VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.5% → 99.1%

Time: 18.9s

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: 6.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ t_1 := e^{t\_0}\\ t_2 := e^{-t\_0}\\ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{t\_1 + t\_2}{t\_1 - t\_2}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (/ PI 4.0) f)) (t_1 (exp t_0)) (t_2 (exp (- t_0))))
   (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ t_1 t_2) (- t_1 t_2)))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) / 4.0) * f;
	double t_1 = exp(t_0);
	double t_2 = exp(-t_0);
	return -((1.0 / (((double) M_PI) / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
}

Java

public static double code(double f) {
	double t_0 = (Math.PI / 4.0) * f;
	double t_1 = Math.exp(t_0);
	double t_2 = Math.exp(-t_0);
	return -((1.0 / (Math.PI / 4.0)) * Math.log(((t_1 + t_2) / (t_1 - t_2))));
}

Python

def code(f):
	t_0 = (math.pi / 4.0) * f
	t_1 = math.exp(t_0)
	t_2 = math.exp(-t_0)
	return -((1.0 / (math.pi / 4.0)) * math.log(((t_1 + t_2) / (t_1 - t_2))))

Julia

function code(f)
	t_0 = Float64(Float64(pi / 4.0) * f)
	t_1 = exp(t_0)
	t_2 = exp(Float64(-t_0))
	return Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(t_1 + t_2) / Float64(t_1 - t_2)))))
end

MATLAB

function tmp = code(f)
	t_0 = (pi / 4.0) * f;
	t_1 = exp(t_0);
	t_2 = exp(-t_0);
	tmp = -((1.0 / (pi / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]}, Block[{t$95$1 = N[Exp[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-t$95$0)], $MachinePrecision]}, (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(t$95$1 + t$95$2), $MachinePrecision] / N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Alternative 1: 99.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.5%

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

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

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

   2. distribute-neg-frac2 N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

3. Simplified 6.5%

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

   1. frac-2neg N/A

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

   2. metadata-eval N/A

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

   3. div-inv N/A

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

   4. associate-/r* N/A

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

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

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

6. Applied egg-rr 99.4%

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

Alternative 2: 98.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.5%

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

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

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

   2. distribute-neg-frac2 N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

3. Simplified 6.5%

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

   1. frac-2neg N/A

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

   2. metadata-eval N/A

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

   3. div-inv N/A

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

   4. associate-/r* N/A

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

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

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

6. Applied egg-rr 99.4%

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

   1. div-inv N/A

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

   2. clear-num N/A

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

   3. metadata-eval N/A

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

   4. associate-*l/ N/A

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

   5. metadata-eval N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   13. PI-lowering-PI.f64 99.3%

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

8. Applied egg-rr 99.3%

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

Alternative 3: 98.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.5%

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

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

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

   2. distribute-neg-frac2 N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

3. Simplified 6.5%

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

   1. clear-num N/A

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

   2. frac-2neg N/A

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

   3. associate-/r/ N/A

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

   4. distribute-frac-neg2 N/A

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

   5. clear-num N/A

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

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

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

6. Applied egg-rr 99.2%

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

Alternative 4: 96.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\frac{\frac{2}{\pi \cdot 0.5} + f \cdot \left(f \cdot \left(0.0625 \cdot \left(\pi \cdot 2\right) + -2 \cdot \left(\left(2 \cdot \left(\pi \cdot 2\right)\right) \cdot 0.005208333333333333\right)\right)\right)}{f}\right)}{\frac{\pi}{-4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (log
   (/
    (+
     (/ 2.0 (* PI 0.5))
     (*
      f
      (*
       f
       (+
        (* 0.0625 (* PI 2.0))
        (* -2.0 (* (* 2.0 (* PI 2.0)) 0.005208333333333333))))))
    f))
  (/ PI -4.0)))

C

double code(double f) {
	return log((((2.0 / (((double) M_PI) * 0.5)) + (f * (f * ((0.0625 * (((double) M_PI) * 2.0)) + (-2.0 * ((2.0 * (((double) M_PI) * 2.0)) * 0.005208333333333333)))))) / f)) / (((double) M_PI) / -4.0);
}

Java

public static double code(double f) {
	return Math.log((((2.0 / (Math.PI * 0.5)) + (f * (f * ((0.0625 * (Math.PI * 2.0)) + (-2.0 * ((2.0 * (Math.PI * 2.0)) * 0.005208333333333333)))))) / f)) / (Math.PI / -4.0);
}

Python

def code(f):
	return math.log((((2.0 / (math.pi * 0.5)) + (f * (f * ((0.0625 * (math.pi * 2.0)) + (-2.0 * ((2.0 * (math.pi * 2.0)) * 0.005208333333333333)))))) / f)) / (math.pi / -4.0)

Julia

function code(f)
	return Float64(log(Float64(Float64(Float64(2.0 / Float64(pi * 0.5)) + Float64(f * Float64(f * Float64(Float64(0.0625 * Float64(pi * 2.0)) + Float64(-2.0 * Float64(Float64(2.0 * Float64(pi * 2.0)) * 0.005208333333333333)))))) / f)) / Float64(pi / -4.0))
end

MATLAB

function tmp = code(f)
	tmp = log((((2.0 / (pi * 0.5)) + (f * (f * ((0.0625 * (pi * 2.0)) + (-2.0 * ((2.0 * (pi * 2.0)) * 0.005208333333333333)))))) / f)) / (pi / -4.0);
end

Wolfram

code[f_] := N[(N[Log[N[(N[(N[(2.0 / N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] + N[(f * N[(f * N[(N[(0.0625 * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(2.0 * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision] * 0.005208333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision] / N[(Pi / -4.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.5%

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

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

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

   2. distribute-neg-frac2 N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

3. Simplified 6.5%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{e^{\frac{\pi \cdot f}{4}} + e^{\frac{\pi \cdot f}{-4}}}{e^{\frac{\pi \cdot f}{4}} - e^{\frac{\pi \cdot f}{-4}}}\right)}{\frac{\pi}{-4}}} \]
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 96.7%

   \[\leadsto \frac{\log \color{blue}{\left(\frac{\frac{2}{\pi \cdot 0.5} + f \cdot \left(f \cdot \left(0.0625 \cdot \left(\pi \cdot 2\right) + -2 \cdot \left(\left(2 \cdot \left(\pi \cdot 2\right)\right) \cdot 0.005208333333333333\right)\right)\right)}{f}\right)}}{\frac{\pi}{-4}} \]
8. Add Preprocessing

Alternative 5: 96.0% accurate, 4.9× speedup

Math

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

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

Derivation

1. Initial program 6.5%

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

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

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

   2. distribute-neg-frac2 N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

3. Simplified 6.5%

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

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

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

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

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

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

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

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

   10. metadata-eval N/A

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

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

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

   12. PI-lowering-PI.f64 96.0%

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

7. Simplified 96.0%

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

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

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

   5. PI-lowering-PI.f64 96.0%

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

9. Applied egg-rr 96.0%

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

Alternative 6: 95.9% accurate, 4.9× speedup

Math

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

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(Float64(-4.0 / pi) * log(Float64(4.0 / Float64(f * pi))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.5%

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

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

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

   2. distribute-neg-frac2 N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

3. Simplified 6.5%

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

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

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

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

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

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

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

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

   10. metadata-eval N/A

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

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

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

   12. PI-lowering-PI.f64 96.0%

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

7. Simplified 96.0%

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

   1. diff-log N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. clear-num N/A

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

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

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

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

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

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

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

   8. *-commutative N/A

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

   9. associate-/r* N/A

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

   10. metadata-eval N/A

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

   11. diff-log N/A

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

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

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

   13. associate-/l/ N/A

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

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

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

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

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

   16. PI-lowering-PI.f64 95.8%

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

9. Applied egg-rr 95.8%

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

Alternative 7: 0.0% accurate, 106.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0}{0}}{\pi} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (/ 0.0 0.0) PI))

C

double code(double f) {
	return (0.0 / 0.0) / ((double) M_PI);
}

Java

public static double code(double f) {
	return (0.0 / 0.0) / Math.PI;
}

Python

def code(f):
	return (0.0 / 0.0) / math.pi

Julia

function code(f)
	return Float64(Float64(0.0 / 0.0) / pi)
end

MATLAB

function tmp = code(f)
	tmp = (0.0 / 0.0) / pi;
end

Wolfram

code[f_] := N[(N[(0.0 / 0.0), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 6.5%

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

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

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

   2. distribute-neg-frac2 N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

3. Simplified 6.5%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. associate-/r* N/A

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

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

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

6. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{\frac{\log \left(\frac{1}{\tanh \left(\frac{f}{\frac{4}{\pi}}\right)}\right)}{-0.25}}{\pi}} \]
7. Step-by-step derivation

   1. inv-pow N/A

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

   2. sqr-pow N/A

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

   3. log-prod N/A

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

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

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

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

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

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

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

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

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

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

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

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

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

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

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

   13. metadata-eval N/A

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

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

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

8. Applied egg-rr 99.4%

   \[\leadsto \frac{\frac{\color{blue}{\log \left({\tanh \left(\frac{f}{\frac{4}{\pi}}\right)}^{-0.5}\right) + \log \left({\tanh \left(\frac{f}{\frac{4}{\pi}}\right)}^{-0.5}\right)}}{-0.25}}{\pi} \]
9. Step-by-step derivation

   1. flip-+ N/A

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

   2. +-inverses N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{0}{\log \left({\tanh \left(\frac{f}{\frac{4}{\mathsf{PI}\left(\right)}}\right)}^{\frac{-1}{2}}\right) - \log \left({\tanh \left(\frac{f}{\frac{4}{\mathsf{PI}\left(\right)}}\right)}^{\frac{-1}{2}}\right)}}{\frac{-1}{4}}\right), \mathsf{PI.f64}\left(\right)\right) \]

   3. +-inverses N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{0}{0}}{\frac{-1}{4}}\right), \mathsf{PI.f64}\left(\right)\right) \]

   4. associate-/l/ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{0}{\frac{-1}{4} \cdot 0}\right), \mathsf{PI.f64}\left(\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{0}{0}\right), \mathsf{PI.f64}\left(\right)\right) \]

   6. /-lowering-/.f64 0.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(0, 0\right), \mathsf{PI.f64}\left(\right)\right) \]

10. Applied egg-rr 0.0%

    \[\leadsto \frac{\color{blue}{\frac{0}{0}}}{\pi} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024288 
(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)))))))))