VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.6% → 98.9%

Time: 19.4s

Alternatives: 8

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

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

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 6.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. 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 99.4%

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

Alternative 2: 98.8% 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 6.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. 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 99.4%

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

   1. associate-/r/ N/A

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

   2. *-commutative N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   12. /-lowering-/.f64 99.3%

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

6. Applied egg-rr 99.3%

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

Alternative 3: 96.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot 0.125 + -2 \cdot \left(\pi \cdot 0.020833333333333332\right)\\ t_1 := t\_0 \cdot \left(f \cdot f\right)\\ \frac{\log \left(\frac{\frac{\frac{16}{\pi \cdot \pi} - t\_0 \cdot \left(\left(f \cdot f\right) \cdot t\_1\right)}{\frac{4}{\pi} - t\_1}}{f}\right)}{\frac{\pi}{-4}} \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (+ (* PI 0.125) (* -2.0 (* PI 0.020833333333333332))))
        (t_1 (* t_0 (* f f))))
   (/
    (log
     (/
      (/ (- (/ 16.0 (* PI PI)) (* t_0 (* (* f f) t_1))) (- (/ 4.0 PI) t_1))
      f))
    (/ PI -4.0))))

C

double code(double f) {
	double t_0 = (((double) M_PI) * 0.125) + (-2.0 * (((double) M_PI) * 0.020833333333333332));
	double t_1 = t_0 * (f * f);
	return log(((((16.0 / (((double) M_PI) * ((double) M_PI))) - (t_0 * ((f * f) * t_1))) / ((4.0 / ((double) M_PI)) - t_1)) / f)) / (((double) M_PI) / -4.0);
}

Java

public static double code(double f) {
	double t_0 = (Math.PI * 0.125) + (-2.0 * (Math.PI * 0.020833333333333332));
	double t_1 = t_0 * (f * f);
	return Math.log(((((16.0 / (Math.PI * Math.PI)) - (t_0 * ((f * f) * t_1))) / ((4.0 / Math.PI) - t_1)) / f)) / (Math.PI / -4.0);
}

Python

def code(f):
	t_0 = (math.pi * 0.125) + (-2.0 * (math.pi * 0.020833333333333332))
	t_1 = t_0 * (f * f)
	return math.log(((((16.0 / (math.pi * math.pi)) - (t_0 * ((f * f) * t_1))) / ((4.0 / math.pi) - t_1)) / f)) / (math.pi / -4.0)

Julia

function code(f)
	t_0 = Float64(Float64(pi * 0.125) + Float64(-2.0 * Float64(pi * 0.020833333333333332)))
	t_1 = Float64(t_0 * Float64(f * f))
	return Float64(log(Float64(Float64(Float64(Float64(16.0 / Float64(pi * pi)) - Float64(t_0 * Float64(Float64(f * f) * t_1))) / Float64(Float64(4.0 / pi) - t_1)) / f)) / Float64(pi / -4.0))
end

MATLAB

function tmp = code(f)
	t_0 = (pi * 0.125) + (-2.0 * (pi * 0.020833333333333332));
	t_1 = t_0 * (f * f);
	tmp = log(((((16.0 / (pi * pi)) - (t_0 * ((f * f) * t_1))) / ((4.0 / pi) - t_1)) / f)) / (pi / -4.0);
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi * 0.125), $MachinePrecision] + N[(-2.0 * N[(Pi * 0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(f * f), $MachinePrecision]), $MachinePrecision]}, N[(N[Log[N[(N[(N[(N[(16.0 / N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[(N[(f * f), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(4.0 / Pi), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision] / N[(Pi / -4.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 6.9%

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

   1. distribute-lft-neg-in 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 6.9%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{e^{\pi \cdot \frac{f}{4}} + e^{\frac{\pi \cdot f}{-4}}}{e^{\pi \cdot \frac{f}{4}} - e^{\frac{\pi \cdot f}{-4}}}\right)}{\frac{\pi}{-4}}} \]
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 95.4%

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

9. Applied egg-rr 95.4%

   \[\leadsto \frac{\log \left(\frac{\color{blue}{\frac{\frac{\frac{-4}{\pi}}{\frac{\pi}{-4}} - \left(\left(\pi \cdot 0.125 + \frac{-2 \cdot \left(\left(\pi \cdot \pi\right) \cdot 0.020833333333333332\right)}{\pi}\right) \cdot \left(f \cdot f\right)\right) \cdot \left(\left(\pi \cdot 0.125 + \frac{-2 \cdot \left(\left(\pi \cdot \pi\right) \cdot 0.020833333333333332\right)}{\pi}\right) \cdot \left(f \cdot f\right)\right)}{\frac{4}{\pi} - \left(\pi \cdot 0.125 + \frac{-2 \cdot \left(\left(\pi \cdot \pi\right) \cdot 0.020833333333333332\right)}{\pi}\right) \cdot \left(f \cdot f\right)}}}{f}\right)}{\frac{\pi}{-4}} \]
10. Step-by-step derivation

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{-4}{\mathsf{PI}\left(\right)}}{\frac{\mathsf{PI}\left(\right)}{-4}} - \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{8} + \frac{-2 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{48}\right)}{\mathsf{PI}\left(\right)}\right) \cdot \left(f \cdot f\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{8} + \frac{-2 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{48}\right)}{\mathsf{PI}\left(\right)}\right) \cdot \left(f \cdot f\right)\right)\right), \left(\frac{4}{\mathsf{PI}\left(\right)} - \left(\mathsf{PI}\left(\right) \cdot \frac{1}{8} + \frac{-2 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{48}\right)}{\mathsf{PI}\left(\right)}\right) \cdot \left(f \cdot f\right)\right)\right), f\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -4\right)\right) \]

11. Applied egg-rr 95.4%

    \[\leadsto \frac{\log \left(\frac{\color{blue}{\frac{\frac{16}{\pi \cdot \pi} - \left(\pi \cdot 0.125 + -2 \cdot \left(\pi \cdot 0.020833333333333332\right)\right) \cdot \left(\left(f \cdot f\right) \cdot \left(\left(\pi \cdot 0.125 + -2 \cdot \left(\pi \cdot 0.020833333333333332\right)\right) \cdot \left(f \cdot f\right)\right)\right)}{\frac{4}{\pi} - \left(\pi \cdot 0.125 + -2 \cdot \left(\pi \cdot 0.020833333333333332\right)\right) \cdot \left(f \cdot f\right)}}}{f}\right)}{\frac{\pi}{-4}} \]
12. Add Preprocessing

Alternative 4: 96.5% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\frac{f}{\left(\pi \cdot 0.125 + -2 \cdot \left(\pi \cdot 0.020833333333333332\right)\right) \cdot \left(f \cdot f\right) + \frac{4}{\pi}}\right)}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (log
   (/
    f
    (+
     (* (+ (* PI 0.125) (* -2.0 (* PI 0.020833333333333332))) (* f f))
     (/ 4.0 PI))))
  (/ PI 4.0)))

C

double code(double f) {
	return log((f / ((((((double) M_PI) * 0.125) + (-2.0 * (((double) M_PI) * 0.020833333333333332))) * (f * f)) + (4.0 / ((double) M_PI))))) / (((double) M_PI) / 4.0);
}

Java

public static double code(double f) {
	return Math.log((f / ((((Math.PI * 0.125) + (-2.0 * (Math.PI * 0.020833333333333332))) * (f * f)) + (4.0 / Math.PI)))) / (Math.PI / 4.0);
}

Python

def code(f):
	return math.log((f / ((((math.pi * 0.125) + (-2.0 * (math.pi * 0.020833333333333332))) * (f * f)) + (4.0 / math.pi)))) / (math.pi / 4.0)

Julia

function code(f)
	return Float64(log(Float64(f / Float64(Float64(Float64(Float64(pi * 0.125) + Float64(-2.0 * Float64(pi * 0.020833333333333332))) * Float64(f * f)) + Float64(4.0 / pi)))) / Float64(pi / 4.0))
end

MATLAB

function tmp = code(f)
	tmp = log((f / ((((pi * 0.125) + (-2.0 * (pi * 0.020833333333333332))) * (f * f)) + (4.0 / pi)))) / (pi / 4.0);
end

Wolfram

code[f_] := N[(N[Log[N[(f / N[(N[(N[(N[(Pi * 0.125), $MachinePrecision] + N[(-2.0 * N[(Pi * 0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(f * f), $MachinePrecision]), $MachinePrecision] + N[(4.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.9%

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

   1. distribute-lft-neg-in 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 6.9%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{e^{\pi \cdot \frac{f}{4}} + e^{\frac{\pi \cdot f}{-4}}}{e^{\pi \cdot \frac{f}{4}} - e^{\frac{\pi \cdot f}{-4}}}\right)}{\frac{\pi}{-4}}} \]
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 95.4%

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

9. Applied egg-rr 95.4%

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

11. Applied egg-rr 95.4%

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

12. Final simplification 95.4%

    \[\leadsto \frac{\log \left(\frac{f}{\left(\pi \cdot 0.125 + -2 \cdot \left(\pi \cdot 0.020833333333333332\right)\right) \cdot \left(f \cdot f\right) + \frac{4}{\pi}}\right)}{\frac{\pi}{4}} \]
13. Add Preprocessing

Alternative 5: 96.5% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. distribute-lft-neg-in 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 6.9%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{e^{\pi \cdot \frac{f}{4}} + e^{\frac{\pi \cdot f}{-4}}}{e^{\pi \cdot \frac{f}{4}} - e^{\frac{\pi \cdot f}{-4}}}\right)}{\frac{\pi}{-4}}} \]
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 95.4%

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

8. Taylor expanded in f around 0

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

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

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

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

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

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

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

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

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

   7. distribute-rgt-out N/A

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

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

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

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

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

   10. metadata-eval N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

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

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

   14. PI-lowering-PI.f64 95.4%

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

10. Simplified 95.4%

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

11. Final simplification 95.4%

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

Alternative 6: 96.0% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. distribute-lft-neg-in 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 6.9%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{e^{\pi \cdot \frac{f}{4}} + e^{\frac{\pi \cdot f}{-4}}}{e^{\pi \cdot \frac{f}{4}} - e^{\frac{\pi \cdot f}{-4}}}\right)}{\frac{\pi}{-4}}} \]
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 94.9%

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

   1. clear-num N/A

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

   2. associate-*r* N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. div-inv N/A

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

   7. clear-num N/A

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

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

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

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

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

   10. PI-lowering-PI.f64 94.9%

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

9. Applied egg-rr 94.9%

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

Alternative 7: 95.8% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. distribute-lft-neg-in 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 6.9%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{e^{\pi \cdot \frac{f}{4}} + e^{\frac{\pi \cdot f}{-4}}}{e^{\pi \cdot \frac{f}{4}} - e^{\frac{\pi \cdot f}{-4}}}\right)}{\frac{\pi}{-4}}} \]
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 94.9%

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

   1. associate-/r/ N/A

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

   2. *-commutative N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

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

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

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

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

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

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

   8. clear-num N/A

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

   9. associate-*r* N/A

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

   10. associate-/l* N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. div-inv N/A

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

   14. clear-num N/A

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

   15. associate-/l/ N/A

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

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

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

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

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

   18. /-lowering-/.f64 N/A

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

   19. PI-lowering-PI.f64 94.8%

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

9. Applied egg-rr 94.8%

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

Alternative 8: 95.8% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
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 6.9%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{e^{\pi \cdot \frac{f}{4}} + e^{\frac{\pi \cdot f}{-4}}}{e^{\pi \cdot \frac{f}{4}} - e^{\frac{\pi \cdot f}{-4}}}\right)}{\frac{\pi}{-4}}} \]
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 94.9%

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

   1. div-inv N/A

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

   2. associate-/r/ N/A

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

   3. metadata-eval N/A

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

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

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

   5. associate-/r/ N/A

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

   6. *-commutative N/A

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

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

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

   8. clear-num N/A

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

   9. distribute-neg-frac N/A

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

   10. metadata-eval N/A

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

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

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

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

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

   13. clear-num N/A

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

   14. associate-*r* N/A

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

   15. associate-/l* N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. div-inv N/A

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

   19. clear-num N/A

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

   20. associate-/l/ N/A

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

9. Applied egg-rr 94.8%

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

10. Final simplification 94.8%

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

Reproduce

herbie shell --seed 2024145 
(FPCore (f)
  :name "VandenBroeck and Keller, Equation (20)"
  :precision binary64
  (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f)))) (- (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f)))))))))