VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.3% → 98.9%

Time: 18.6s

Alternatives: 5

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: 7.3% 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, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.5%

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

   1. *-commutative N/A

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

   2. un-div-inv N/A

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

   3. distribute-neg-frac N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{4} \cdot f\right)}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{4} \cdot f\right)}}\right)\right)\right), \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{4}\right)}\right) \]

4. Applied egg-rr 99.1%

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

Alternative 2: 96.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\frac{\frac{2}{\pi \cdot 0.5} + f \cdot \left(f \cdot \left(0.0625 \cdot \left(\pi \cdot 2\right) + -2 \cdot \left(\left(\pi \cdot \left(\pi \cdot 2\right)\right) \cdot \frac{0.010416666666666666}{\pi}\right)\right)\right)}{f}\right)}{\frac{\pi}{-4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (log
   (/
    (+
     (/ 2.0 (* PI 0.5))
     (*
      f
      (*
       f
       (+
        (* 0.0625 (* PI 2.0))
        (* -2.0 (* (* PI (* PI 2.0)) (/ 0.010416666666666666 PI)))))))
    f))
  (/ PI -4.0)))

C

double code(double f) {
	return log((((2.0 / (((double) M_PI) * 0.5)) + (f * (f * ((0.0625 * (((double) M_PI) * 2.0)) + (-2.0 * ((((double) M_PI) * (((double) M_PI) * 2.0)) * (0.010416666666666666 / ((double) M_PI)))))))) / f)) / (((double) M_PI) / -4.0);
}

Java

public static double code(double f) {
	return Math.log((((2.0 / (Math.PI * 0.5)) + (f * (f * ((0.0625 * (Math.PI * 2.0)) + (-2.0 * ((Math.PI * (Math.PI * 2.0)) * (0.010416666666666666 / Math.PI))))))) / f)) / (Math.PI / -4.0);
}

Python

def code(f):
	return math.log((((2.0 / (math.pi * 0.5)) + (f * (f * ((0.0625 * (math.pi * 2.0)) + (-2.0 * ((math.pi * (math.pi * 2.0)) * (0.010416666666666666 / math.pi))))))) / f)) / (math.pi / -4.0)

Julia

function code(f)
	return Float64(log(Float64(Float64(Float64(2.0 / Float64(pi * 0.5)) + Float64(f * Float64(f * Float64(Float64(0.0625 * Float64(pi * 2.0)) + Float64(-2.0 * Float64(Float64(pi * Float64(pi * 2.0)) * Float64(0.010416666666666666 / pi))))))) / f)) / Float64(pi / -4.0))
end

MATLAB

function tmp = code(f)
	tmp = log((((2.0 / (pi * 0.5)) + (f * (f * ((0.0625 * (pi * 2.0)) + (-2.0 * ((pi * (pi * 2.0)) * (0.010416666666666666 / pi))))))) / f)) / (pi / -4.0);
end

Wolfram

code[f_] := N[(N[Log[N[(N[(N[(2.0 / N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] + N[(f * N[(f * N[(N[(0.0625 * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(Pi * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision] * N[(0.010416666666666666 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision] / N[(Pi / -4.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.5%

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

   1. distribute-lft-neg-in N/A

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

   2. distribute-neg-frac2 N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{4} \cdot f\right)}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{4} \cdot f\right)}}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{4}\right)\right)}\right) \]

3. Simplified 8.5%

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

5. Taylor expanded in f around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(\frac{f \cdot \left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} + f \cdot \left(\frac{1}{16} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} - 2 \cdot \frac{\frac{1}{384} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{384} \cdot {\mathsf{PI}\left(\right)}^{3}}{{\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right)\right) + 2 \cdot \frac{1}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}{f}\right)}\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

7. Simplified 97.1%

   \[\leadsto \frac{\log \color{blue}{\left(\frac{\frac{2}{\pi \cdot 0.5} + f \cdot \left(f \cdot \left(0.0625 \cdot \left(\pi \cdot 2\right) + -2 \cdot \left(\left(\pi \cdot \left(\pi \cdot 2\right)\right) \cdot \frac{0.010416666666666666}{\pi}\right)\right)\right)}{f}\right)}}{\frac{\pi}{-4}} \]
8. Add Preprocessing

Alternative 3: 95.5% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.5%

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

   1. *-commutative N/A

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

   2. un-div-inv N/A

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

   3. distribute-neg-frac N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{4} \cdot f\right)}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{4} \cdot f\right)}}\right)\right)\right), \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{4}\right)}\right) \]

4. Applied egg-rr 99.1%

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

5. Taylor expanded in f around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)}\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 4\right)\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 4\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(f \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 4\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 4\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(f, \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 4\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(f, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 4\right)\right) \]

   6. PI-lowering-PI.f64 97.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(f, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 4\right)\right) \]

7. Simplified 97.0%

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

8. Final simplification 97.0%

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

Alternative 4: 95.4% accurate, 4.9× speedup

Math

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

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(-4.0 / Float64(pi / log(Float64(4.0 / Float64(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[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.5%

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

   1. distribute-lft-neg-in N/A

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

   2. distribute-neg-frac2 N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{4} \cdot f\right)}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{4} \cdot f\right)}}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{4}\right)\right)}\right) \]

3. Simplified 8.5%

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

5. Taylor expanded in f around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(\frac{2}{f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}\right)}\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot f\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   3. distribute-rgt-out-- N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{4} - \frac{-1}{4}\right)\right) \cdot f\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot f\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} \cdot f\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot \left(\left(1 \cdot \frac{1}{2}\right) \cdot f\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   7. *-inverses N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot \left(\left(\frac{\mathsf{PI}\left(\right)}{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}\right) \cdot f\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot \left(\left(\frac{\mathsf{PI}\left(\right)}{\mathsf{PI}\left(\right)} \cdot \frac{\frac{1}{4}}{\frac{1}{2}}\right) \cdot f\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   9. times-frac N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot \left(\frac{\mathsf{PI}\left(\right) \cdot \frac{1}{4}}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} \cdot f\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot \left(\frac{\frac{1}{4} \cdot \mathsf{PI}\left(\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} \cdot f\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot \left(\frac{\frac{1}{4} \cdot \mathsf{PI}\left(\right)}{\mathsf{PI}\left(\right) \cdot \left(\frac{1}{4} - \frac{-1}{4}\right)} \cdot f\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   12. distribute-rgt-out-- N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot \left(\frac{\frac{1}{4} \cdot \mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} \cdot f\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   13. associate-*r/ N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot \left(\left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}\right) \cdot f\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}\right) \cdot f\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}\right) \cdot f\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

7. Simplified 96.6%

   \[\leadsto \frac{\log \color{blue}{\left(\frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)}}{\frac{\pi}{-4}} \]
8. Step-by-step derivation

   1. associate-/r* N/A

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

   2. associate-/l/ N/A

      \[\leadsto \frac{\log \left(\frac{\frac{\frac{2}{\mathsf{PI}\left(\right)}}{\frac{1}{2}}}{f}\right)}{\frac{\mathsf{PI}\left(\right)}{-4}} \]

   3. associate-/r* N/A

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

   4. diff-log N/A

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

   5. frac-2neg N/A

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

   6. div-inv N/A

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

   7. distribute-neg-frac2 N/A

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

   8. metadata-eval N/A

      \[\leadsto \left(\mathsf{neg}\left(\left(\log \left(\frac{2}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right) - \log f\right)\right)\right) \cdot \frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \]

   9. clear-num N/A

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

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

       \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\left(\log \left(\frac{2}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right) - \log f\right)\right)\right), \color{blue}{\left(\frac{4}{\mathsf{PI}\left(\right)}\right)}\right) \]

9. Applied egg-rr 96.4%

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

    1. clear-num N/A

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

    2. add-cbrt-cube N/A

       \[\leadsto \log \left(\frac{\mathsf{PI}\left(\right)}{\frac{4}{f}}\right) \cdot \frac{1}{\frac{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}{4}} \]

    3. associate-*r* N/A

       \[\leadsto \log \left(\frac{\mathsf{PI}\left(\right)}{\frac{4}{f}}\right) \cdot \frac{1}{\frac{\sqrt[3]{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}}{4}} \]

    4. un-div-inv N/A

       \[\leadsto \frac{\log \left(\frac{\mathsf{PI}\left(\right)}{\frac{4}{f}}\right)}{\color{blue}{\frac{\sqrt[3]{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}}{4}}} \]

    5. clear-num N/A

       \[\leadsto \frac{\log \left(\frac{1}{\frac{\frac{4}{f}}{\mathsf{PI}\left(\right)}}\right)}{\frac{\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}}}{4}} \]

    6. neg-log N/A

       \[\leadsto \frac{\mathsf{neg}\left(\log \left(\frac{\frac{4}{f}}{\mathsf{PI}\left(\right)}\right)\right)}{\frac{\color{blue}{\sqrt[3]{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}}}{4}} \]

    7. diff-log N/A

       \[\leadsto \frac{\mathsf{neg}\left(\left(\log \left(\frac{4}{f}\right) - \log \mathsf{PI}\left(\right)\right)\right)}{\frac{\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}}}{4}} \]

    8. associate-*r* N/A

       \[\leadsto \frac{\mathsf{neg}\left(\left(\log \left(\frac{4}{f}\right) - \log \mathsf{PI}\left(\right)\right)\right)}{\frac{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}{4}} \]

    9. add-cbrt-cube N/A

       \[\leadsto \frac{\mathsf{neg}\left(\left(\log \left(\frac{4}{f}\right) - \log \mathsf{PI}\left(\right)\right)\right)}{\frac{\mathsf{PI}\left(\right)}{4}} \]

    10. metadata-eval N/A

        \[\leadsto \frac{\mathsf{neg}\left(\left(\log \left(\frac{4}{f}\right) - \log \mathsf{PI}\left(\right)\right)\right)}{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(-4\right)}} \]

    11. distribute-neg-frac2 N/A

        \[\leadsto \frac{\mathsf{neg}\left(\left(\log \left(\frac{4}{f}\right) - \log \mathsf{PI}\left(\right)\right)\right)}{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{-4}\right)} \]

    12. frac-2neg N/A

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

    13. associate-/r/ N/A

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

    14. *-commutative N/A

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

11. Applied egg-rr 96.5%

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

Alternative 5: 95.3% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.5%

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

   1. distribute-lft-neg-in N/A

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

   2. distribute-neg-frac2 N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{4} \cdot f\right)}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{4} \cdot f\right)}}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{4}\right)\right)}\right) \]

3. Simplified 8.5%

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

5. Taylor expanded in f around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(\frac{2}{f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}\right)}\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot f\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   3. distribute-rgt-out-- N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{4} - \frac{-1}{4}\right)\right) \cdot f\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot f\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} \cdot f\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot \left(\left(1 \cdot \frac{1}{2}\right) \cdot f\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   7. *-inverses N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot \left(\left(\frac{\mathsf{PI}\left(\right)}{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}\right) \cdot f\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot \left(\left(\frac{\mathsf{PI}\left(\right)}{\mathsf{PI}\left(\right)} \cdot \frac{\frac{1}{4}}{\frac{1}{2}}\right) \cdot f\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   9. times-frac N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot \left(\frac{\mathsf{PI}\left(\right) \cdot \frac{1}{4}}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} \cdot f\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot \left(\frac{\frac{1}{4} \cdot \mathsf{PI}\left(\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} \cdot f\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot \left(\frac{\frac{1}{4} \cdot \mathsf{PI}\left(\right)}{\mathsf{PI}\left(\right) \cdot \left(\frac{1}{4} - \frac{-1}{4}\right)} \cdot f\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   12. distribute-rgt-out-- N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot \left(\frac{\frac{1}{4} \cdot \mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} \cdot f\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   13. associate-*r/ N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot \left(\left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}\right) \cdot f\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}\right) \cdot f\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}\right) \cdot f\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

7. Simplified 96.6%

   \[\leadsto \frac{\log \color{blue}{\left(\frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)}}{\frac{\pi}{-4}} \]
8. Step-by-step derivation

   1. associate-/r* N/A

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

   2. associate-/l/ N/A

      \[\leadsto \frac{\log \left(\frac{\frac{\frac{2}{\mathsf{PI}\left(\right)}}{\frac{1}{2}}}{f}\right)}{\frac{\mathsf{PI}\left(\right)}{-4}} \]

   3. associate-/r* N/A

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

   4. diff-log N/A

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

   5. frac-2neg N/A

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

   6. div-inv N/A

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

   7. distribute-neg-frac2 N/A

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

   8. metadata-eval N/A

      \[\leadsto \left(\mathsf{neg}\left(\left(\log \left(\frac{2}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right) - \log f\right)\right)\right) \cdot \frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \]

   9. clear-num N/A

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

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

       \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\left(\log \left(\frac{2}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right) - \log f\right)\right)\right), \color{blue}{\left(\frac{4}{\mathsf{PI}\left(\right)}\right)}\right) \]

9. Applied egg-rr 96.4%

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

Reproduce

herbie shell --seed 2024149 
(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)))))))))