot \left({\pi}^{4} \cdot 0.25\right) - \pi \cdot \left({\pi}^{3} \cdot 0.005208333333333333\right)}{\color{blue}{\left(\left(\pi \cdot 0.5\right) \cdot {\pi}^{2}\right) \cdot 0.25}}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]

   6. Applied egg-rr 1.0

      \[\leadsto -4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \color{blue}{\left(\left(\left({\pi}^{4} \cdot 0.015625 + {\pi}^{4} \cdot -0.005208333333333333\right) \cdot \frac{8}{{\pi}^{3}}\right) \cdot \left(\pi \cdot 0.25\right) + \left(\left({\pi}^{4} \cdot 0.015625 + {\pi}^{4} \cdot -0.005208333333333333\right) \cdot \frac{8}{{\pi}^{3}}\right) \cdot \left(\pi \cdot 0.25\right)\right)}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]

   7. Simplified 1.0

      \[\leadsto -4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \color{blue}{\frac{\pi \cdot \left({\pi}^{4} \cdot 0.041666666666666664\right)}{{\pi}^{3}}}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]
      Proof

      [Start] 1.0	\[ -4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \left(\left(\left({\pi}^{4} \cdot 0.015625 + {\pi}^{4} \cdot -0.005208333333333333\right) \cdot \frac{8}{{\pi}^{3}}\right) \cdot \left(\pi \cdot 0.25\right) + \left(\left({\pi}^{4} \cdot 0.015625 + {\pi}^{4} \cdot -0.005208333333333333\right) \cdot \frac{8}{{\pi}^{3}}\right) \cdot \left(\pi \cdot 0.25\right)\right)\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]
      distribute-lft-out [=>] 1.0	\[ -4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \color{blue}{\left(\left({\pi}^{4} \cdot 0.015625 + {\pi}^{4} \cdot -0.005208333333333333\right) \cdot \frac{8}{{\pi}^{3}}\right) \cdot \left(\pi \cdot 0.25 + \pi \cdot 0.25\right)}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]
      distribute-lft-out [=>] 1.0	\[ -4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \left(\left({\pi}^{4} \cdot 0.015625 + {\pi}^{4} \cdot -0.005208333333333333\right) \cdot \frac{8}{{\pi}^{3}}\right) \cdot \color{blue}{\left(\pi \cdot \left(0.25 + 0.25\right)\right)}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]
      metadata-eval [=>] 1.0	\[ -4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \left(\left({\pi}^{4} \cdot 0.015625 + {\pi}^{4} \cdot -0.005208333333333333\right) \cdot \frac{8}{{\pi}^{3}}\right) \cdot \left(\pi \cdot \color{blue}{0.5}\right)\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]
      associate-*r* [<=] 1.0	\[ -4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \color{blue}{\left({\pi}^{4} \cdot 0.015625 + {\pi}^{4} \cdot -0.005208333333333333\right) \cdot \left(\frac{8}{{\pi}^{3}} \cdot \left(\pi \cdot 0.5\right)\right)}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]
      associate-*l/ [=>] 1.0	\[ -4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \left({\pi}^{4} \cdot 0.015625 + {\pi}^{4} \cdot -0.005208333333333333\right) \cdot \color{blue}{\frac{8 \cdot \left(\pi \cdot 0.5\right)}{{\pi}^{3}}}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]
      associate-*r/ [=>] 1.0	\[ -4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \color{blue}{\frac{\left({\pi}^{4} \cdot 0.015625 + {\pi}^{4} \cdot -0.005208333333333333\right) \cdot \left(8 \cdot \left(\pi \cdot 0.5\right)\right)}{{\pi}^{3}}}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]

   8. Taylor expanded in f around inf 1.0

      \[\leadsto -4 \cdot \color{blue}{\frac{\log \left(\frac{1}{f}\right) + \log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}{\pi}} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \frac{\pi \cdot \left({\pi}^{4} \cdot 0.041666666666666664\right)}{{\pi}^{3}}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]

   9. Simplified 1.0

      \[\leadsto -4 \cdot \color{blue}{\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \frac{\pi \cdot \left({\pi}^{4} \cdot 0.041666666666666664\right)}{{\pi}^{3}}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]
      Proof

      [Start] 1.0	\[ -4 \cdot \frac{\log \left(\frac{1}{f}\right) + \log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \frac{\pi \cdot \left({\pi}^{4} \cdot 0.041666666666666664\right)}{{\pi}^{3}}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]
      +-commutative [=>] 1.0	\[ -4 \cdot \frac{\color{blue}{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + \log \left(\frac{1}{f}\right)}}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \frac{\pi \cdot \left({\pi}^{4} \cdot 0.041666666666666664\right)}{{\pi}^{3}}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]
      log-rec [=>] 1.0	\[ -4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + \color{blue}{\left(-\log f\right)}}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \frac{\pi \cdot \left({\pi}^{4} \cdot 0.041666666666666664\right)}{{\pi}^{3}}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]
      unsub-neg [=>] 1.0	\[ -4 \cdot \frac{\color{blue}{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) - \log f}}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \frac{\pi \cdot \left({\pi}^{4} \cdot 0.041666666666666664\right)}{{\pi}^{3}}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]
      distribute-rgt-out-- [=>] 1.0	\[ -4 \cdot \frac{\log \left(\frac{2}{\color{blue}{\pi \cdot \left(0.25 - -0.25\right)}}\right) - \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \frac{\pi \cdot \left({\pi}^{4} \cdot 0.041666666666666664\right)}{{\pi}^{3}}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]
      *-commutative [=>] 1.0	\[ -4 \cdot \frac{\log \left(\frac{2}{\color{blue}{\left(0.25 - -0.25\right) \cdot \pi}}\right) - \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \frac{\pi \cdot \left({\pi}^{4} \cdot 0.041666666666666664\right)}{{\pi}^{3}}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]
      associate-/r* [=>] 1.0	\[ -4 \cdot \frac{\log \color{blue}{\left(\frac{\frac{2}{0.25 - -0.25}}{\pi}\right)} - \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \frac{\pi \cdot \left({\pi}^{4} \cdot 0.041666666666666664\right)}{{\pi}^{3}}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]
      metadata-eval [=>] 1.0	\[ -4 \cdot \frac{\log \left(\frac{\frac{2}{\color{blue}{0.5}}}{\pi}\right) - \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \frac{\pi \cdot \left({\pi}^{4} \cdot 0.041666666666666664\right)}{{\pi}^{3}}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]
      metadata-eval [=>] 1.0	\[ -4 \cdot \frac{\log \left(\frac{\color{blue}{4}}{\pi}\right) - \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \frac{\pi \cdot \left({\pi}^{4} \cdot 0.041666666666666664\right)}{{\pi}^{3}}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right) \]

   if 100 < (*.f64 (/.f64 (PI.f64) 4) f)

   1. Initial program 59.5

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

   2. Simplified 61.0

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

      [Start] 59.5	\[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
      *-commutative [=>] 59.5	\[ -\color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \frac{1}{\frac{\pi}{4}}} \]
      distribute-rgt-neg-in [=>] 59.5	\[ \color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \left(-\frac{1}{\frac{\pi}{4}}\right)} \]

   3. Taylor expanded in f around 0 63.8

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

   4. Simplified 63.8

      \[\leadsto \log \left(\frac{{\left(e^{f}\right)}^{\left(\frac{\pi}{4}\right)} + {\left(e^{-0.25 \cdot \pi}\right)}^{f}}{\color{blue}{\left(\pi \cdot 0.5\right) \cdot f}}\right) \cdot \frac{-4}{\pi} \]
      Proof

      [Start] 63.8	\[ \log \left(\frac{{\left(e^{f}\right)}^{\left(\frac{\pi}{4}\right)} + {\left(e^{-0.25 \cdot \pi}\right)}^{f}}{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f}\right) \cdot \frac{-4}{\pi} \]
      distribute-rgt-out-- [=>] 63.8	\[ \log \left(\frac{{\left(e^{f}\right)}^{\left(\frac{\pi}{4}\right)} + {\left(e^{-0.25 \cdot \pi}\right)}^{f}}{\color{blue}{\left(\pi \cdot \left(0.25 - -0.25\right)\right)} \cdot f}\right) \cdot \frac{-4}{\pi} \]
      metadata-eval [=>] 63.8	\[ \log \left(\frac{{\left(e^{f}\right)}^{\left(\frac{\pi}{4}\right)} + {\left(e^{-0.25 \cdot \pi}\right)}^{f}}{\left(\pi \cdot \color{blue}{0.5}\right) \cdot f}\right) \cdot \frac{-4}{\pi} \]

   5. Applied egg-rr 63.8

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

   6. Applied egg-rr 64.0

      \[\leadsto \color{blue}{\frac{1}{\frac{\pi}{\log \left(\frac{\frac{0}{0}}{f \cdot \pi}\right) \cdot -4}}} \]

   7. Simplified 2.9

      \[\leadsto \color{blue}{0 \cdot \left(-4 \cdot \left(0 - \log \left(f \cdot \pi\right)\right)\right)} \]
      Proof

      [Start] 64.0	\[ \frac{1}{\frac{\pi}{\log \left(\frac{\frac{0}{0}}{f \cdot \pi}\right) \cdot -4}} \]
      associate-/r/ [=>] 64.0	\[ \color{blue}{\frac{1}{\pi} \cdot \left(\log \left(\frac{\frac{0}{0}}{f \cdot \pi}\right) \cdot -4\right)} \]
      *-inverses [<=] 64.0	\[ \frac{\color{blue}{\frac{0}{0}}}{\pi} \cdot \left(\log \left(\frac{\frac{0}{0}}{f \cdot \pi}\right) \cdot -4\right) \]
      div0 [=>] 64.0	\[ \frac{\color{blue}{0}}{\pi} \cdot \left(\log \left(\frac{\frac{0}{0}}{f \cdot \pi}\right) \cdot -4\right) \]
      metadata-eval [<=] 64.0	\[ \frac{\color{blue}{-4 \cdot 0}}{\pi} \cdot \left(\log \left(\frac{\frac{0}{0}}{f \cdot \pi}\right) \cdot -4\right) \]
      associate-*l/ [<=] 64.0	\[ \color{blue}{\left(\frac{-4}{\pi} \cdot 0\right)} \cdot \left(\log \left(\frac{\frac{0}{0}}{f \cdot \pi}\right) \cdot -4\right) \]
      mul0-rgt [=>] 64.0	\[ \color{blue}{0} \cdot \left(\log \left(\frac{\frac{0}{0}}{f \cdot \pi}\right) \cdot -4\right) \]
      *-commutative [=>] 64.0	\[ 0 \cdot \color{blue}{\left(-4 \cdot \log \left(\frac{\frac{0}{0}}{f \cdot \pi}\right)\right)} \]
      *-inverses [=>] 2.9	\[ 0 \cdot \left(-4 \cdot \log \left(\frac{\color{blue}{1}}{f \cdot \pi}\right)\right) \]
      log-div [=>] 2.9	\[ 0 \cdot \left(-4 \cdot \color{blue}{\left(\log 1 - \log \left(f \cdot \pi\right)\right)}\right) \]
      metadata-eval [=>] 2.9	\[ 0 \cdot \left(-4 \cdot \left(\color{blue}{0} - \log \left(f \cdot \pi\right)\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 100:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(\pi \cdot 0.25 + \pi \cdot 0.25\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{\pi \cdot 0.25 + \pi \cdot 0.25} + 0.25 \cdot \frac{\pi}{\pi \cdot 0.25 + \pi \cdot 0.25}\right)}^{2}\right) + \frac{\pi \cdot \left({\pi}^{4} \cdot 0.041666666666666664\right)}{{\pi}^{3}}\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(\pi \cdot 0.25 + \pi \cdot 0.25\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{\pi \cdot 0.25 + \pi \cdot 0.25} + 0.25 \cdot \frac{\pi}{\pi \cdot 0.25 + \pi \cdot 0.25}\right)\right)}{\pi}\right)\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \left(4 \cdot \log \left(\pi \cdot f\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	163012

\[\begin{array}{l} t_0 := e^{\frac{\pi}{4} \cdot f}\\ t_1 := e^{\frac{\pi}{4} \cdot \left(-f\right)}\\ \mathbf{if}\;\frac{t_0 + t_1}{t_0 - t_1} \leq \infty:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{{\left(e^{f}\right)}^{\left(\frac{\pi}{4}\right)} + {\left(e^{\pi \cdot -0.25}\right)}^{f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\pi \cdot 0.5, f, {f}^{3} \cdot \left({\pi}^{3} \cdot 0.005208333333333333\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \left(4 \cdot \log \left(\pi \cdot f\right)\right)\\ \end{array} \]

Alternative 2
Error	0.9
Cost	143044

\[\begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 100:\\ \;\;\;\;\log \left(\frac{{\left(e^{f}\right)}^{\left(\frac{\pi}{4}\right)} + {\left(e^{\pi \cdot -0.25}\right)}^{f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\pi \cdot 0.5, f, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{7} \cdot \left({\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}\right)\right)\right)\right)}\right) \cdot \frac{-4}{\pi}\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \left(4 \cdot \log \left(\pi \cdot f\right)\right)\\ \end{array} \]

Alternative 3
Error	1.5
Cost	32772

\[\begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 0.02:\\ \;\;\;\;\frac{-4 \cdot \left(\log \left(\frac{4}{\pi}\right) - \log f\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \left(4 \cdot \log \left(\pi \cdot f\right)\right)\\ \end{array} \]

Alternative 4
Error	1.5
Cost	19780

\[\begin{array}{l} \mathbf{if}\;f \leq 1.25:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{4}{\pi \cdot f}\right)\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \left(4 \cdot \log \left(\pi \cdot f\right)\right)\\ \end{array} \]

Alternative 5
Error	1.5
Cost	19780

\[\begin{array}{l} \mathbf{if}\;f \leq 1.25:\\ \;\;\;\;\frac{-4}{\frac{\pi}{\log \left(\frac{4}{\pi \cdot f}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \left(4 \cdot \log \left(\pi \cdot f\right)\right)\\ \end{array} \]

Alternative 6
Error	1.5
Cost	19780

\[\begin{array}{l} \mathbf{if}\;f \leq 1.25:\\ \;\;\;\;\frac{-4}{\frac{\pi}{\log \left(\frac{\frac{4}{\pi}}{f}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \left(4 \cdot \log \left(\pi \cdot f\right)\right)\\ \end{array} \]

Alternative 7
Error	1.4
Cost	19780

\[\begin{array}{l} \mathbf{if}\;f \leq 1.25:\\ \;\;\;\;\frac{\log \left(\frac{\pi}{4} \cdot f\right)}{\pi \cdot 0.25}\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \left(4 \cdot \log \left(\pi \cdot f\right)\right)\\ \end{array} \]

Alternative 8
Error	60.8
Cost	13184

\[0 \cdot \left(4 \cdot \log \left(\pi \cdot f\right)\right) \]

Alternative 9
Error	64.0
Cost	13056

\[\frac{-4}{\pi} \cdot \mathsf{log1p}\left(-2\right) \]

Reproduce

herbie shell --seed 2023012 
(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)))))))))