VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.0% → 97.3%

Time: 43.2s

Alternatives: 6

Speedup: 3.3×

• could not determine a ground truth

  1. f = -2.3071037950419194e+150

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 7.0% 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: 97.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{\frac{4}{f}}\\ \frac{-\log \left(\frac{\cosh t_0}{\sinh t_0}\right)}{\pi \cdot 0.25} \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (/ PI (/ 4.0 f))))
   (/ (- (log (/ (cosh t_0) (sinh t_0)))) (* PI 0.25))))

C

double code(double f) {
	double t_0 = ((double) M_PI) / (4.0 / f);
	return -log((cosh(t_0) / sinh(t_0))) / (((double) M_PI) * 0.25);
}

Java

public static double code(double f) {
	double t_0 = Math.PI / (4.0 / f);
	return -Math.log((Math.cosh(t_0) / Math.sinh(t_0))) / (Math.PI * 0.25);
}

Python

def code(f):
	t_0 = math.pi / (4.0 / f)
	return -math.log((math.cosh(t_0) / math.sinh(t_0))) / (math.pi * 0.25)

Julia

function code(f)
	t_0 = Float64(pi / Float64(4.0 / f))
	return Float64(Float64(-log(Float64(cosh(t_0) / sinh(t_0)))) / Float64(pi * 0.25))
end

MATLAB

function tmp = code(f)
	t_0 = pi / (4.0 / f);
	tmp = -log((cosh(t_0) / sinh(t_0))) / (pi * 0.25);
end

Wolfram

code[f_] := Block[{t$95$0 = N[(Pi / N[(4.0 / f), $MachinePrecision]), $MachinePrecision]}, N[((-N[Log[N[(N[Cosh[t$95$0], $MachinePrecision] / N[Sinh[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.4%

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

   1. expm1-log1p-u 6.4%

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

   2. expm1-udef 6.4%

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

3. Applied egg-rr 95.4%

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

   1. expm1-def 95.4%

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

   2. expm1-log1p 96.7%

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

   3. *-commutative 96.7%

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

   4. times-frac 96.7%

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

   5. metadata-eval 96.7%

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

   6. *-lft-identity 96.7%

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

   7. associate-/l* 96.7%

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

   8. associate-/l* 96.7%

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

5. Simplified 96.7%

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

6. Final simplification 96.7%

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

Alternative 2: 96.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

   1. expm1-log1p-u 6.4%

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

   2. expm1-udef 6.4%

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

3. Applied egg-rr 95.4%

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

   1. expm1-def 95.4%

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

   2. expm1-log1p 96.7%

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

   3. *-commutative 96.7%

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

   4. times-frac 96.7%

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

   5. metadata-eval 96.7%

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

   6. *-lft-identity 96.7%

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

   7. associate-/l* 96.7%

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

   8. associate-/l* 96.7%

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

5. Simplified 96.7%

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

   1. add-cbrt-cube 32.6%

      \[\leadsto -\frac{\log \left(\frac{\cosh \left(\frac{\pi}{\frac{4}{f}}\right)}{\color{blue}{\sqrt[3]{\left(\sinh \left(\frac{\pi}{\frac{4}{f}}\right) \cdot \sinh \left(\frac{\pi}{\frac{4}{f}}\right)\right) \cdot \sinh \left(\frac{\pi}{\frac{4}{f}}\right)}}}\right)}{0.25 \cdot \pi} \]

   2. pow3 32.6%

      \[\leadsto -\frac{\log \left(\frac{\cosh \left(\frac{\pi}{\frac{4}{f}}\right)}{\sqrt[3]{\color{blue}{{\sinh \left(\frac{\pi}{\frac{4}{f}}\right)}^{3}}}}\right)}{0.25 \cdot \pi} \]

   3. associate-/l* 32.6%

      \[\leadsto -\frac{\log \left(\frac{\cosh \left(\frac{\pi}{\frac{4}{f}}\right)}{\sqrt[3]{{\sinh \color{blue}{\left(\frac{\pi \cdot f}{4}\right)}}^{3}}}\right)}{0.25 \cdot \pi} \]

   4. associate-*l/ 32.6%

      \[\leadsto -\frac{\log \left(\frac{\cosh \left(\frac{\pi}{\frac{4}{f}}\right)}{\sqrt[3]{{\sinh \color{blue}{\left(\frac{\pi}{4} \cdot f\right)}}^{3}}}\right)}{0.25 \cdot \pi} \]

   5. *-commutative 32.6%

      \[\leadsto -\frac{\log \left(\frac{\cosh \left(\frac{\pi}{\frac{4}{f}}\right)}{\sqrt[3]{{\sinh \color{blue}{\left(f \cdot \frac{\pi}{4}\right)}}^{3}}}\right)}{0.25 \cdot \pi} \]

   6. div-inv 32.6%

      \[\leadsto -\frac{\log \left(\frac{\cosh \left(\frac{\pi}{\frac{4}{f}}\right)}{\sqrt[3]{{\sinh \left(f \cdot \color{blue}{\left(\pi \cdot \frac{1}{4}\right)}\right)}^{3}}}\right)}{0.25 \cdot \pi} \]

   7. metadata-eval 32.6%

      \[\leadsto -\frac{\log \left(\frac{\cosh \left(\frac{\pi}{\frac{4}{f}}\right)}{\sqrt[3]{{\sinh \left(f \cdot \left(\pi \cdot \color{blue}{0.25}\right)\right)}^{3}}}\right)}{0.25 \cdot \pi} \]

7. Applied egg-rr 32.6%

   \[\leadsto -\frac{\log \left(\frac{\cosh \left(\frac{\pi}{\frac{4}{f}}\right)}{\color{blue}{\sqrt[3]{{\sinh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}^{3}}}}\right)}{0.25 \cdot \pi} \]

8. Taylor expanded in f around 0 95.5%

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

   1. *-commutative 95.5%

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

   2. *-commutative 95.5%

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

   3. associate-*l* 95.5%

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

10. Simplified 95.5%

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

11. Final simplification 95.5%

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

Alternative 3: 96.3% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

   1. expm1-log1p-u 6.4%

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

   2. expm1-udef 6.4%

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

3. Applied egg-rr 95.4%

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

   1. expm1-def 95.4%

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

   2. expm1-log1p 96.7%

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

   3. *-commutative 96.7%

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

   4. times-frac 96.7%

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

   5. metadata-eval 96.7%

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

   6. *-lft-identity 96.7%

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

   7. associate-/l* 96.7%

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

   8. associate-/l* 96.7%

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

5. Simplified 96.7%

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

6. Taylor expanded in f around 0 95.4%

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

   1. add-sqr-sqrt 95.0%

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

   2. sqrt-unprod 95.5%

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

   3. pow2 95.5%

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

   4. frac-2neg 95.5%

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

   5. neg-log 95.5%

      \[\leadsto -\sqrt{{\left(\frac{\color{blue}{\log \left(\frac{1}{\frac{4}{f \cdot \pi}}\right)}}{-0.25 \cdot \pi}\right)}^{2}} \]

   6. clear-num 95.5%

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

   7. div-inv 95.5%

      \[\leadsto -\sqrt{{\left(\frac{\log \color{blue}{\left(\left(f \cdot \pi\right) \cdot \frac{1}{4}\right)}}{-0.25 \cdot \pi}\right)}^{2}} \]

   8. metadata-eval 95.5%

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

   9. *-commutative 95.5%

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

   10. associate-*r* 95.5%

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

   11. *-commutative 95.5%

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

   12. distribute-rgt-neg-in 95.5%

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

   13. metadata-eval 95.5%

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

8. Applied egg-rr 95.5%

   \[\leadsto -\color{blue}{\sqrt{{\left(\frac{\log \left(\pi \cdot \left(f \cdot 0.25\right)\right)}{\pi \cdot -0.25}\right)}^{2}}} \]
9. Step-by-step derivation

   1. unpow2 95.5%

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

   2. rem-sqrt-square 95.5%

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

10. Simplified 95.5%

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

11. Final simplification 95.5%

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

Alternative 4: 96.1% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

   1. expm1-log1p-u 6.4%

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

   2. expm1-udef 6.4%

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

3. Applied egg-rr 95.4%

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

   1. expm1-def 95.4%

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

   2. expm1-log1p 96.7%

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

   3. *-commutative 96.7%

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

   4. times-frac 96.7%

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

   5. metadata-eval 96.7%

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

   6. *-lft-identity 96.7%

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

   7. associate-/l* 96.7%

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

   8. associate-/l* 96.7%

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

5. Simplified 96.7%

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

6. Taylor expanded in f around 0 95.4%

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

7. Taylor expanded in f around 0 95.4%

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

   1. *-commutative 95.4%

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

   2. mul-1-neg 95.4%

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

   3. log-rec 95.4%

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

   4. +-commutative 95.4%

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

   5. log-rec 95.4%

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

   6. unsub-neg 95.4%

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

   7. log-div 95.4%

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

   8. associate-/r* 95.4%

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

   9. metadata-eval 95.4%

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

   10. times-frac 95.4%

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

   11. *-commutative 95.4%

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

   12. *-lft-identity 95.4%

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

   13. associate-*r/ 95.3%

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

   14. *-commutative 95.3%

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

   15. *-commutative 95.3%

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

9. Simplified 95.3%

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

10. Final simplification 95.3%

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

Alternative 5: 96.2% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

   1. expm1-log1p-u 6.4%

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

   2. expm1-udef 6.4%

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

3. Applied egg-rr 95.4%

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

   1. expm1-def 95.4%

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

   2. expm1-log1p 96.7%

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

   3. *-commutative 96.7%

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

   4. times-frac 96.7%

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

   5. metadata-eval 96.7%

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

   6. *-lft-identity 96.7%

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

   7. associate-/l* 96.7%

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

   8. associate-/l* 96.7%

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

5. Simplified 96.7%

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

6. Taylor expanded in f around 0 95.4%

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

7. Final simplification 95.4%

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

Alternative 6: 0.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \log 0 \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (log 0.0) (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(0.0) * (-1.0 / (((double) M_PI) / 4.0));
}

Java

public static double code(double f) {
	return Math.log(0.0) * (-1.0 / (Math.PI / 4.0));
}

Python

def code(f):
	return math.log(0.0) * (-1.0 / (math.pi / 4.0))

Julia

function code(f)
	return Float64(log(0.0) * Float64(-1.0 / Float64(pi / 4.0)))
end

MATLAB

function tmp = code(f)
	tmp = log(0.0) * (-1.0 / (pi / 4.0));
end

Wolfram

code[f_] := N[(N[Log[0.0], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.4%

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

2. Taylor expanded in f around 0 95.3%

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

   1. distribute-rgt-out-- 95.3%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{2}{\color{blue}{\left(\pi \cdot \left(0.25 - -0.25\right)\right)} \cdot f}\right) \]

   2. metadata-eval 95.3%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{2}{\left(\pi \cdot \color{blue}{0.5}\right) \cdot f}\right) \]

4. Simplified 95.3%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{2}{\left(\pi \cdot 0.5\right) \cdot f}\right)} \]
5. Step-by-step derivation

   1. add-log-exp 3.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\log \left(e^{\frac{2}{\left(\pi \cdot 0.5\right) \cdot f}}\right)} \]

   2. associate-*l* 3.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \log \left(e^{\frac{2}{\color{blue}{\pi \cdot \left(0.5 \cdot f\right)}}}\right) \]

6. Applied egg-rr 3.4%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\log \left(e^{\frac{2}{\pi \cdot \left(0.5 \cdot f\right)}}\right)} \]

7. Taylor expanded in f around inf 0.7%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \log \color{blue}{1} \]

8. Final simplification 0.7%

   \[\leadsto \log 0 \cdot \frac{-1}{\frac{\pi}{4}} \]

Reproduce

herbie shell --seed 2023252 
(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)))))))))