VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.6% → 98.9%

Time: 21.4s

Alternatives: 7

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, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.7%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. 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-frac2 N/A

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

   4. div-inv N/A

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

4. Applied egg-rr 99.1%

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

Alternative 2: 98.7% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.7%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. 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-frac2 N/A

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

   4. div-inv N/A

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

4. Applied egg-rr 99.1%

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

6. Applied egg-rr 98.9%

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

Alternative 3: 96.3% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double f) {
	return (Math.log((((4.0 / Math.PI) + (Math.PI * (0.08333333333333333 * (f * f)))) / f)) / Math.PI) / -0.25;
}

Python

def code(f):
	return (math.log((((4.0 / math.pi) + (math.pi * (0.08333333333333333 * (f * f)))) / f)) / math.pi) / -0.25

Julia

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

MATLAB

function tmp = code(f)
	tmp = (log((((4.0 / pi) + (pi * (0.08333333333333333 * (f * f)))) / f)) / pi) / -0.25;
end

Wolfram

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

Derivation

1. Initial program 7.7%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. 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 7.7%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{e^{\frac{\pi \cdot f}{4}} + e^{\frac{\pi \cdot f}{-4}}}{e^{\frac{\pi \cdot f}{4}} - e^{\frac{\pi \cdot f}{-4}}}\right)}{\frac{\pi}{-4}}} \]
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 95.0%

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

8. Taylor expanded in f around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right)\right), \mathsf{*.f64}\left(f, \color{blue}{\left(f \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right), f\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right)\right), \mathsf{*.f64}\left(f, \left(\left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right) \cdot f\right)\right)\right), f\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   2. distribute-rgt-out N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right)\right), \mathsf{*.f64}\left(f, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{24} + \frac{1}{8}\right)\right) \cdot f\right)\right)\right), f\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right)\right), \mathsf{*.f64}\left(f, \left(\mathsf{PI}\left(\right) \cdot \left(\left(\frac{-1}{24} + \frac{1}{8}\right) \cdot f\right)\right)\right)\right), f\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right)\right), \mathsf{*.f64}\left(f, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\left(\frac{-1}{24} + \frac{1}{8}\right) \cdot f\right)\right)\right)\right), f\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right)\right), \mathsf{*.f64}\left(f, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\left(\frac{-1}{24} + \frac{1}{8}\right) \cdot f\right)\right)\right)\right), f\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right)\right), \mathsf{*.f64}\left(f, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\left(\frac{-1}{24} + \frac{1}{8}\right), f\right)\right)\right)\right), f\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   7. metadata-eval 95.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right)\right), \mathsf{*.f64}\left(f, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\frac{1}{12}, f\right)\right)\right)\right), f\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

10. Simplified 95.0%

    \[\leadsto \frac{\log \left(\frac{\frac{2}{\pi \cdot 0.5} + f \cdot \color{blue}{\left(\pi \cdot \left(0.08333333333333333 \cdot f\right)\right)}}{f}\right)}{\frac{\pi}{-4}} \]
11. Step-by-step derivation

    1. div-inv N/A

       \[\leadsto \frac{\log \left(\frac{\frac{2}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + f \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{12} \cdot f\right)\right)}{f}\right)}{\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{-4}}} \]

    2. metadata-eval N/A

       \[\leadsto \frac{\log \left(\frac{\frac{2}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + f \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{12} \cdot f\right)\right)}{f}\right)}{\mathsf{PI}\left(\right) \cdot \frac{-1}{4}} \]

    3. metadata-eval N/A

       \[\leadsto \frac{\log \left(\frac{\frac{2}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + f \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{12} \cdot f\right)\right)}{f}\right)}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)} \]

    4. associate-/r* N/A

       \[\leadsto \frac{\frac{\log \left(\frac{\frac{2}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + f \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{12} \cdot f\right)\right)}{f}\right)}{\mathsf{PI}\left(\right)}}{\color{blue}{\mathsf{neg}\left(\frac{1}{4}\right)}} \]

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

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{\log \left(\frac{\frac{2}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + f \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{12} \cdot f\right)\right)}{f}\right)}{\mathsf{PI}\left(\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right) \]

12. Applied egg-rr 95.0%

    \[\leadsto \color{blue}{\frac{\frac{\log \left(\frac{\frac{4}{\pi} + \pi \cdot \left(0.08333333333333333 \cdot \left(f \cdot f\right)\right)}{f}\right)}{\pi}}{-0.25}} \]
13. Add Preprocessing

Alternative 4: 96.2% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.7%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. 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 7.7%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{e^{\frac{\pi \cdot f}{4}} + e^{\frac{\pi \cdot f}{-4}}}{e^{\frac{\pi \cdot f}{4}} - e^{\frac{\pi \cdot f}{-4}}}\right)}{\frac{\pi}{-4}}} \]
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 95.0%

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

8. Taylor expanded in f around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right)\right), \mathsf{*.f64}\left(f, \color{blue}{\left(f \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right), f\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right)\right), \mathsf{*.f64}\left(f, \left(\left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right) \cdot f\right)\right)\right), f\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   2. distribute-rgt-out N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right)\right), \mathsf{*.f64}\left(f, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{24} + \frac{1}{8}\right)\right) \cdot f\right)\right)\right), f\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right)\right), \mathsf{*.f64}\left(f, \left(\mathsf{PI}\left(\right) \cdot \left(\left(\frac{-1}{24} + \frac{1}{8}\right) \cdot f\right)\right)\right)\right), f\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right)\right), \mathsf{*.f64}\left(f, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\left(\frac{-1}{24} + \frac{1}{8}\right) \cdot f\right)\right)\right)\right), f\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right)\right), \mathsf{*.f64}\left(f, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\left(\frac{-1}{24} + \frac{1}{8}\right) \cdot f\right)\right)\right)\right), f\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right)\right), \mathsf{*.f64}\left(f, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\left(\frac{-1}{24} + \frac{1}{8}\right), f\right)\right)\right)\right), f\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

   7. metadata-eval 95.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right)\right), \mathsf{*.f64}\left(f, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\frac{1}{12}, f\right)\right)\right)\right), f\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

10. Simplified 95.0%

    \[\leadsto \frac{\log \left(\frac{\frac{2}{\pi \cdot 0.5} + f \cdot \color{blue}{\left(\pi \cdot \left(0.08333333333333333 \cdot f\right)\right)}}{f}\right)}{\frac{\pi}{-4}} \]
11. Step-by-step derivation

    1. associate-/r/ N/A

       \[\leadsto \frac{\log \left(\frac{\frac{2}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + f \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{12} \cdot f\right)\right)}{f}\right)}{\mathsf{PI}\left(\right)} \cdot \color{blue}{-4} \]

    2. *-commutative N/A

       \[\leadsto -4 \cdot \color{blue}{\frac{\log \left(\frac{\frac{2}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + f \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{12} \cdot f\right)\right)}{f}\right)}{\mathsf{PI}\left(\right)}} \]

    3. clear-num N/A

       \[\leadsto -4 \cdot \frac{1}{\color{blue}{\frac{\mathsf{PI}\left(\right)}{\log \left(\frac{\frac{2}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + f \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{12} \cdot f\right)\right)}{f}\right)}}} \]

    4. un-div-inv N/A

       \[\leadsto \frac{-4}{\color{blue}{\frac{\mathsf{PI}\left(\right)}{\log \left(\frac{\frac{2}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + f \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{12} \cdot f\right)\right)}{f}\right)}}} \]

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

       \[\leadsto \mathsf{/.f64}\left(-4, \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{\log \left(\frac{\frac{2}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + f \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{12} \cdot f\right)\right)}{f}\right)}\right)}\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\log \left(\frac{\frac{2}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + f \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{12} \cdot f\right)\right)}{f}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \log \color{blue}{\left(\frac{\frac{2}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + f \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{12} \cdot f\right)\right)}{f}\right)}\right)\right) \]

    8. log-lowering-log.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{log.f64}\left(\left(\frac{\frac{2}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + f \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{12} \cdot f\right)\right)}{f}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + f \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{12} \cdot f\right)\right)\right), f\right)\right)\right)\right) \]

12. Applied egg-rr 94.8%

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

Alternative 5: 95.7% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\log \left(\frac{\pi}{2} \cdot \frac{0.5}{\frac{1}{f}}\right)}{\pi}}{0.25} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/ (/ (log (* (/ PI 2.0) (/ 0.5 (/ 1.0 f)))) PI) 0.25))

C

double code(double f) {
	return (log(((((double) M_PI) / 2.0) * (0.5 / (1.0 / f)))) / ((double) M_PI)) / 0.25;
}

Java

public static double code(double f) {
	return (Math.log(((Math.PI / 2.0) * (0.5 / (1.0 / f)))) / Math.PI) / 0.25;
}

Python

def code(f):
	return (math.log(((math.pi / 2.0) * (0.5 / (1.0 / f)))) / math.pi) / 0.25

Julia

function code(f)
	return Float64(Float64(log(Float64(Float64(pi / 2.0) * Float64(0.5 / Float64(1.0 / f)))) / pi) / 0.25)
end

MATLAB

function tmp = code(f)
	tmp = (log(((pi / 2.0) * (0.5 / (1.0 / f)))) / pi) / 0.25;
end

Wolfram

code[f_] := N[(N[(N[Log[N[(N[(Pi / 2.0), $MachinePrecision] * N[(0.5 / N[(1.0 / f), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] / 0.25), $MachinePrecision]

Derivation

1. Initial program 7.7%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. 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 7.7%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{e^{\frac{\pi \cdot f}{4}} + e^{\frac{\pi \cdot f}{-4}}}{e^{\frac{\pi \cdot f}{4}} - e^{\frac{\pi \cdot f}{-4}}}\right)}{\frac{\pi}{-4}}} \]
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. *-lowering-*.f64 N/A

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

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

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

   5. metadata-eval N/A

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

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

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

   7. PI-lowering-PI.f64 94.3%

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

7. Simplified 94.3%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. clear-num N/A

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

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

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

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

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

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

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

   7. associate-/r* N/A

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

   8. div-inv N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. metadata-eval N/A

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

   12. frac-times N/A

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

   13. metadata-eval N/A

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

   14. associate-/l/ N/A

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

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

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

   16. div-inv N/A

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

   17. frac-times N/A

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

   18. metadata-eval N/A

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

   19. /-lowering-/.f64 N/A

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

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

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

   21. PI-lowering-PI.f64 94.2%

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

9. Applied egg-rr 94.2%

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

11. Applied egg-rr 94.3%

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

    1. div-inv N/A

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

    2. associate-/r* N/A

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

    3. div-inv N/A

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

    4. metadata-eval N/A

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

    5. metadata-eval N/A

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

    6. associate-*l* N/A

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

    7. div-inv N/A

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

    8. times-frac N/A

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

    9. /-rgt-identity N/A

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

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

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

    11. metadata-eval N/A

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

    12. div-inv N/A

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

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

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

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

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

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

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

    16. /-lowering-/.f64 94.3%

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

13. Applied egg-rr 94.3%

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

Alternative 6: 95.7% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.7%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. 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 7.7%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{e^{\frac{\pi \cdot f}{4}} + e^{\frac{\pi \cdot f}{-4}}}{e^{\frac{\pi \cdot f}{4}} - e^{\frac{\pi \cdot f}{-4}}}\right)}{\frac{\pi}{-4}}} \]
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. *-lowering-*.f64 N/A

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

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

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

   5. metadata-eval N/A

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

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

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

   7. PI-lowering-PI.f64 94.3%

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

7. Simplified 94.3%

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

   1. div-inv N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. associate-/r* N/A

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

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

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

9. Applied egg-rr 94.3%

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

10. Final simplification 94.3%

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

Alternative 7: 95.6% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.7%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. 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 7.7%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{e^{\frac{\pi \cdot f}{4}} + e^{\frac{\pi \cdot f}{-4}}}{e^{\frac{\pi \cdot f}{4}} - e^{\frac{\pi \cdot f}{-4}}}\right)}{\frac{\pi}{-4}}} \]
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. *-lowering-*.f64 N/A

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

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

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

   5. metadata-eval N/A

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

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

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

   7. PI-lowering-PI.f64 94.3%

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

7. Simplified 94.3%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. clear-num N/A

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

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

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

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

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

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

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

   7. associate-/r* N/A

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

   8. div-inv N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. metadata-eval N/A

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

   12. frac-times N/A

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

   13. metadata-eval N/A

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

   14. associate-/l/ N/A

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

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

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

   16. div-inv N/A

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

   17. frac-times N/A

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

   18. metadata-eval N/A

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

   19. /-lowering-/.f64 N/A

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

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

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

   21. PI-lowering-PI.f64 94.2%

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

9. Applied egg-rr 94.2%

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

10. Final simplification 94.2%

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

Reproduce

herbie shell --seed 2024164 
(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)))))))))