VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 99.1%

Time: 20.3s

Alternatives: 7

Speedup: 3.3×

• could not determine a ground truth

  1. f = 2.00000000000000007e154

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ t_1 := e^{t_0}\\ t_2 := e^{-t_0}\\ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{t_1 + t_2}{t_1 - t_2}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (/ PI 4.0) f)) (t_1 (exp t_0)) (t_2 (exp (- t_0))))
   (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ t_1 t_2) (- t_1 t_2)))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) / 4.0) * f;
	double t_1 = exp(t_0);
	double t_2 = exp(-t_0);
	return -((1.0 / (((double) M_PI) / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
}

Java

public static double code(double f) {
	double t_0 = (Math.PI / 4.0) * f;
	double t_1 = Math.exp(t_0);
	double t_2 = Math.exp(-t_0);
	return -((1.0 / (Math.PI / 4.0)) * Math.log(((t_1 + t_2) / (t_1 - t_2))));
}

Python

def code(f):
	t_0 = (math.pi / 4.0) * f
	t_1 = math.exp(t_0)
	t_2 = math.exp(-t_0)
	return -((1.0 / (math.pi / 4.0)) * math.log(((t_1 + t_2) / (t_1 - t_2))))

Julia

function code(f)
	t_0 = Float64(Float64(pi / 4.0) * f)
	t_1 = exp(t_0)
	t_2 = exp(Float64(-t_0))
	return Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(t_1 + t_2) / Float64(t_1 - t_2)))))
end

MATLAB

function tmp = code(f)
	t_0 = (pi / 4.0) * f;
	t_1 = exp(t_0);
	t_2 = exp(-t_0);
	tmp = -((1.0 / (pi / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]}, Block[{t$95$1 = N[Exp[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-t$95$0)], $MachinePrecision]}, (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(t$95$1 + t$95$2), $MachinePrecision] / N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Initial Program: 6.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ t_1 := e^{t_0}\\ t_2 := e^{-t_0}\\ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{t_1 + t_2}{t_1 - t_2}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (/ PI 4.0) f)) (t_1 (exp t_0)) (t_2 (exp (- t_0))))
   (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ t_1 t_2) (- t_1 t_2)))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) / 4.0) * f;
	double t_1 = exp(t_0);
	double t_2 = exp(-t_0);
	return -((1.0 / (((double) M_PI) / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
}

Java

public static double code(double f) {
	double t_0 = (Math.PI / 4.0) * f;
	double t_1 = Math.exp(t_0);
	double t_2 = Math.exp(-t_0);
	return -((1.0 / (Math.PI / 4.0)) * Math.log(((t_1 + t_2) / (t_1 - t_2))));
}

Python

def code(f):
	t_0 = (math.pi / 4.0) * f
	t_1 = math.exp(t_0)
	t_2 = math.exp(-t_0)
	return -((1.0 / (math.pi / 4.0)) * math.log(((t_1 + t_2) / (t_1 - t_2))))

Julia

function code(f)
	t_0 = Float64(Float64(pi / 4.0) * f)
	t_1 = exp(t_0)
	t_2 = exp(Float64(-t_0))
	return Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(t_1 + t_2) / Float64(t_1 - t_2)))))
end

MATLAB

function tmp = code(f)
	t_0 = (pi / 4.0) * f;
	t_1 = exp(t_0);
	t_2 = exp(-t_0);
	tmp = -((1.0 / (pi / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]}, Block[{t$95$1 = N[Exp[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-t$95$0)], $MachinePrecision]}, (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(t$95$1 + t$95$2), $MachinePrecision] / N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Alternative 1: 99.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 2 \cdot 10^{-32}:\\ \;\;\;\;4 \cdot \frac{\log f - \log \left(\frac{4}{\pi}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{{\left(\frac{1}{\tanh \left(f \cdot \left(\pi \cdot -0.25\right)\right)}\right)}^{2}}\right) \cdot \frac{-1}{\frac{\pi}{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= (* (/ PI 4.0) f) 2e-32)
   (* 4.0 (/ (- (log f) (log (/ 4.0 PI))) PI))
   (*
    (log (sqrt (pow (/ 1.0 (tanh (* f (* PI -0.25)))) 2.0)))
    (/ -1.0 (/ PI 4.0)))))

C

double code(double f) {
	double tmp;
	if (((((double) M_PI) / 4.0) * f) <= 2e-32) {
		tmp = 4.0 * ((log(f) - log((4.0 / ((double) M_PI)))) / ((double) M_PI));
	} else {
		tmp = log(sqrt(pow((1.0 / tanh((f * (((double) M_PI) * -0.25)))), 2.0))) * (-1.0 / (((double) M_PI) / 4.0));
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (((Math.PI / 4.0) * f) <= 2e-32) {
		tmp = 4.0 * ((Math.log(f) - Math.log((4.0 / Math.PI))) / Math.PI);
	} else {
		tmp = Math.log(Math.sqrt(Math.pow((1.0 / Math.tanh((f * (Math.PI * -0.25)))), 2.0))) * (-1.0 / (Math.PI / 4.0));
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if ((math.pi / 4.0) * f) <= 2e-32:
		tmp = 4.0 * ((math.log(f) - math.log((4.0 / math.pi))) / math.pi)
	else:
		tmp = math.log(math.sqrt(math.pow((1.0 / math.tanh((f * (math.pi * -0.25)))), 2.0))) * (-1.0 / (math.pi / 4.0))
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (Float64(Float64(pi / 4.0) * f) <= 2e-32)
		tmp = Float64(4.0 * Float64(Float64(log(f) - log(Float64(4.0 / pi))) / pi));
	else
		tmp = Float64(log(sqrt((Float64(1.0 / tanh(Float64(f * Float64(pi * -0.25)))) ^ 2.0))) * Float64(-1.0 / Float64(pi / 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (((pi / 4.0) * f) <= 2e-32)
		tmp = 4.0 * ((log(f) - log((4.0 / pi))) / pi);
	else
		tmp = log(sqrt(((1.0 / tanh((f * (pi * -0.25)))) ^ 2.0))) * (-1.0 / (pi / 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision], 2e-32], N[(4.0 * N[(N[(N[Log[f], $MachinePrecision] - N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[Sqrt[N[Power[N[(1.0 / N[Tanh[N[(f * N[(Pi * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (PI.f64) 4) f) < 2.00000000000000011e-32

   1. Initial program 3.1%

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

   2. Taylor expanded in f around 0 99.3%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{2}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}\right)} \]
   3. Step-by-step derivation

      1. associate-/r* 99.3%

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

      2. distribute-rgt-out-- 99.3%

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

      3. metadata-eval 99.3%

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

   4. Simplified 99.3%

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

   5. Taylor expanded in f around 0 99.6%

      \[\leadsto -\color{blue}{4 \cdot \frac{\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f}{\pi}} \]
   6. Step-by-step derivation

      1. associate-*r/ 99.6%

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

      2. neg-mul-1 99.6%

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

      3. sub-neg 99.6%

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

      4. associate-*r/ 99.6%

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

   7. Simplified 99.6%

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

   if 2.00000000000000011e-32 < (*.f64 (/.f64 (PI.f64) 4) f)

   1. Initial program 33.0%

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

   3. Applied egg-rr 93.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\sqrt{{\left(\frac{1}{\tanh \left(\left(\pi \cdot -0.25\right) \cdot f\right)}\right)}^{2}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 2 \cdot 10^{-32}:\\ \;\;\;\;4 \cdot \frac{\log f - \log \left(\frac{4}{\pi}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{{\left(\frac{1}{\tanh \left(f \cdot \left(\pi \cdot -0.25\right)\right)}\right)}^{2}}\right) \cdot \frac{-1}{\frac{\pi}{4}}\\ \end{array} \]

Alternative 2: 96.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{-\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{4}{\pi \cdot f}\right)\right)}{\pi \cdot 0.25} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/ (- (log (fma f (* PI 0.08333333333333333) (/ 4.0 (* PI f))))) (* PI 0.25)))

C

double code(double f) {
	return -log(fma(f, (((double) M_PI) * 0.08333333333333333), (4.0 / (((double) M_PI) * f)))) / (((double) M_PI) * 0.25);
}

Julia

function code(f)
	return Float64(Float64(-log(fma(f, Float64(pi * 0.08333333333333333), Float64(4.0 / Float64(pi * f))))) / Float64(pi * 0.25))
end

Wolfram

code[f_] := N[((-N[Log[N[(f * N[(Pi * 0.08333333333333333), $MachinePrecision] + N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.3%

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

2. Taylor expanded in f around 0 96.1%

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

4. Simplified 96.1%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, 1 \cdot \frac{\pi}{0.5}, \frac{{\pi}^{3}}{\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}} \cdot -2\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)} \]
5. Step-by-step derivation

   1. *-un-lft-identity 96.1%

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

   2. fma-udef 96.1%

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

   3. div-inv 96.1%

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

   4. metadata-eval 96.1%

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

   5. associate-/r/ 96.1%

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

   6. unpow-prod-down 96.1%

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

   7. metadata-eval 96.1%

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

6. Applied egg-rr 96.1%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{0.0625 \cdot \left(\pi \cdot 2\right) + \left(\frac{{\pi}^{3}}{{\pi}^{2} \cdot 0.25} \cdot 0.005208333333333333\right) \cdot -2}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]
7. Step-by-step derivation

   1. +-commutative 96.1%

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

   2. *-commutative 96.1%

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

   3. fma-def 96.1%

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

   4. associate-*l/ 96.1%

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

   5. times-frac 96.1%

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

   6. cube-unmult 96.1%

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

   7. unpow2 96.1%

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

   8. associate-/l* 96.1%

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

   9. *-inverses 96.1%

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

   10. /-rgt-identity 96.1%

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

   11. metadata-eval 96.1%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(-2, \pi \cdot \color{blue}{0.020833333333333332}, 0.0625 \cdot \left(\pi \cdot 2\right)\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

   12. *-commutative 96.1%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(-2, \pi \cdot 0.020833333333333332, \color{blue}{\left(\pi \cdot 2\right) \cdot 0.0625}\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

   13. associate-*l* 96.1%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(-2, \pi \cdot 0.020833333333333332, \color{blue}{\pi \cdot \left(2 \cdot 0.0625\right)}\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

   14. metadata-eval 96.1%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(-2, \pi \cdot 0.020833333333333332, \pi \cdot \color{blue}{0.125}\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

8. Simplified 96.1%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\mathsf{fma}\left(-2, \pi \cdot 0.020833333333333332, \pi \cdot 0.125\right)}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]
9. Step-by-step derivation

   1. associate-*l/ 96.2%

      \[\leadsto -\color{blue}{\frac{1 \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(-2, \pi \cdot 0.020833333333333332, \pi \cdot 0.125\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\frac{\pi}{4}}} \]

   2. *-un-lft-identity 96.2%

      \[\leadsto -\frac{\color{blue}{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(-2, \pi \cdot 0.020833333333333332, \pi \cdot 0.125\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}}{\frac{\pi}{4}} \]

   3. div-inv 96.2%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(-2, \pi \cdot 0.020833333333333332, \pi \cdot 0.125\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\color{blue}{\pi \cdot \frac{1}{4}}} \]

   4. metadata-eval 96.2%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(-2, \pi \cdot 0.020833333333333332, \pi \cdot 0.125\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\pi \cdot \color{blue}{0.25}} \]

10. Applied egg-rr 96.2%

    \[\leadsto -\color{blue}{\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(-2, \pi \cdot 0.020833333333333332, \pi \cdot 0.125\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\pi \cdot 0.25}} \]
11. Step-by-step derivation

12. Simplified 96.2%

    \[\leadsto -\color{blue}{\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{4}{f \cdot \pi}\right)\right)}{\pi \cdot 0.25}} \]

13. Final simplification 96.2%

    \[\leadsto \frac{-\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{4}{\pi \cdot f}\right)\right)}{\pi \cdot 0.25} \]

Alternative 3: 95.8% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (f) :precision binary64 (* 4.0 (/ (- (log f) (log (/ 4.0 PI))) PI)))

C

double code(double f) {
	return 4.0 * ((log(f) - log((4.0 / ((double) M_PI)))) / ((double) M_PI));
}

Java

public static double code(double f) {
	return 4.0 * ((Math.log(f) - Math.log((4.0 / Math.PI))) / Math.PI);
}

Python

def code(f):
	return 4.0 * ((math.log(f) - math.log((4.0 / math.pi))) / math.pi)

Julia

function code(f)
	return Float64(4.0 * Float64(Float64(log(f) - log(Float64(4.0 / pi))) / pi))
end

MATLAB

function tmp = code(f)
	tmp = 4.0 * ((log(f) - log((4.0 / pi))) / pi);
end

Wolfram

code[f_] := N[(4.0 * N[(N[(N[Log[f], $MachinePrecision] - N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.3%

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

2. Taylor expanded in f around 0 95.2%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{2}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}\right)} \]
3. Step-by-step derivation

   1. associate-/r* 95.2%

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

   2. distribute-rgt-out-- 95.2%

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

   3. metadata-eval 95.2%

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

4. Simplified 95.2%

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

5. Taylor expanded in f around 0 95.4%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f}{\pi}} \]
6. Step-by-step derivation

   1. associate-*r/ 95.4%

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

   2. neg-mul-1 95.4%

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

   3. sub-neg 95.4%

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

   4. associate-*r/ 95.4%

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

7. Simplified 95.4%

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

8. Final simplification 95.4%

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

Alternative 4: 1.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\pi \cdot \left(f \cdot 0.08333333333333333\right)\right)}{\pi} \cdot \left(-4\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (* (/ (log (* PI (* f 0.08333333333333333))) PI) (- 4.0)))

C

double code(double f) {
	return (log((((double) M_PI) * (f * 0.08333333333333333))) / ((double) M_PI)) * -4.0;
}

Java

public static double code(double f) {
	return (Math.log((Math.PI * (f * 0.08333333333333333))) / Math.PI) * -4.0;
}

Python

def code(f):
	return (math.log((math.pi * (f * 0.08333333333333333))) / math.pi) * -4.0

Julia

function code(f)
	return Float64(Float64(log(Float64(pi * Float64(f * 0.08333333333333333))) / pi) * Float64(-4.0))
end

MATLAB

function tmp = code(f)
	tmp = (log((pi * (f * 0.08333333333333333))) / pi) * -4.0;
end

Wolfram

code[f_] := N[(N[(N[Log[N[(Pi * N[(f * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * (-4.0)), $MachinePrecision]

Derivation

1. Initial program 7.3%

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

2. Taylor expanded in f around 0 96.1%

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

4. Simplified 96.1%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, 1 \cdot \frac{\pi}{0.5}, \frac{{\pi}^{3}}{\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}} \cdot -2\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)} \]
5. Step-by-step derivation

   1. *-un-lft-identity 96.1%

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

   2. fma-udef 96.1%

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

   3. div-inv 96.1%

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

   4. metadata-eval 96.1%

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

   5. associate-/r/ 96.1%

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

   6. unpow-prod-down 96.1%

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

   7. metadata-eval 96.1%

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

6. Applied egg-rr 96.1%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{0.0625 \cdot \left(\pi \cdot 2\right) + \left(\frac{{\pi}^{3}}{{\pi}^{2} \cdot 0.25} \cdot 0.005208333333333333\right) \cdot -2}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]
7. Step-by-step derivation

   1. +-commutative 96.1%

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

   2. *-commutative 96.1%

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

   3. fma-def 96.1%

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

   4. associate-*l/ 96.1%

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

   5. times-frac 96.1%

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

   6. cube-unmult 96.1%

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

   7. unpow2 96.1%

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

   8. associate-/l* 96.1%

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

   9. *-inverses 96.1%

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

   10. /-rgt-identity 96.1%

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

   11. metadata-eval 96.1%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(-2, \pi \cdot \color{blue}{0.020833333333333332}, 0.0625 \cdot \left(\pi \cdot 2\right)\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

   12. *-commutative 96.1%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(-2, \pi \cdot 0.020833333333333332, \color{blue}{\left(\pi \cdot 2\right) \cdot 0.0625}\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

   13. associate-*l* 96.1%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(-2, \pi \cdot 0.020833333333333332, \color{blue}{\pi \cdot \left(2 \cdot 0.0625\right)}\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

   14. metadata-eval 96.1%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(-2, \pi \cdot 0.020833333333333332, \pi \cdot \color{blue}{0.125}\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

8. Simplified 96.1%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\mathsf{fma}\left(-2, \pi \cdot 0.020833333333333332, \pi \cdot 0.125\right)}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]
9. Step-by-step derivation

   1. associate-*l/ 96.2%

      \[\leadsto -\color{blue}{\frac{1 \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(-2, \pi \cdot 0.020833333333333332, \pi \cdot 0.125\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\frac{\pi}{4}}} \]

   2. *-un-lft-identity 96.2%

      \[\leadsto -\frac{\color{blue}{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(-2, \pi \cdot 0.020833333333333332, \pi \cdot 0.125\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}}{\frac{\pi}{4}} \]

   3. div-inv 96.2%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(-2, \pi \cdot 0.020833333333333332, \pi \cdot 0.125\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\color{blue}{\pi \cdot \frac{1}{4}}} \]

   4. metadata-eval 96.2%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(-2, \pi \cdot 0.020833333333333332, \pi \cdot 0.125\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\pi \cdot \color{blue}{0.25}} \]

10. Applied egg-rr 96.2%

    \[\leadsto -\color{blue}{\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(-2, \pi \cdot 0.020833333333333332, \pi \cdot 0.125\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\pi \cdot 0.25}} \]
11. Step-by-step derivation

12. Simplified 96.2%

    \[\leadsto -\color{blue}{\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{4}{f \cdot \pi}\right)\right)}{\pi \cdot 0.25}} \]

13. Taylor expanded in f around inf 1.6%

    \[\leadsto -\color{blue}{4 \cdot \frac{\log \left(0.08333333333333333 \cdot \pi\right) + -1 \cdot \log \left(\frac{1}{f}\right)}{\pi}} \]
14. Step-by-step derivation

    1. log-prod 1.6%

       \[\leadsto -4 \cdot \frac{\color{blue}{\left(\log 0.08333333333333333 + \log \pi\right)} + -1 \cdot \log \left(\frac{1}{f}\right)}{\pi} \]

    2. +-commutative 1.6%

       \[\leadsto -4 \cdot \frac{\color{blue}{\left(\log \pi + \log 0.08333333333333333\right)} + -1 \cdot \log \left(\frac{1}{f}\right)}{\pi} \]

    3. mul-1-neg 1.6%

       \[\leadsto -4 \cdot \frac{\left(\log \pi + \log 0.08333333333333333\right) + \color{blue}{\left(-\log \left(\frac{1}{f}\right)\right)}}{\pi} \]

    4. log-rec 1.6%

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

    5. remove-double-neg 1.6%

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

    6. log-prod 1.6%

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

    7. log-prod 1.6%

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

    8. associate-*l* 1.6%

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

    9. *-commutative 1.6%

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

15. Simplified 1.6%

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

16. Final simplification 1.6%

    \[\leadsto \frac{\log \left(\pi \cdot \left(f \cdot 0.08333333333333333\right)\right)}{\pi} \cdot \left(-4\right) \]

Alternative 5: 95.6% accurate, 3.3× speedup

Math

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

FPCore

(FPCore (f) :precision binary64 (* (log (/ 4.0 (* PI f))) (/ (- 4.0) PI)))

C

double code(double f) {
	return log((4.0 / (((double) M_PI) * f))) * (-4.0 / ((double) M_PI));
}

Java

public static double code(double f) {
	return Math.log((4.0 / (Math.PI * f))) * (-4.0 / Math.PI);
}

Python

def code(f):
	return math.log((4.0 / (math.pi * f))) * (-4.0 / math.pi)

Julia

function code(f)
	return Float64(log(Float64(4.0 / Float64(pi * f))) * Float64(Float64(-4.0) / pi))
end

MATLAB

function tmp = code(f)
	tmp = log((4.0 / (pi * f))) * (-4.0 / pi);
end

Wolfram

code[f_] := N[(N[Log[N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-4.0) / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.3%

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

2. Taylor expanded in f around 0 95.2%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{2}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}\right)} \]
3. Step-by-step derivation

   1. associate-/r* 95.2%

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

   2. distribute-rgt-out-- 95.2%

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

   3. metadata-eval 95.2%

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

4. Simplified 95.2%

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

5. Taylor expanded in f around 0 95.4%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f}{\pi}} \]
6. Step-by-step derivation

   1. associate-*r/ 95.4%

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

   2. neg-mul-1 95.4%

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

   3. sub-neg 95.4%

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

   4. associate-/l* 95.3%

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

   5. remove-double-neg 95.3%

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

   6. log-rec 95.3%

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

   7. mul-1-neg 95.3%

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

   8. mul-1-neg 95.3%

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

   9. log-rec 95.3%

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

   10. remove-double-neg 95.3%

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

7. Simplified 95.3%

   \[\leadsto -\color{blue}{\frac{4}{\frac{\pi}{\log \left(\frac{\frac{4}{\pi}}{f}\right)}}} \]
8. Step-by-step derivation

   1. associate-/r/ 95.2%

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

   2. div-inv 95.2%

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

   3. frac-times 95.2%

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

   4. metadata-eval 95.2%

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

9. Applied egg-rr 95.2%

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

10. Final simplification 95.2%

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

Alternative 6: 95.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{-4}{\frac{\pi}{\log \left(\frac{\frac{4}{\pi}}{f}\right)}} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (- 4.0) (/ PI (log (/ (/ 4.0 PI) f)))))

C

double code(double f) {
	return -4.0 / (((double) M_PI) / log(((4.0 / ((double) M_PI)) / f)));
}

Java

public static double code(double f) {
	return -4.0 / (Math.PI / Math.log(((4.0 / Math.PI) / f)));
}

Python

def code(f):
	return -4.0 / (math.pi / math.log(((4.0 / math.pi) / f)))

Julia

function code(f)
	return Float64(Float64(-4.0) / Float64(pi / log(Float64(Float64(4.0 / pi) / f))))
end

MATLAB

function tmp = code(f)
	tmp = -4.0 / (pi / log(((4.0 / pi) / f)));
end

Wolfram

code[f_] := N[((-4.0) / N[(Pi / N[Log[N[(N[(4.0 / Pi), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.3%

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

2. Taylor expanded in f around 0 95.2%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{2}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}\right)} \]
3. Step-by-step derivation

   1. associate-/r* 95.2%

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

   2. distribute-rgt-out-- 95.2%

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

   3. metadata-eval 95.2%

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

4. Simplified 95.2%

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

5. Taylor expanded in f around 0 95.4%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f}{\pi}} \]
6. Step-by-step derivation

   1. associate-*r/ 95.4%

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

   2. neg-mul-1 95.4%

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

   3. sub-neg 95.4%

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

   4. associate-/l* 95.3%

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

   5. remove-double-neg 95.3%

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

   6. log-rec 95.3%

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

   7. mul-1-neg 95.3%

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

   8. mul-1-neg 95.3%

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

   9. log-rec 95.3%

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

   10. remove-double-neg 95.3%

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

7. Simplified 95.3%

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

8. Final simplification 95.3%

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

Alternative 7: 95.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{-\log \left(\frac{4}{\pi \cdot f}\right)}{\pi \cdot 0.25} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (- (log (/ 4.0 (* PI f)))) (* PI 0.25)))

C

double code(double f) {
	return -log((4.0 / (((double) M_PI) * f))) / (((double) M_PI) * 0.25);
}

Java

public static double code(double f) {
	return -Math.log((4.0 / (Math.PI * f))) / (Math.PI * 0.25);
}

Python

def code(f):
	return -math.log((4.0 / (math.pi * f))) / (math.pi * 0.25)

Julia

function code(f)
	return Float64(Float64(-log(Float64(4.0 / Float64(pi * f)))) / Float64(pi * 0.25))
end

MATLAB

function tmp = code(f)
	tmp = -log((4.0 / (pi * f))) / (pi * 0.25);
end

Wolfram

code[f_] := N[((-N[Log[N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.3%

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

2. Taylor expanded in f around 0 95.2%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{2}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}\right)} \]
3. Step-by-step derivation

   1. associate-/r* 95.2%

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

   2. distribute-rgt-out-- 95.2%

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

   3. metadata-eval 95.2%

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

4. Simplified 95.2%

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

5. Taylor expanded in f around 0 95.4%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f}{\pi}} \]
6. Step-by-step derivation

   1. associate-*r/ 95.4%

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

   2. neg-mul-1 95.4%

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

   3. sub-neg 95.4%

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

   4. associate-/l* 95.3%

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

   5. remove-double-neg 95.3%

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

   6. log-rec 95.3%

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

   7. mul-1-neg 95.3%

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

   8. mul-1-neg 95.3%

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

   9. log-rec 95.3%

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

   10. remove-double-neg 95.3%

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

7. Simplified 95.3%

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

8. Taylor expanded in f around 0 95.4%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f}{\pi}} \]
9. Step-by-step derivation

   1. *-commutative 95.4%

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

   2. mul-1-neg 95.4%

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

   3. log-rec 95.4%

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

   4. log-prod 95.4%

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

   5. metadata-eval 95.4%

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

   6. associate-/r* 95.4%

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

   7. *-commutative 95.4%

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

   8. associate-*l/ 95.4%

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

   9. associate-*r/ 95.4%

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

   10. metadata-eval 95.4%

       \[\leadsto -\frac{\log \left(\frac{\frac{\color{blue}{2}}{f}}{\pi \cdot 0.5}\right)}{\pi} \cdot 4 \]

   11. associate-/r* 95.4%

       \[\leadsto -\frac{\log \color{blue}{\left(\frac{2}{f \cdot \left(\pi \cdot 0.5\right)}\right)}}{\pi} \cdot 4 \]

   12. metadata-eval 95.4%

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

   13. times-frac 95.4%

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

10. Simplified 95.4%

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

11. Final simplification 95.4%

    \[\leadsto \frac{-\log \left(\frac{4}{\pi \cdot f}\right)}{\pi \cdot 0.25} \]

Reproduce

herbie shell --seed 2023292 
(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)))))))))