VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 99.0%

Time: 31.2s

Alternatives: 6

Speedup: 3.3×

• could not determine a ground truth

  1. f = -8.470293959832365e+128

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: 6.9% 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: 99.0% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (/ (- (log (/ 1.0 (tanh (* f (* PI 0.25)))))) (* PI 0.25)))

C

double code(double f) {
	return -log((1.0 / tanh((f * (((double) M_PI) * 0.25))))) / (((double) M_PI) * 0.25);
}

Java

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

Python

def code(f):
	return -math.log((1.0 / math.tanh((f * (math.pi * 0.25))))) / (math.pi * 0.25)

Julia

function code(f)
	return Float64(Float64(-log(Float64(1.0 / tanh(Float64(f * Float64(pi * 0.25)))))) / Float64(pi * 0.25))
end

MATLAB

function tmp = code(f)
	tmp = -log((1.0 / tanh((f * (pi * 0.25))))) / (pi * 0.25);
end

Wolfram

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

Derivation

1. Initial program 9.4%

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

   1. expm1-log1p-u 9.4%

      \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\right)\right)} \]

   2. expm1-udef 9.4%

      \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\right)} - 1\right)} \]

3. Applied egg-rr 97.6%

   \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{2 \cdot \sinh \left(\frac{\pi \cdot f}{4}\right)}\right)}{\pi \cdot 0.25}\right)} - 1\right)} \]
4. Step-by-step derivation

   1. expm1-def 97.6%

      \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{2 \cdot \sinh \left(\frac{\pi \cdot f}{4}\right)}\right)}{\pi \cdot 0.25}\right)\right)} \]

   2. expm1-log1p 98.9%

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

   3. times-frac 98.9%

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

   4. metadata-eval 98.9%

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

   5. *-lft-identity 98.9%

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

   6. associate-/l* 98.9%

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

   7. associate-/r/ 98.9%

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

   8. associate-/l* 98.9%

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

   9. associate-/r/ 98.9%

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

   10. *-commutative 98.9%

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

5. Simplified 98.9%

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

   1. cosh-def 98.9%

      \[\leadsto -\frac{\log \left(\frac{\color{blue}{\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{2}}}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right)}{0.25 \cdot \pi} \]

   2. cosh-undef 98.9%

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

   3. associate-*l/ 98.9%

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

   4. associate-*l/ 98.9%

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

   5. associate-/r* 98.9%

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

   6. expm1-log1p-u 98.9%

      \[\leadsto -\frac{\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{2 \cdot \sinh \left(\frac{\pi \cdot f}{4}\right)}\right)\right)\right)}}{0.25 \cdot \pi} \]

   7. expm1-udef 98.9%

      \[\leadsto -\frac{\log \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{2 \cdot \sinh \left(\frac{\pi \cdot f}{4}\right)}\right)} - 1\right)}}{0.25 \cdot \pi} \]

7. Applied egg-rr 99.3%

   \[\leadsto -\frac{\log \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\tanh \left(0.25 \cdot \left(\pi \cdot f\right)\right)}\right)} - 1\right)}}{0.25 \cdot \pi} \]
8. Step-by-step derivation

   1. expm1-def 99.3%

      \[\leadsto -\frac{\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\tanh \left(0.25 \cdot \left(\pi \cdot f\right)\right)}\right)\right)\right)}}{0.25 \cdot \pi} \]

   2. expm1-log1p 99.3%

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

   3. associate-*r* 99.3%

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

   4. *-commutative 99.3%

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

9. Simplified 99.3%

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

10. Final simplification 99.3%

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

Alternative 2: 99.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.4%

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

   1. expm1-log1p-u 9.4%

      \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\right)\right)} \]

   2. expm1-udef 9.4%

      \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\right)} - 1\right)} \]

3. Applied egg-rr 97.6%

   \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{2 \cdot \sinh \left(\frac{\pi \cdot f}{4}\right)}\right)}{\pi \cdot 0.25}\right)} - 1\right)} \]
4. Step-by-step derivation

   1. expm1-def 97.6%

      \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{2 \cdot \sinh \left(\frac{\pi \cdot f}{4}\right)}\right)}{\pi \cdot 0.25}\right)\right)} \]

   2. expm1-log1p 98.9%

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

   3. times-frac 98.9%

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

   4. metadata-eval 98.9%

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

   5. *-lft-identity 98.9%

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

   6. associate-/l* 98.9%

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

   7. associate-/r/ 98.9%

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

   8. associate-/l* 98.9%

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

   9. associate-/r/ 98.9%

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

   10. *-commutative 98.9%

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

5. Simplified 98.9%

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

7. Applied egg-rr 98.0%

   \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-\log \tanh \left(0.25 \cdot \left(\pi \cdot f\right)\right)}{\pi \cdot 0.25}\right)} - 1\right)} \]
8. Step-by-step derivation

   1. expm1-def 98.0%

      \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\log \tanh \left(0.25 \cdot \left(\pi \cdot f\right)\right)}{\pi \cdot 0.25}\right)\right)} \]

   2. expm1-log1p 99.3%

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

   3. neg-mul-1 99.3%

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

   4. *-commutative 99.3%

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

   5. times-frac 99.3%

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

   6. metadata-eval 99.3%

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

   7. *-commutative 99.3%

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

   8. *-commutative 99.3%

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

   9. associate-*l* 99.3%

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

9. Simplified 99.3%

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

10. Final simplification 99.3%

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

Alternative 3: 95.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.4%

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

   1. distribute-lft-neg-in 9.4%

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

   2. *-commutative 9.4%

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

   3. associate-/r/ 9.4%

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

   4. associate-*l/ 9.4%

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

   5. metadata-eval 9.4%

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

   6. distribute-neg-frac 9.4%

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

3. Simplified 9.4%

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

4. Taylor expanded in f around 0 96.6%

   \[\leadsto \color{blue}{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f\right)} \cdot \frac{-4}{\pi} \]
5. Step-by-step derivation

   1. mul-1-neg 96.6%

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

   2. unsub-neg 96.6%

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

   3. distribute-rgt-out-- 96.6%

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

   4. metadata-eval 96.6%

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

6. Simplified 96.6%

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

   1. add-exp-log 95.3%

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

   2. diff-log 95.3%

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

8. Applied egg-rr 95.3%

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

   1. add-exp-log 96.5%

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

   2. *-commutative 96.5%

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

   3. associate-/r* 96.5%

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

   4. metadata-eval 96.5%

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

   5. associate-/r* 96.5%

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

   6. associate-/l/ 96.6%

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

   7. add-sqr-sqrt 96.1%

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

   8. sqrt-unprod 96.7%

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

   9. pow2 96.7%

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

   10. associate-/l/ 96.7%

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

   11. associate-/r* 96.7%

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

10. Applied egg-rr 96.7%

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

    1. unpow2 96.7%

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

    2. rem-sqrt-square 96.7%

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

    3. associate-/l/ 96.7%

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

12. Simplified 96.7%

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

13. Final simplification 96.7%

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

Alternative 4: 95.8% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.4%

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

   1. expm1-log1p-u 9.4%

      \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\right)\right)} \]

   2. expm1-udef 9.4%

      \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\right)} - 1\right)} \]

3. Applied egg-rr 97.6%

   \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{2 \cdot \sinh \left(\frac{\pi \cdot f}{4}\right)}\right)}{\pi \cdot 0.25}\right)} - 1\right)} \]
4. Step-by-step derivation

   1. expm1-def 97.6%

      \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{2 \cdot \sinh \left(\frac{\pi \cdot f}{4}\right)}\right)}{\pi \cdot 0.25}\right)\right)} \]

   2. expm1-log1p 98.9%

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

   3. times-frac 98.9%

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

   4. metadata-eval 98.9%

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

   5. *-lft-identity 98.9%

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

   6. associate-/l* 98.9%

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

   7. associate-/r/ 98.9%

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

   8. associate-/l* 98.9%

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

   9. associate-/r/ 98.9%

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

   10. *-commutative 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in f around 0 96.6%

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

   1. associate-/r* 96.6%

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

8. Simplified 96.6%

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

9. Final simplification 96.6%

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

Alternative 5: 95.7% accurate, 3.3× speedup

Math

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

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

Derivation

1. Initial program 9.4%

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

   1. distribute-lft-neg-in 9.4%

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

   2. *-commutative 9.4%

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

   3. associate-/r/ 9.4%

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

   4. associate-*l/ 9.4%

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

   5. metadata-eval 9.4%

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

   6. distribute-neg-frac 9.4%

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

3. Simplified 9.4%

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

4. Taylor expanded in f around 0 96.6%

   \[\leadsto \color{blue}{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f\right)} \cdot \frac{-4}{\pi} \]
5. Step-by-step derivation

   1. mul-1-neg 96.6%

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

   2. unsub-neg 96.6%

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

   3. distribute-rgt-out-- 96.6%

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

   4. metadata-eval 96.6%

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

6. Simplified 96.6%

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

7. Taylor expanded in f around 0 96.6%

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

   1. associate-*r/ 96.6%

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

   2. sub-neg 96.6%

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

   3. distribute-rgt-in 96.6%

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

   4. metadata-eval 96.6%

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

   5. associate-/r* 96.6%

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

   6. *-commutative 96.6%

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

   7. distribute-rgt-in 96.6%

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

   8. unsub-neg 96.6%

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

   9. log-div 96.6%

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

   10. associate-*l/ 96.5%

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

   11. *-commutative 96.5%

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

   12. associate-/r* 96.5%

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

   13. metadata-eval 96.5%

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

9. Simplified 96.6%

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

10. Final simplification 96.6%

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

Alternative 6: 95.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.4%

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

   1. distribute-lft-neg-in 9.4%

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

   2. *-commutative 9.4%

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

   3. associate-/r/ 9.4%

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

   4. associate-*l/ 9.4%

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

   5. metadata-eval 9.4%

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

   6. distribute-neg-frac 9.4%

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

3. Simplified 9.4%

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

4. Taylor expanded in f around 0 96.6%

   \[\leadsto \color{blue}{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f\right)} \cdot \frac{-4}{\pi} \]
5. Step-by-step derivation

   1. mul-1-neg 96.6%

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

   2. unsub-neg 96.6%

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

   3. distribute-rgt-out-- 96.6%

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

   4. metadata-eval 96.6%

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

6. Simplified 96.6%

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

7. Taylor expanded in f around 0 96.6%

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

   1. associate-*r/ 96.6%

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

   2. sub-neg 96.6%

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

   3. distribute-rgt-in 96.6%

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

   4. metadata-eval 96.6%

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

   5. associate-/r* 96.6%

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

   6. *-commutative 96.6%

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

   7. distribute-rgt-in 96.6%

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

   8. unsub-neg 96.6%

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

   9. log-div 96.6%

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

   10. associate-*l/ 96.5%

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

   11. *-commutative 96.5%

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

   12. associate-/r* 96.5%

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

   13. metadata-eval 96.5%

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

9. Simplified 96.6%

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

    1. *-commutative 96.6%

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

    2. associate-*r/ 96.6%

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

    3. associate-/l/ 96.6%

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

    4. associate-/r* 96.6%

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

11. Applied egg-rr 96.6%

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

12. Final simplification 96.6%

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

Reproduce

herbie shell --seed 2023256 
(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)))))))))