VandenBroeck and Keller, Equation (20)

Average Error: 4.2% → 97.9%

Time: 28.8s

Precision: binary64

Cost: 45508.00

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{\log \left(\frac{4}{\pi \cdot f}\right)}{\pi}, \left(f \cdot f\right) \cdot \left(\pi \cdot -0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-{\left({\left(e^{f}\right)}^{\pi}\right)}^{-0.25}\right) \cdot \frac{4}{\pi}\\ \end{array} \]

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
 (if (<= (* (/ PI 4.0) f) 0.0002)
   (fma
    -4.0
    (/ (log (/ 4.0 (* PI f))) PI)
    (* (* f f) (* PI -0.08333333333333333)))
   (* (log1p (- (pow (pow (exp f) PI) -0.25))) (/ 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 tmp;
	if (((((double) M_PI) / 4.0) * f) <= 0.0002) {
		tmp = fma(-4.0, (log((4.0 / (((double) M_PI) * f))) / ((double) M_PI)), ((f * f) * (((double) M_PI) * -0.08333333333333333)));
	} else {
		tmp = log1p(-pow(pow(exp(f), ((double) M_PI)), -0.25)) * (4.0 / ((double) M_PI));
	}
	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)
	tmp = 0.0
	if (Float64(Float64(pi / 4.0) * f) <= 0.0002)
		tmp = fma(-4.0, Float64(log(Float64(4.0 / Float64(pi * f))) / pi), Float64(Float64(f * f) * Float64(pi * -0.08333333333333333)));
	else
		tmp = Float64(log1p(Float64(-((exp(f) ^ pi) ^ -0.25))) * Float64(4.0 / pi));
	end
	return 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_] := If[LessEqual[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision], 0.0002], N[(-4.0 * N[(N[Log[N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] + N[(N[(f * f), $MachinePrecision] * N[(Pi * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + (-N[Power[N[Power[N[Exp[f], $MachinePrecision], Pi], $MachinePrecision], -0.25], $MachinePrecision])], $MachinePrecision] * N[(4.0 / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (PI.f64) 4) f) < 2.0000000000000001e-4

   1. Initial program 2.7

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

   2. Simplified 2.7

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

      [Start] 2.7	\[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
      *-commutative [=>] 2.7	\[ -\color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \frac{1}{\frac{\pi}{4}}} \]
      distribute-rgt-neg-in [=>] 2.7	\[ \color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \left(-\frac{1}{\frac{\pi}{4}}\right)} \]

   3. Applied egg-rr 2.7

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

   4. Taylor expanded in f around 0 99.3

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

   5. Simplified 99.3

      \[\leadsto \log \left(\frac{e^{\frac{\pi \cdot f}{4}} + e^{-0.25 \cdot \left(\pi \cdot f\right)}}{\color{blue}{\mathsf{fma}\left(\pi \cdot 0.5, f, {f}^{3} \cdot \left({\pi}^{3} \cdot 0.005208333333333333\right)\right)}}\right) \cdot \frac{-4}{\pi} \]
      Proof

      [Start] 99.3	\[ \log \left(\frac{e^{\frac{\pi \cdot f}{4}} + e^{-0.25 \cdot \left(\pi \cdot f\right)}}{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}\right) \cdot \frac{-4}{\pi} \]
      fma-def [=>] 99.3	\[ \log \left(\frac{e^{\frac{\pi \cdot f}{4}} + e^{-0.25 \cdot \left(\pi \cdot f\right)}}{\color{blue}{\mathsf{fma}\left(0.25 \cdot \pi - -0.25 \cdot \pi, f, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}}\right) \cdot \frac{-4}{\pi} \]
      distribute-rgt-out-- [=>] 99.3	\[ \log \left(\frac{e^{\frac{\pi \cdot f}{4}} + e^{-0.25 \cdot \left(\pi \cdot f\right)}}{\mathsf{fma}\left(\color{blue}{\pi \cdot \left(0.25 - -0.25\right)}, f, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right) \cdot \frac{-4}{\pi} \]
      metadata-eval [=>] 99.3	\[ \log \left(\frac{e^{\frac{\pi \cdot f}{4}} + e^{-0.25 \cdot \left(\pi \cdot f\right)}}{\mathsf{fma}\left(\pi \cdot \color{blue}{0.5}, f, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right) \cdot \frac{-4}{\pi} \]
      distribute-rgt-out-- [=>] 99.3	\[ \log \left(\frac{e^{\frac{\pi \cdot f}{4}} + e^{-0.25 \cdot \left(\pi \cdot f\right)}}{\mathsf{fma}\left(\pi \cdot 0.5, f, {f}^{3} \cdot \color{blue}{\left({\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)}\right)}\right) \cdot \frac{-4}{\pi} \]
      metadata-eval [=>] 99.3	\[ \log \left(\frac{e^{\frac{\pi \cdot f}{4}} + e^{-0.25 \cdot \left(\pi \cdot f\right)}}{\mathsf{fma}\left(\pi \cdot 0.5, f, {f}^{3} \cdot \left({\pi}^{3} \cdot \color{blue}{0.005208333333333333}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   6. Taylor expanded in f around 0 99.4

      \[\leadsto \color{blue}{\left(0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) \cdot \pi\right)\right)\right) + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + \left(\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f\right)\right)\right)} \cdot \frac{-4}{\pi} \]

   7. Simplified 99.4

      \[\leadsto \color{blue}{\left(\left(\log \left(\frac{4}{\pi}\right) - \log f\right) + \mathsf{fma}\left(0.5, \left(f \cdot f\right) \cdot \mathsf{fma}\left(0.5, {\pi}^{2} \cdot 0.08333333333333333, 0\right), f \cdot 0\right)\right)} \cdot \frac{-4}{\pi} \]
      Proof

      [Start] 99.4	\[ \left(0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) \cdot \pi\right)\right)\right) + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + \left(\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f\right)\right)\right) \cdot \frac{-4}{\pi} \]
      associate-+r+ [=>] 99.4	\[ \color{blue}{\left(\left(0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) \cdot \pi\right)\right)\right) + 0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right)\right) + \left(\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f\right)\right)} \cdot \frac{-4}{\pi} \]
      +-commutative [=>] 99.4	\[ \color{blue}{\left(\left(\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f\right) + \left(0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) \cdot \pi\right)\right)\right) + 0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right)\right)\right)} \cdot \frac{-4}{\pi} \]
      mul-1-neg [=>] 99.4	\[ \left(\left(\log \left(\frac{4}{\pi}\right) + \color{blue}{\left(-\log f\right)}\right) + \left(0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) \cdot \pi\right)\right)\right) + 0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right)\right)\right) \cdot \frac{-4}{\pi} \]
      unsub-neg [=>] 99.4	\[ \left(\color{blue}{\left(\log \left(\frac{4}{\pi}\right) - \log f\right)} + \left(0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) \cdot \pi\right)\right)\right) + 0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right)\right)\right) \cdot \frac{-4}{\pi} \]
      fma-def [=>] 99.4	\[ \left(\left(\log \left(\frac{4}{\pi}\right) - \log f\right) + \color{blue}{\mathsf{fma}\left(0.5, {f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) \cdot \pi\right)\right), 0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   8. Taylor expanded in f around 0 99.5

      \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} + -0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)} \]

   9. Simplified 99.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, \frac{\log \left(\frac{4}{\pi \cdot f}\right)}{\pi}, \left(f \cdot f\right) \cdot \left(\pi \cdot -0.08333333333333333\right)\right)} \]
      Proof

      [Start] 99.5	\[ -4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} + -0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right) \]
      fma-def [=>] 99.5	\[ \color{blue}{\mathsf{fma}\left(-4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)\right)} \]
      log-div [=>] 99.5	\[ \mathsf{fma}\left(-4, \frac{\color{blue}{\left(\log 4 - \log \pi\right)} - \log f}{\pi}, -0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)\right) \]
      associate--l- [=>] 99.4	\[ \mathsf{fma}\left(-4, \frac{\color{blue}{\log 4 - \left(\log \pi + \log f\right)}}{\pi}, -0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)\right) \]
      log-prod [<=] 99.4	\[ \mathsf{fma}\left(-4, \frac{\log 4 - \color{blue}{\log \left(\pi \cdot f\right)}}{\pi}, -0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)\right) \]
      *-commutative [<=] 99.4	\[ \mathsf{fma}\left(-4, \frac{\log 4 - \log \color{blue}{\left(f \cdot \pi\right)}}{\pi}, -0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)\right) \]
      log-div [<=] 99.4	\[ \mathsf{fma}\left(-4, \frac{\color{blue}{\log \left(\frac{4}{f \cdot \pi}\right)}}{\pi}, -0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)\right) \]
      *-commutative [=>] 99.4	\[ \mathsf{fma}\left(-4, \frac{\log \left(\frac{4}{\color{blue}{\pi \cdot f}}\right)}{\pi}, -0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)\right) \]
      *-commutative [=>] 99.4	\[ \mathsf{fma}\left(-4, \frac{\log \left(\frac{4}{\pi \cdot f}\right)}{\pi}, \color{blue}{\left({f}^{2} \cdot \pi\right) \cdot -0.08333333333333333}\right) \]
      associate-*l* [=>] 99.4	\[ \mathsf{fma}\left(-4, \frac{\log \left(\frac{4}{\pi \cdot f}\right)}{\pi}, \color{blue}{{f}^{2} \cdot \left(\pi \cdot -0.08333333333333333\right)}\right) \]
      unpow2 [=>] 99.4	\[ \mathsf{fma}\left(-4, \frac{\log \left(\frac{4}{\pi \cdot f}\right)}{\pi}, \color{blue}{\left(f \cdot f\right)} \cdot \left(\pi \cdot -0.08333333333333333\right)\right) \]

   if 2.0000000000000001e-4 < (*.f64 (/.f64 (PI.f64) 4) f)

   1. Initial program 40.0

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

   2. Simplified 40.0

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

      [Start] 40.0	\[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
      *-commutative [=>] 40.0	\[ -\color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \frac{1}{\frac{\pi}{4}}} \]
      distribute-rgt-neg-in [=>] 40.0	\[ \color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \left(-\frac{1}{\frac{\pi}{4}}\right)} \]

   3. Taylor expanded in f around 0 8.7

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

   4. Applied egg-rr 0.0

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

   5. Simplified 61.2

      \[\leadsto \color{blue}{\left(-\mathsf{log1p}\left(-{\left({\left(e^{f}\right)}^{\pi}\right)}^{-0.25}\right)\right)} \cdot \frac{-4}{\pi} \]
      Proof

      [Start] 0.0	\[ \left(\log \left({\left({\left({\left(e^{f}\right)}^{\pi}\right)}^{-0.25}\right)}^{2} - {\left({\left({\left(e^{f}\right)}^{\pi}\right)}^{-0.25}\right)}^{2}\right) - \left(\log \left({\left({\left(e^{f}\right)}^{\pi}\right)}^{-0.25} - {\left({\left(e^{f}\right)}^{\pi}\right)}^{-0.25}\right) - \log \left(\frac{1}{1 - {\left({\left(e^{f}\right)}^{\pi}\right)}^{-0.25}}\right)\right)\right) \cdot \frac{-4}{\pi} \]
      +-inverses [=>] 0.0	\[ \left(\log \color{blue}{0} - \left(\log \left({\left({\left(e^{f}\right)}^{\pi}\right)}^{-0.25} - {\left({\left(e^{f}\right)}^{\pi}\right)}^{-0.25}\right) - \log \left(\frac{1}{1 - {\left({\left(e^{f}\right)}^{\pi}\right)}^{-0.25}}\right)\right)\right) \cdot \frac{-4}{\pi} \]
      +-inverses [<=] 0.0	\[ \left(\log \color{blue}{\left({\left({\left(e^{f}\right)}^{\pi}\right)}^{-0.25} - {\left({\left(e^{f}\right)}^{\pi}\right)}^{-0.25}\right)} - \left(\log \left({\left({\left(e^{f}\right)}^{\pi}\right)}^{-0.25} - {\left({\left(e^{f}\right)}^{\pi}\right)}^{-0.25}\right) - \log \left(\frac{1}{1 - {\left({\left(e^{f}\right)}^{\pi}\right)}^{-0.25}}\right)\right)\right) \cdot \frac{-4}{\pi} \]
      associate-+l- [<=] 0.0	\[ \color{blue}{\left(\left(\log \left({\left({\left(e^{f}\right)}^{\pi}\right)}^{-0.25} - {\left({\left(e^{f}\right)}^{\pi}\right)}^{-0.25}\right) - \log \left({\left({\left(e^{f}\right)}^{\pi}\right)}^{-0.25} - {\left({\left(e^{f}\right)}^{\pi}\right)}^{-0.25}\right)\right) + \log \left(\frac{1}{1 - {\left({\left(e^{f}\right)}^{\pi}\right)}^{-0.25}}\right)\right)} \cdot \frac{-4}{\pi} \]
      +-inverses [=>] 61.0	\[ \left(\color{blue}{0} + \log \left(\frac{1}{1 - {\left({\left(e^{f}\right)}^{\pi}\right)}^{-0.25}}\right)\right) \cdot \frac{-4}{\pi} \]
      log-rec [=>] 61.0	\[ \left(0 + \color{blue}{\left(-\log \left(1 - {\left({\left(e^{f}\right)}^{\pi}\right)}^{-0.25}\right)\right)}\right) \cdot \frac{-4}{\pi} \]
      sub-neg [<=] 61.0	\[ \color{blue}{\left(0 - \log \left(1 - {\left({\left(e^{f}\right)}^{\pi}\right)}^{-0.25}\right)\right)} \cdot \frac{-4}{\pi} \]
      sub0-neg [=>] 61.0	\[ \color{blue}{\left(-\log \left(1 - {\left({\left(e^{f}\right)}^{\pi}\right)}^{-0.25}\right)\right)} \cdot \frac{-4}{\pi} \]
      sub-neg [=>] 61.0	\[ \left(-\log \color{blue}{\left(1 + \left(-{\left({\left(e^{f}\right)}^{\pi}\right)}^{-0.25}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

3. Recombined 2 regimes into one program.

4. Final simplification 97.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{\log \left(\frac{4}{\pi \cdot f}\right)}{\pi}, \left(f \cdot f\right) \cdot \left(\pi \cdot -0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-{\left({\left(e^{f}\right)}^{\pi}\right)}^{-0.25}\right) \cdot \frac{4}{\pi}\\ \end{array} \]

Alternatives

Alternative 1
Error	98.3%
Cost	32900.00

\[\begin{array}{l} \mathbf{if}\;f \leq 220:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{\log \left(\frac{4}{\pi \cdot f}\right)}{\pi}, \left(f \cdot f\right) \cdot \left(\pi \cdot -0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{\pi} \cdot 0\\ \end{array} \]

Alternative 2
Error	97.8%
Cost	26180.00

\[\begin{array}{l} \mathbf{if}\;f \leq 1.26:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{\pi} \cdot 0\\ \end{array} \]

Alternative 3
Error	97.6%
Cost	20036.00

\[\begin{array}{l} \mathbf{if}\;f \leq 1.26:\\ \;\;\;\;-4 \cdot \frac{\frac{1}{\frac{1}{\log \left(\frac{\frac{4}{f}}{\pi}\right)}}}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{\pi} \cdot 0\\ \end{array} \]

Alternative 4
Error	97.6%
Cost	19780.00

\[\begin{array}{l} \mathbf{if}\;f \leq 1.26:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{\frac{4}{\pi}}{f}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{\pi} \cdot 0\\ \end{array} \]

Alternative 5
Error	97.6%
Cost	19780.00

\[\begin{array}{l} \mathbf{if}\;f \leq 1.26:\\ \;\;\;\;\frac{-4}{\frac{\pi}{\log \left(\frac{\frac{4}{f}}{\pi}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{\pi} \cdot 0\\ \end{array} \]

Alternative 6
Error	30.9%
Cost	19652.00

\[\begin{array}{l} \mathbf{if}\;f \leq 0.32:\\ \;\;\;\;\log \left(\pi \cdot f\right) \cdot \frac{4}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{\pi} \cdot 0\\ \end{array} \]

Alternative 7
Error	7.4%
Cost	13252.00

\[\begin{array}{l} \mathbf{if}\;f \leq 230:\\ \;\;\;\;\frac{-16}{f \cdot {\pi}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{\pi} \cdot 0\\ \end{array} \]

Alternative 8
Error	4.3%
Cost	6784.00

\[-0.08333333333333333 \cdot \left(\pi \cdot \left(f \cdot f\right)\right) \]

Alternative 9
Error	5.1%
Cost	6656.00

\[\frac{-4}{\pi} \cdot 0 \]

Reproduce

herbie shell --seed 2023093 
(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)))))))))