VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 99.0%

Time: 18.2s

Alternatives: 5

Speedup: 4.9×

• could not determine a ground truth

  1. f = 1474397023.13528

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: 6.7% 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

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

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 7.9%

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

   1. *-commutative N/A

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

   2. un-div-inv N/A

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

   3. div-inv N/A

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

   4. associate-/r* N/A

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

4. Applied egg-rr 98.9%

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

Alternative 2: 96.5% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := f \cdot \left(f \cdot \pi\right)\\ \frac{4 \cdot \log \left(\frac{f}{\frac{16}{\pi \cdot \pi} - \left(t\_0 \cdot t\_0\right) \cdot 0.006944444444444444} \cdot \left(\frac{4}{\pi} - f \cdot \left(f \cdot \left(\pi \cdot 0.08333333333333333\right)\right)\right)\right)}{\pi} \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* f (* f PI))))
   (/
    (*
     4.0
     (log
      (*
       (/ f (- (/ 16.0 (* PI PI)) (* (* t_0 t_0) 0.006944444444444444)))
       (- (/ 4.0 PI) (* f (* f (* PI 0.08333333333333333)))))))
    PI)))

C

double code(double f) {
	double t_0 = f * (f * ((double) M_PI));
	return (4.0 * log(((f / ((16.0 / (((double) M_PI) * ((double) M_PI))) - ((t_0 * t_0) * 0.006944444444444444))) * ((4.0 / ((double) M_PI)) - (f * (f * (((double) M_PI) * 0.08333333333333333))))))) / ((double) M_PI);
}

Java

public static double code(double f) {
	double t_0 = f * (f * Math.PI);
	return (4.0 * Math.log(((f / ((16.0 / (Math.PI * Math.PI)) - ((t_0 * t_0) * 0.006944444444444444))) * ((4.0 / Math.PI) - (f * (f * (Math.PI * 0.08333333333333333))))))) / Math.PI;
}

Python

def code(f):
	t_0 = f * (f * math.pi)
	return (4.0 * math.log(((f / ((16.0 / (math.pi * math.pi)) - ((t_0 * t_0) * 0.006944444444444444))) * ((4.0 / math.pi) - (f * (f * (math.pi * 0.08333333333333333))))))) / math.pi

Julia

function code(f)
	t_0 = Float64(f * Float64(f * pi))
	return Float64(Float64(4.0 * log(Float64(Float64(f / Float64(Float64(16.0 / Float64(pi * pi)) - Float64(Float64(t_0 * t_0) * 0.006944444444444444))) * Float64(Float64(4.0 / pi) - Float64(f * Float64(f * Float64(pi * 0.08333333333333333))))))) / pi)
end

MATLAB

function tmp = code(f)
	t_0 = f * (f * pi);
	tmp = (4.0 * log(((f / ((16.0 / (pi * pi)) - ((t_0 * t_0) * 0.006944444444444444))) * ((4.0 / pi) - (f * (f * (pi * 0.08333333333333333))))))) / pi;
end

Wolfram

code[f_] := Block[{t$95$0 = N[(f * N[(f * Pi), $MachinePrecision]), $MachinePrecision]}, N[(N[(4.0 * N[Log[N[(N[(f / N[(N[(16.0 / N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.006944444444444444), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(4.0 / Pi), $MachinePrecision] - N[(f * N[(f * N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]]

Derivation

1. Initial program 7.9%

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

4. Applied egg-rr 8.2%

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

5. Taylor expanded in f around 0

   \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 4\right)\right), \mathsf{log.f64}\left(\color{blue}{\left(\frac{4 \cdot \left({f}^{2} \cdot \left(\left(\frac{-1}{48} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}} + \frac{1}{16} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}\right) - \frac{1}{48} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right) + 4 \cdot \frac{1}{\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}}{f}\right)}\right)\right)\right) \]

7. Simplified 96.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{\frac{4}{\pi} + \left(\pi \cdot 0.0625 + \pi \cdot -0.041666666666666664\right) \cdot \left(4 \cdot \left(f \cdot f\right)\right)}{f}\right)} \]
8. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{neg}\left(\frac{4}{\mathsf{PI}\left(\right)} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)} + \left(\mathsf{PI}\left(\right) \cdot \frac{1}{16} + \mathsf{PI}\left(\right) \cdot \frac{-1}{24}\right) \cdot \left(4 \cdot \left(f \cdot f\right)\right)}{f}\right)\right) \]

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

      \[\leadsto \frac{4}{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)} + \left(\mathsf{PI}\left(\right) \cdot \frac{1}{16} + \mathsf{PI}\left(\right) \cdot \frac{-1}{24}\right) \cdot \left(4 \cdot \left(f \cdot f\right)\right)}{f}\right)\right)\right)} \]

   3. associate-*l/ N/A

      \[\leadsto \frac{4 \cdot \left(\mathsf{neg}\left(\log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)} + \left(\mathsf{PI}\left(\right) \cdot \frac{1}{16} + \mathsf{PI}\left(\right) \cdot \frac{-1}{24}\right) \cdot \left(4 \cdot \left(f \cdot f\right)\right)}{f}\right)\right)\right)}{\color{blue}{\mathsf{PI}\left(\right)}} \]

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

      \[\leadsto \mathsf{/.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)} + \left(\mathsf{PI}\left(\right) \cdot \frac{1}{16} + \mathsf{PI}\left(\right) \cdot \frac{-1}{24}\right) \cdot \left(4 \cdot \left(f \cdot f\right)\right)}{f}\right)\right)\right)\right), \color{blue}{\mathsf{PI}\left(\right)}\right) \]

9. Applied egg-rr 96.7%

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

    1. flip-+ N/A

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

    2. associate-/r/ N/A

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

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

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

11. Applied egg-rr 96.7%

    \[\leadsto \frac{4 \cdot \log \color{blue}{\left(\frac{f}{\frac{16}{\pi \cdot \pi} - \left(\left(f \cdot \left(f \cdot \pi\right)\right) \cdot \left(f \cdot \left(f \cdot \pi\right)\right)\right) \cdot 0.006944444444444444} \cdot \left(\frac{4}{\pi} - f \cdot \left(f \cdot \left(\pi \cdot 0.08333333333333333\right)\right)\right)\right)}}{\pi} \]
12. Add Preprocessing

Alternative 3: 96.5% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.9%

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

4. Applied egg-rr 8.2%

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

5. Taylor expanded in f around 0

   \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 4\right)\right), \mathsf{log.f64}\left(\color{blue}{\left(\frac{4 \cdot \left({f}^{2} \cdot \left(\left(\frac{-1}{48} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}} + \frac{1}{16} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}\right) - \frac{1}{48} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right) + 4 \cdot \frac{1}{\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}}{f}\right)}\right)\right)\right) \]

7. Simplified 96.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{\frac{4}{\pi} + \left(\pi \cdot 0.0625 + \pi \cdot -0.041666666666666664\right) \cdot \left(4 \cdot \left(f \cdot f\right)\right)}{f}\right)} \]
8. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{neg}\left(\frac{4}{\mathsf{PI}\left(\right)} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)} + \left(\mathsf{PI}\left(\right) \cdot \frac{1}{16} + \mathsf{PI}\left(\right) \cdot \frac{-1}{24}\right) \cdot \left(4 \cdot \left(f \cdot f\right)\right)}{f}\right)\right) \]

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

      \[\leadsto \frac{4}{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)} + \left(\mathsf{PI}\left(\right) \cdot \frac{1}{16} + \mathsf{PI}\left(\right) \cdot \frac{-1}{24}\right) \cdot \left(4 \cdot \left(f \cdot f\right)\right)}{f}\right)\right)\right)} \]

   3. associate-*l/ N/A

      \[\leadsto \frac{4 \cdot \left(\mathsf{neg}\left(\log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)} + \left(\mathsf{PI}\left(\right) \cdot \frac{1}{16} + \mathsf{PI}\left(\right) \cdot \frac{-1}{24}\right) \cdot \left(4 \cdot \left(f \cdot f\right)\right)}{f}\right)\right)\right)}{\color{blue}{\mathsf{PI}\left(\right)}} \]

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

      \[\leadsto \mathsf{/.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)} + \left(\mathsf{PI}\left(\right) \cdot \frac{1}{16} + \mathsf{PI}\left(\right) \cdot \frac{-1}{24}\right) \cdot \left(4 \cdot \left(f \cdot f\right)\right)}{f}\right)\right)\right)\right), \color{blue}{\mathsf{PI}\left(\right)}\right) \]

9. Applied egg-rr 96.7%

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

10. Final simplification 96.7%

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

Alternative 4: 96.0% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.9%

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

   1. *-commutative N/A

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

   2. un-div-inv N/A

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

   3. div-inv N/A

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

   4. associate-/r* N/A

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

4. Applied egg-rr 98.9%

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

5. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

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

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

   6. PI-lowering-PI.f64 96.0%

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

7. Simplified 96.0%

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

8. Final simplification 96.0%

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

Alternative 5: 95.9% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.9%

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

3. Taylor expanded in f around 0

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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

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

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

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

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

   8. metadata-eval N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. metadata-eval N/A

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

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

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

   13. PI-lowering-PI.f64 95.9%

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

5. Simplified 95.9%

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

   1. clear-num N/A

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

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

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

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

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

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

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

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

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

   6. neg-log N/A

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

   7. clear-num N/A

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

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

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

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

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

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

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

   11. PI-lowering-PI.f64 95.9%

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

7. Applied egg-rr 95.9%

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

Reproduce

herbie shell --seed 2024191 
(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)))))))))