VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 99.1%

Time: 19.1s

Alternatives: 7

Speedup: 4.8×

• could not determine a ground truth

  1. f = 1006855438.9242038

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, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\frac{1}{\frac{1}{\tanh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}}\right)}{\pi \cdot 0.25} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/ (log (/ 1.0 (/ 1.0 (tanh (* PI (* 0.25 f)))))) (* PI 0.25)))

C

double code(double f) {
	return log((1.0 / (1.0 / tanh((((double) M_PI) * (0.25 * f)))))) / (((double) M_PI) * 0.25);
}

Java

public static double code(double f) {
	return Math.log((1.0 / (1.0 / Math.tanh((Math.PI * (0.25 * f)))))) / (Math.PI * 0.25);
}

Python

def code(f):
	return math.log((1.0 / (1.0 / math.tanh((math.pi * (0.25 * f)))))) / (math.pi * 0.25)

Julia

function code(f)
	return Float64(log(Float64(1.0 / Float64(1.0 / tanh(Float64(pi * Float64(0.25 * f)))))) / Float64(pi * 0.25))
end

MATLAB

function tmp = code(f)
	tmp = log((1.0 / (1.0 / tanh((pi * (0.25 * f)))))) / (pi * 0.25);
end

Wolfram

code[f_] := N[(N[Log[N[(1.0 / N[(1.0 / N[Tanh[N[(Pi * N[(0.25 * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.0%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

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

   2. un-div-inv N/A

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

   3. distribute-neg-frac N/A

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

   4. /-lowering-/.f64 N/A

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

4. Applied egg-rr 99.5%

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

   1. tanh-def-b N/A

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

   2. clear-num N/A

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

   3. /-lowering-/.f64 N/A

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

   4. clear-num N/A

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

   5. tanh-def-b N/A

      \[\leadsto \frac{\log \left(\frac{1}{\frac{1}{\color{blue}{\tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right)}}}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \frac{\log \left(\frac{1}{\color{blue}{\frac{1}{\tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right)}}}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   7. tanh-lowering-tanh.f64 N/A

      \[\leadsto \frac{\log \left(\frac{1}{\frac{1}{\color{blue}{\tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right)}}}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   8. associate-*l* N/A

      \[\leadsto \frac{\log \left(\frac{1}{\frac{1}{\tanh \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{4} \cdot f\right)\right)}}}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \frac{\log \left(\frac{1}{\frac{1}{\tanh \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{4} \cdot f\right)\right)}}}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   10. PI-lowering-PI.f64 N/A

       \[\leadsto \frac{\log \left(\frac{1}{\frac{1}{\tanh \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{1}{4} \cdot f\right)\right)}}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   11. *-lowering-*.f64 99.5

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

6. Applied egg-rr 99.5%

   \[\leadsto \frac{\log \color{blue}{\left(\frac{1}{\frac{1}{\tanh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}}\right)}}{\pi \cdot 0.25} \]
7. Add Preprocessing

Alternative 2: 99.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.0%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

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

   2. un-div-inv N/A

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

   3. distribute-neg-frac N/A

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

   4. /-lowering-/.f64 N/A

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

4. Applied egg-rr 99.5%

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

5. Final simplification 99.5%

   \[\leadsto \frac{\log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\pi \cdot 0.25} \]
6. Add Preprocessing

Alternative 3: 96.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\frac{\mathsf{fma}\left(f, \mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.08333333333333333, 0\right), 0\right), \frac{4}{\pi}\right)}{f}\right)}{\mathsf{fma}\left(\pi, -0.25, 0\right)} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (log (/ (fma f (fma f (fma PI 0.08333333333333333 0.0) 0.0) (/ 4.0 PI)) f))
  (fma PI -0.25 0.0)))

C

double code(double f) {
	return log((fma(f, fma(f, fma(((double) M_PI), 0.08333333333333333, 0.0), 0.0), (4.0 / ((double) M_PI))) / f)) / fma(((double) M_PI), -0.25, 0.0);
}

Julia

function code(f)
	return Float64(log(Float64(fma(f, fma(f, fma(pi, 0.08333333333333333, 0.0), 0.0), Float64(4.0 / pi)) / f)) / fma(pi, -0.25, 0.0))
end

Wolfram

code[f_] := N[(N[Log[N[(N[(f * N[(f * N[(Pi * 0.08333333333333333 + 0.0), $MachinePrecision] + 0.0), $MachinePrecision] + N[(4.0 / Pi), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision] / N[(Pi * -0.25 + 0.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.0%

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

3. Taylor expanded in f around 0

   \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \color{blue}{\left(\frac{f \cdot \left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} + f \cdot \left(\frac{1}{16} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} - 2 \cdot \frac{\frac{1}{384} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{384} \cdot {\mathsf{PI}\left(\right)}^{3}}{{\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right)\right) + 2 \cdot \frac{1}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}{f}\right)}\right) \]

5. Simplified 96.5%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{\mathsf{fma}\left(f, \mathsf{fma}\left(f, \mathsf{fma}\left(\pi \cdot 0.020833333333333332, -2, \mathsf{fma}\left(\pi, 0.125, 0\right)\right), 0\right), \frac{4}{\pi}\right)}{f}\right)} \]
6. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{4}{\mathsf{PI}\left(\right)}} \cdot \log \left(\frac{f \cdot \left(f \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{48}\right) \cdot -2 + \left(\mathsf{PI}\left(\right) \cdot \frac{1}{8} + 0\right)\right) + 0\right) + \frac{4}{\mathsf{PI}\left(\right)}}{f}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\log \left(\frac{f \cdot \left(f \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{48}\right) \cdot -2 + \left(\mathsf{PI}\left(\right) \cdot \frac{1}{8} + 0\right)\right) + 0\right) + \frac{4}{\mathsf{PI}\left(\right)}}{f}\right) \cdot \frac{4}{\mathsf{PI}\left(\right)}}\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{f \cdot \left(f \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{48}\right) \cdot -2 + \left(\mathsf{PI}\left(\right) \cdot \frac{1}{8} + 0\right)\right) + 0\right) + \frac{4}{\mathsf{PI}\left(\right)}}{f}\right) \cdot \color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}}}\right) \]

   4. div-inv N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{f \cdot \left(f \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{48}\right) \cdot -2 + \left(\mathsf{PI}\left(\right) \cdot \frac{1}{8} + 0\right)\right) + 0\right) + \frac{4}{\mathsf{PI}\left(\right)}}{f}\right) \cdot \frac{1}{\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}}}\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{f \cdot \left(f \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{48}\right) \cdot -2 + \left(\mathsf{PI}\left(\right) \cdot \frac{1}{8} + 0\right)\right) + 0\right) + \frac{4}{\mathsf{PI}\left(\right)}}{f}\right) \cdot \frac{1}{\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{4}}}\right) \]

7. Applied egg-rr 96.7%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{\mathsf{fma}\left(f, \mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.08333333333333333, 0\right), 0\right), \frac{4}{\pi}\right)}{f}\right)}{\mathsf{fma}\left(\pi, -0.25, 0\right)}} \]
8. Add Preprocessing

Alternative 4: 96.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \log \left(\frac{f}{\mathsf{fma}\left(f, \mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.08333333333333333, 0\right), 0\right), \frac{4}{\pi}\right)}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (*
   4.0
   (log (/ f (fma f (fma f (fma PI 0.08333333333333333 0.0) 0.0) (/ 4.0 PI)))))
  PI))

C

double code(double f) {
	return (4.0 * log((f / fma(f, fma(f, fma(((double) M_PI), 0.08333333333333333, 0.0), 0.0), (4.0 / ((double) M_PI)))))) / ((double) M_PI);
}

Julia

function code(f)
	return Float64(Float64(4.0 * log(Float64(f / fma(f, fma(f, fma(pi, 0.08333333333333333, 0.0), 0.0), Float64(4.0 / pi))))) / pi)
end

Wolfram

code[f_] := N[(N[(4.0 * N[Log[N[(f / N[(f * N[(f * N[(Pi * 0.08333333333333333 + 0.0), $MachinePrecision] + 0.0), $MachinePrecision] + N[(4.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 8.0%

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

3. Taylor expanded in f around 0

   \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \color{blue}{\left(\frac{f \cdot \left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} + f \cdot \left(\frac{1}{16} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} - 2 \cdot \frac{\frac{1}{384} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{384} \cdot {\mathsf{PI}\left(\right)}^{3}}{{\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right)\right) + 2 \cdot \frac{1}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}{f}\right)}\right) \]

5. Simplified 96.5%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{\mathsf{fma}\left(f, \mathsf{fma}\left(f, \mathsf{fma}\left(\pi \cdot 0.020833333333333332, -2, \mathsf{fma}\left(\pi, 0.125, 0\right)\right), 0\right), \frac{4}{\pi}\right)}{f}\right)} \]
6. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{4}{\mathsf{PI}\left(\right)}} \cdot \log \left(\frac{f \cdot \left(f \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{48}\right) \cdot -2 + \left(\mathsf{PI}\left(\right) \cdot \frac{1}{8} + 0\right)\right) + 0\right) + \frac{4}{\mathsf{PI}\left(\right)}}{f}\right)\right) \]

   2. distribute-rgt-neg-in N/A

      \[\leadsto \color{blue}{\frac{4}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{neg}\left(\log \left(\frac{f \cdot \left(f \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{48}\right) \cdot -2 + \left(\mathsf{PI}\left(\right) \cdot \frac{1}{8} + 0\right)\right) + 0\right) + \frac{4}{\mathsf{PI}\left(\right)}}{f}\right)\right)\right)} \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{4 \cdot \left(\mathsf{neg}\left(\log \left(\frac{f \cdot \left(f \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{48}\right) \cdot -2 + \left(\mathsf{PI}\left(\right) \cdot \frac{1}{8} + 0\right)\right) + 0\right) + \frac{4}{\mathsf{PI}\left(\right)}}{f}\right)\right)\right)}{\mathsf{PI}\left(\right)}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{4 \cdot \left(\mathsf{neg}\left(\log \left(\frac{f \cdot \left(f \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{48}\right) \cdot -2 + \left(\mathsf{PI}\left(\right) \cdot \frac{1}{8} + 0\right)\right) + 0\right) + \frac{4}{\mathsf{PI}\left(\right)}}{f}\right)\right)\right)}{\mathsf{PI}\left(\right)}} \]

7. Applied egg-rr 96.7%

   \[\leadsto \color{blue}{\frac{4 \cdot \log \left(\frac{f}{\mathsf{fma}\left(f, \mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.08333333333333333, 0\right), 0\right), \frac{4}{\pi}\right)}\right)}{\pi}} \]
8. Add Preprocessing

Alternative 5: 96.6% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(f \cdot \left(\pi \cdot \mathsf{fma}\left(-0.005208333333333333 \cdot \left(f \cdot f\right), \pi \cdot \pi, 0.25\right)\right)\right)}{\pi \cdot 0.25} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (log (* f (* PI (fma (* -0.005208333333333333 (* f f)) (* PI PI) 0.25))))
  (* PI 0.25)))

C

double code(double f) {
	return log((f * (((double) M_PI) * fma((-0.005208333333333333 * (f * f)), (((double) M_PI) * ((double) M_PI)), 0.25)))) / (((double) M_PI) * 0.25);
}

Julia

function code(f)
	return Float64(log(Float64(f * Float64(pi * fma(Float64(-0.005208333333333333 * Float64(f * f)), Float64(pi * pi), 0.25)))) / Float64(pi * 0.25))
end

Wolfram

code[f_] := N[(N[Log[N[(f * N[(Pi * N[(N[(-0.005208333333333333 * N[(f * f), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.0%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

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

   2. un-div-inv N/A

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

   3. distribute-neg-frac N/A

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

   4. /-lowering-/.f64 N/A

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

4. Applied egg-rr 99.5%

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

5. Taylor expanded in f around 0

   \[\leadsto \frac{\log \color{blue}{\left(f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) + f \cdot \left(\frac{1}{2} \cdot \left(\frac{-1}{16} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{16} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + f \cdot \left(\frac{1}{2} \cdot \left(\frac{-1}{4} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{16} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{32} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + \left(\frac{-1}{128} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{1}{192} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) - \frac{1}{128} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)}}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

7. Simplified 96.6%

   \[\leadsto \frac{\log \color{blue}{\left(f \cdot \mathsf{fma}\left(\mathsf{fma}\left(\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -0.03125\right), -0.125, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -0.009114583333333334\right), f \cdot f, 0.25 \cdot \pi\right)\right)}}{\pi \cdot 0.25} \]

8. Taylor expanded in f around inf

   \[\leadsto \frac{\log \left(f \cdot \color{blue}{\left({f}^{2} \cdot \left(\frac{-7}{768} \cdot {\mathsf{PI}\left(\right)}^{3} + \left(\frac{1}{256} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{{f}^{2}}\right)\right)\right)}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \frac{\log \left(f \cdot \color{blue}{\left({f}^{2} \cdot \left(\frac{-7}{768} \cdot {\mathsf{PI}\left(\right)}^{3} + \left(\frac{1}{256} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{{f}^{2}}\right)\right)\right)}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   2. unpow2 N/A

      \[\leadsto \frac{\log \left(f \cdot \left(\color{blue}{\left(f \cdot f\right)} \cdot \left(\frac{-7}{768} \cdot {\mathsf{PI}\left(\right)}^{3} + \left(\frac{1}{256} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{{f}^{2}}\right)\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \frac{\log \left(f \cdot \left(\color{blue}{\left(f \cdot f\right)} \cdot \left(\frac{-7}{768} \cdot {\mathsf{PI}\left(\right)}^{3} + \left(\frac{1}{256} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{{f}^{2}}\right)\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   4. associate-+r+ N/A

      \[\leadsto \frac{\log \left(f \cdot \left(\left(f \cdot f\right) \cdot \color{blue}{\left(\left(\frac{-7}{768} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{1}{256} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{{f}^{2}}\right)}\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   5. distribute-rgt-out N/A

      \[\leadsto \frac{\log \left(f \cdot \left(\left(f \cdot f\right) \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{3} \cdot \left(\frac{-7}{768} + \frac{1}{256}\right)} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{{f}^{2}}\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\log \left(f \cdot \left(\left(f \cdot f\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \color{blue}{\frac{-1}{192}} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{{f}^{2}}\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\log \left(f \cdot \left(\left(f \cdot f\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \color{blue}{\left(\frac{-1}{128} + \frac{1}{384}\right)} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{{f}^{2}}\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   8. cube-mult N/A

      \[\leadsto \frac{\log \left(f \cdot \left(\left(f \cdot f\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left(\frac{-1}{128} + \frac{1}{384}\right) + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{{f}^{2}}\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\log \left(f \cdot \left(\left(f \cdot f\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right) \cdot \left(\frac{-1}{128} + \frac{1}{384}\right) + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{{f}^{2}}\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   10. associate-*l* N/A

       \[\leadsto \frac{\log \left(f \cdot \left(\left(f \cdot f\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{128} + \frac{1}{384}\right)\right)} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{{f}^{2}}\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   11. associate-*r/ N/A

       \[\leadsto \frac{\log \left(f \cdot \left(\left(f \cdot f\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{128} + \frac{1}{384}\right)\right) + \color{blue}{\frac{\frac{1}{4} \cdot \mathsf{PI}\left(\right)}{{f}^{2}}}\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   12. *-commutative N/A

       \[\leadsto \frac{\log \left(f \cdot \left(\left(f \cdot f\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{128} + \frac{1}{384}\right)\right) + \frac{\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}}}{{f}^{2}}\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   13. associate-/l* N/A

       \[\leadsto \frac{\log \left(f \cdot \left(\left(f \cdot f\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{128} + \frac{1}{384}\right)\right) + \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{\frac{1}{4}}{{f}^{2}}}\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   14. distribute-lft-out N/A

       \[\leadsto \frac{\log \left(f \cdot \left(\left(f \cdot f\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{128} + \frac{1}{384}\right) + \frac{\frac{1}{4}}{{f}^{2}}\right)\right)}\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \frac{\log \left(f \cdot \left(\left(f \cdot f\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{128} + \frac{1}{384}\right) + \frac{\frac{1}{4}}{{f}^{2}}\right)\right)}\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   16. PI-lowering-PI.f64 N/A

       \[\leadsto \frac{\log \left(f \cdot \left(\left(f \cdot f\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{128} + \frac{1}{384}\right) + \frac{\frac{1}{4}}{{f}^{2}}\right)\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   17. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{\log \left(f \cdot \left(\left(f \cdot f\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{fma}\left({\mathsf{PI}\left(\right)}^{2}, \frac{-1}{128} + \frac{1}{384}, \frac{\frac{1}{4}}{{f}^{2}}\right)}\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

10. Simplified 42.6%

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

11. Taylor expanded in f around 0

    \[\leadsto \frac{\log \left(f \cdot \color{blue}{\left(\frac{-1}{192} \cdot \left({f}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]
12. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \frac{\log \left(f \cdot \left(\color{blue}{\left(\frac{-1}{192} \cdot {f}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}} + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

    2. unpow3 N/A

       \[\leadsto \frac{\log \left(f \cdot \left(\left(\frac{-1}{192} \cdot {f}^{2}\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)} + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

    3. unpow2 N/A

       \[\leadsto \frac{\log \left(f \cdot \left(\left(\frac{-1}{192} \cdot {f}^{2}\right) \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

    4. associate-*r* N/A

       \[\leadsto \frac{\log \left(f \cdot \left(\color{blue}{\left(\left(\frac{-1}{192} \cdot {f}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \mathsf{PI}\left(\right)} + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

    5. distribute-rgt-out N/A

       \[\leadsto \frac{\log \left(f \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\left(\frac{-1}{192} \cdot {f}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{4}\right)\right)}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \frac{\log \left(f \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\left(\frac{-1}{192} \cdot {f}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{4}\right)\right)}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

    7. PI-lowering-PI.f64 N/A

       \[\leadsto \frac{\log \left(f \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\left(\frac{-1}{192} \cdot {f}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{4}\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

    8. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{\log \left(f \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{192} \cdot {f}^{2}, {\mathsf{PI}\left(\right)}^{2}, \frac{1}{4}\right)}\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \frac{\log \left(f \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{192} \cdot {f}^{2}}, {\mathsf{PI}\left(\right)}^{2}, \frac{1}{4}\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

    10. unpow2 N/A

        \[\leadsto \frac{\log \left(f \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{192} \cdot \color{blue}{\left(f \cdot f\right)}, {\mathsf{PI}\left(\right)}^{2}, \frac{1}{4}\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \frac{\log \left(f \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{192} \cdot \color{blue}{\left(f \cdot f\right)}, {\mathsf{PI}\left(\right)}^{2}, \frac{1}{4}\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

    12. unpow2 N/A

        \[\leadsto \frac{\log \left(f \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{192} \cdot \left(f \cdot f\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, \frac{1}{4}\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

    13. *-lowering-*.f64 N/A

        \[\leadsto \frac{\log \left(f \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{192} \cdot \left(f \cdot f\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, \frac{1}{4}\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

    14. PI-lowering-PI.f64 N/A

        \[\leadsto \frac{\log \left(f \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{192} \cdot \left(f \cdot f\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), \frac{1}{4}\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

    15. PI-lowering-PI.f64 96.6

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

13. Simplified 96.6%

    \[\leadsto \frac{\log \left(f \cdot \color{blue}{\left(\pi \cdot \mathsf{fma}\left(-0.005208333333333333 \cdot \left(f \cdot f\right), \pi \cdot \pi, 0.25\right)\right)}\right)}{\pi \cdot 0.25} \]
14. Add Preprocessing

Alternative 6: 96.2% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.0%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

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

   2. un-div-inv N/A

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

   3. distribute-neg-frac N/A

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

   4. /-lowering-/.f64 N/A

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

4. Applied egg-rr 99.5%

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

5. Taylor expanded in f around 0

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

   1. *-lowering-*.f64 N/A

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

   2. *-lowering-*.f64 N/A

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

   3. PI-lowering-PI.f64 96.0

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

7. Simplified 96.0%

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

8. Final simplification 96.0%

   \[\leadsto \frac{\log \left(0.25 \cdot \left(\pi \cdot f\right)\right)}{\pi \cdot 0.25} \]
9. Add Preprocessing

Alternative 7: 96.0% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.0%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

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

   2. un-div-inv N/A

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

   3. distribute-neg-frac N/A

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

   4. /-lowering-/.f64 N/A

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

4. Applied egg-rr 99.5%

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

5. Taylor expanded in f around 0

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

   1. *-lowering-*.f64 N/A

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

   2. *-lowering-*.f64 N/A

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

   3. PI-lowering-PI.f64 96.0

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

7. Simplified 96.0%

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

   1. clear-num N/A

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

   2. *-commutative N/A

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

   3. associate-/r/ N/A

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

   4. associate-/r* N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. sum-log N/A

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

   10. *-commutative N/A

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

   11. metadata-eval N/A

       \[\leadsto \frac{4}{\mathsf{PI}\left(\right)} \cdot \left(\log f + \log \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{2}\right)}\right)\right) \]

   12. associate-*l* N/A

       \[\leadsto \frac{4}{\mathsf{PI}\left(\right)} \cdot \left(\log f + \log \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{2}\right)}\right) \]

   13. +-rgt-identity N/A

       \[\leadsto \frac{4}{\mathsf{PI}\left(\right)} \cdot \left(\log f + \log \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + 0\right)} \cdot \frac{1}{2}\right)\right) \]

   14. log-prod N/A

       \[\leadsto \frac{4}{\mathsf{PI}\left(\right)} \cdot \color{blue}{\log \left(f \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + 0\right) \cdot \frac{1}{2}\right)\right)} \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \color{blue}{\frac{4}{\mathsf{PI}\left(\right)} \cdot \log \left(f \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + 0\right) \cdot \frac{1}{2}\right)\right)} \]

9. Applied egg-rr 95.9%

   \[\leadsto \color{blue}{\frac{4}{\pi} \cdot \log \left(f \cdot \mathsf{fma}\left(\pi, 0.25, 0\right)\right)} \]
10. Step-by-step derivation

    1. +-rgt-identity N/A

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

    2. associate-*r* N/A

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

    3. *-lowering-*.f64 N/A

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

    4. *-commutative N/A

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

    5. *-lowering-*.f64 N/A

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

    6. PI-lowering-PI.f64 95.9

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

11. Applied egg-rr 95.9%

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

12. Final simplification 95.9%

    \[\leadsto \frac{4}{\pi} \cdot \log \left(0.25 \cdot \left(\pi \cdot f\right)\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(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)))))))))