VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.6% → 98.9%

Time: 20.4s

Alternatives: 6

Speedup: 4.9×

• 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: 6.6% 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: 98.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{\sqrt{\pi}}\\ t\_0 \cdot \left(\log \tanh \left(0.25 \cdot \left(f \cdot \pi\right)\right) \cdot t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (/ 2.0 (sqrt PI))))
   (* t_0 (* (log (tanh (* 0.25 (* f PI)))) t_0))))

C

double code(double f) {
	double t_0 = 2.0 / sqrt(((double) M_PI));
	return t_0 * (log(tanh((0.25 * (f * ((double) M_PI))))) * t_0);
}

Java

public static double code(double f) {
	double t_0 = 2.0 / Math.sqrt(Math.PI);
	return t_0 * (Math.log(Math.tanh((0.25 * (f * Math.PI)))) * t_0);
}

Python

def code(f):
	t_0 = 2.0 / math.sqrt(math.pi)
	return t_0 * (math.log(math.tanh((0.25 * (f * math.pi)))) * t_0)

Julia

function code(f)
	t_0 = Float64(2.0 / sqrt(pi))
	return Float64(t_0 * Float64(log(tanh(Float64(0.25 * Float64(f * pi)))) * t_0))
end

MATLAB

function tmp = code(f)
	t_0 = 2.0 / sqrt(pi);
	tmp = t_0 * (log(tanh((0.25 * (f * pi)))) * t_0);
end

Wolfram

code[f_] := Block[{t$95$0 = N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * N[(N[Log[N[Tanh[N[(0.25 * N[(f * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.9%

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

   1. distribute-lft-neg-in 6.9%

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

   2. distribute-neg-frac2 6.9%

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

   3. associate-*l/ 6.9%

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

   4. *-lft-identity 6.9%

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

3. Simplified 6.9%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{{\left(e^{\frac{\pi}{-4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{f}}{{\left(e^{\frac{\pi}{4}}\right)}^{f} - {\left(e^{\frac{\pi}{-4}}\right)}^{f}}\right)}{\frac{\pi}{-4}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num 6.9%

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

   2. log-div 6.9%

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

   3. metadata-eval 6.9%

      \[\leadsto \frac{\color{blue}{0} - \log \left(\frac{{\left(e^{\frac{\pi}{4}}\right)}^{f} - {\left(e^{\frac{\pi}{-4}}\right)}^{f}}{{\left(e^{\frac{\pi}{-4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{f}}\right)}{\frac{\pi}{-4}} \]

6. Applied egg-rr 6.9%

   \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{{\left(e^{f}\right)}^{\left(\pi \cdot 0.25\right)} - {\left({\left(e^{\pi}\right)}^{-0.25}\right)}^{f}}{{\left({\left(e^{\pi}\right)}^{-0.25}\right)}^{f} + {\left(e^{f}\right)}^{\left(\pi \cdot 0.25\right)}}\right)}}{\frac{\pi}{-4}} \]
7. Step-by-step derivation

   1. neg-sub0 6.9%

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

   2. exp-prod 6.9%

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

   3. *-commutative 6.9%

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

   4. exp-prod 6.9%

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

   5. exp-prod 6.9%

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

   6. metadata-eval 6.9%

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

   7. distribute-rgt-neg-in 6.9%

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

   8. distribute-lft-neg-out 6.9%

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

   9. +-commutative 6.9%

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

8. Simplified 99.4%

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

   1. add-sqr-sqrt 99.4%

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

   2. log-prod 99.4%

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

10. Applied egg-rr 99.4%

    \[\leadsto \frac{-\color{blue}{\left(\log \left(\sqrt{\tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}\right) + \log \left(\sqrt{\tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}\right)\right)}}{\frac{\pi}{-4}} \]
11. Step-by-step derivation

    1. count-2 99.4%

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

    2. associate-*r* 99.4%

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

    3. *-commutative 99.4%

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

12. Simplified 99.4%

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

14. Applied egg-rr 99.4%

    \[\leadsto \color{blue}{\left(\log \tanh \left(0.25 \cdot \left(f \cdot \pi\right)\right) \cdot \frac{2}{\sqrt{\pi}}\right) \cdot \frac{2}{\sqrt{\pi}}} \]

15. Final simplification 99.4%

    \[\leadsto \frac{2}{\sqrt{\pi}} \cdot \left(\log \tanh \left(0.25 \cdot \left(f \cdot \pi\right)\right) \cdot \frac{2}{\sqrt{\pi}}\right) \]
16. Add Preprocessing

Alternative 2: 99.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{8 \cdot \log \left(\sqrt{\tanh \left(0.25 \cdot \left(f \cdot \pi\right)\right)}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/ (* 8.0 (log (sqrt (tanh (* 0.25 (* f PI)))))) PI))

C

double code(double f) {
	return (8.0 * log(sqrt(tanh((0.25 * (f * ((double) M_PI))))))) / ((double) M_PI);
}

Java

public static double code(double f) {
	return (8.0 * Math.log(Math.sqrt(Math.tanh((0.25 * (f * Math.PI)))))) / Math.PI;
}

Python

def code(f):
	return (8.0 * math.log(math.sqrt(math.tanh((0.25 * (f * math.pi)))))) / math.pi

Julia

function code(f)
	return Float64(Float64(8.0 * log(sqrt(tanh(Float64(0.25 * Float64(f * pi)))))) / pi)
end

MATLAB

function tmp = code(f)
	tmp = (8.0 * log(sqrt(tanh((0.25 * (f * pi)))))) / pi;
end

Wolfram

code[f_] := N[(N[(8.0 * N[Log[N[Sqrt[N[Tanh[N[(0.25 * N[(f * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 6.9%

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

   1. distribute-lft-neg-in 6.9%

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

   2. distribute-neg-frac2 6.9%

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

   3. associate-*l/ 6.9%

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

   4. *-lft-identity 6.9%

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

3. Simplified 6.9%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{{\left(e^{\frac{\pi}{-4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{f}}{{\left(e^{\frac{\pi}{4}}\right)}^{f} - {\left(e^{\frac{\pi}{-4}}\right)}^{f}}\right)}{\frac{\pi}{-4}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num 6.9%

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

   2. log-div 6.9%

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

   3. metadata-eval 6.9%

      \[\leadsto \frac{\color{blue}{0} - \log \left(\frac{{\left(e^{\frac{\pi}{4}}\right)}^{f} - {\left(e^{\frac{\pi}{-4}}\right)}^{f}}{{\left(e^{\frac{\pi}{-4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{f}}\right)}{\frac{\pi}{-4}} \]

6. Applied egg-rr 6.9%

   \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{{\left(e^{f}\right)}^{\left(\pi \cdot 0.25\right)} - {\left({\left(e^{\pi}\right)}^{-0.25}\right)}^{f}}{{\left({\left(e^{\pi}\right)}^{-0.25}\right)}^{f} + {\left(e^{f}\right)}^{\left(\pi \cdot 0.25\right)}}\right)}}{\frac{\pi}{-4}} \]
7. Step-by-step derivation

   1. neg-sub0 6.9%

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

   2. exp-prod 6.9%

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

   3. *-commutative 6.9%

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

   4. exp-prod 6.9%

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

   5. exp-prod 6.9%

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

   6. metadata-eval 6.9%

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

   7. distribute-rgt-neg-in 6.9%

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

   8. distribute-lft-neg-out 6.9%

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

   9. +-commutative 6.9%

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

8. Simplified 99.4%

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

   1. add-sqr-sqrt 99.4%

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

   2. log-prod 99.4%

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

10. Applied egg-rr 99.4%

    \[\leadsto \frac{-\color{blue}{\left(\log \left(\sqrt{\tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}\right) + \log \left(\sqrt{\tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}\right)\right)}}{\frac{\pi}{-4}} \]
11. Step-by-step derivation

    1. count-2 99.4%

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

    2. associate-*r* 99.4%

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

    3. *-commutative 99.4%

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

12. Simplified 99.4%

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

13. Taylor expanded in f around inf 6.9%

    \[\leadsto \color{blue}{8 \cdot \frac{\log \left(\sqrt{\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} - \frac{1}{e^{0.25 \cdot \left(f \cdot \pi\right)}}}{e^{0.25 \cdot \left(f \cdot \pi\right)} + \frac{1}{e^{0.25 \cdot \left(f \cdot \pi\right)}}}}\right)}{\pi}} \]
14. Step-by-step derivation

    1. associate-*r/ 6.9%

       \[\leadsto \color{blue}{\frac{8 \cdot \log \left(\sqrt{\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} - \frac{1}{e^{0.25 \cdot \left(f \cdot \pi\right)}}}{e^{0.25 \cdot \left(f \cdot \pi\right)} + \frac{1}{e^{0.25 \cdot \left(f \cdot \pi\right)}}}}\right)}{\pi}} \]

15. Simplified 99.4%

    \[\leadsto \color{blue}{\frac{8 \cdot \log \left(\sqrt{\tanh \left(\left(f \cdot \pi\right) \cdot 0.25\right)}\right)}{\pi}} \]

16. Final simplification 99.4%

    \[\leadsto \frac{8 \cdot \log \left(\sqrt{\tanh \left(0.25 \cdot \left(f \cdot \pi\right)\right)}\right)}{\pi} \]
17. Add Preprocessing

Alternative 3: 99.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. distribute-lft-neg-in 6.9%

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

   2. distribute-neg-frac2 6.9%

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

   3. associate-*l/ 6.9%

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

   4. *-lft-identity 6.9%

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

3. Simplified 6.9%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{{\left(e^{\frac{\pi}{-4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{f}}{{\left(e^{\frac{\pi}{4}}\right)}^{f} - {\left(e^{\frac{\pi}{-4}}\right)}^{f}}\right)}{\frac{\pi}{-4}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num 6.9%

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

   2. log-div 6.9%

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

   3. metadata-eval 6.9%

      \[\leadsto \frac{\color{blue}{0} - \log \left(\frac{{\left(e^{\frac{\pi}{4}}\right)}^{f} - {\left(e^{\frac{\pi}{-4}}\right)}^{f}}{{\left(e^{\frac{\pi}{-4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{f}}\right)}{\frac{\pi}{-4}} \]

6. Applied egg-rr 6.9%

   \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{{\left(e^{f}\right)}^{\left(\pi \cdot 0.25\right)} - {\left({\left(e^{\pi}\right)}^{-0.25}\right)}^{f}}{{\left({\left(e^{\pi}\right)}^{-0.25}\right)}^{f} + {\left(e^{f}\right)}^{\left(\pi \cdot 0.25\right)}}\right)}}{\frac{\pi}{-4}} \]
7. Step-by-step derivation

   1. neg-sub0 6.9%

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

   2. exp-prod 6.9%

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

   3. *-commutative 6.9%

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

   4. exp-prod 6.9%

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

   5. exp-prod 6.9%

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

   6. metadata-eval 6.9%

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

   7. distribute-rgt-neg-in 6.9%

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

   8. distribute-lft-neg-out 6.9%

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

   9. +-commutative 6.9%

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

8. Simplified 99.4%

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

9. Taylor expanded in f around inf 6.8%

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

11. Simplified 99.4%

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

Alternative 4: 96.0% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

3. Simplified 99.3%

   \[\leadsto \color{blue}{\log \left(\frac{1}{\mathsf{expm1}\left(\left(\pi \cdot f\right) \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Taylor expanded in f around 0 94.8%

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

   1. associate-*r/ 94.9%

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

7. Applied egg-rr 94.9%

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

Alternative 5: 95.9% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

3. Simplified 99.3%

   \[\leadsto \color{blue}{\log \left(\frac{1}{\mathsf{expm1}\left(\left(\pi \cdot f\right) \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Taylor expanded in f around 0 94.8%

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

Alternative 6: 1.6% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

3. Simplified 99.3%

   \[\leadsto \color{blue}{\log \left(\frac{1}{\mathsf{expm1}\left(\left(\pi \cdot f\right) \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Taylor expanded in f around 0 94.8%

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

   1. associate-/r* 94.8%

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

7. Simplified 94.8%

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

   1. associate-/r* 94.8%

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

   2. clear-num 94.8%

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

   3. un-div-inv 94.9%

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

   4. add-sqr-sqrt 0.0%

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

   5. sqrt-unprod 1.6%

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

   6. div-inv 1.6%

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

   7. div-inv 1.6%

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

   8. swap-sqr 1.6%

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

   9. metadata-eval 1.6%

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

   10. metadata-eval 1.6%

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

   11. metadata-eval 1.6%

       \[\leadsto \frac{\log \left(\frac{4}{f \cdot \pi}\right)}{\sqrt{\left(\pi \cdot \pi\right) \cdot \color{blue}{0.0625}}} \]

   12. metadata-eval 1.6%

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

   13. swap-sqr 1.6%

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

   14. sqrt-unprod 1.6%

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

   15. add-sqr-sqrt 1.6%

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

   16. *-commutative 1.6%

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

9. Applied egg-rr 1.6%

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

    1. *-commutative 1.6%

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

11. Simplified 1.6%

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

    1. frac-2neg 1.6%

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

    2. distribute-frac-neg2 1.6%

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

    3. neg-log 1.6%

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

    4. clear-num 1.6%

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

    5. associate-/l* 1.6%

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

    6. *-commutative 1.6%

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

13. Applied egg-rr 1.6%

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

    1. distribute-neg-frac 1.6%

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

    2. neg-mul-1 1.6%

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

    3. times-frac 1.6%

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

    4. metadata-eval 1.6%

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

    5. associate-*r/ 1.6%

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

15. Simplified 1.6%

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

Reproduce

herbie shell --seed 2024145 
(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)))))))))