VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.0% → 96.2%

Time: 23.4s

Alternatives: 7

Speedup: 4.8×

• could not determine a ground truth

  1. f = 2.00000000000000007e154

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 96.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{2}{\pi}}\\ \log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{{t\_0}^{2}}{f} \cdot \frac{t\_0}{0.5}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (cbrt (/ 2.0 PI))))
   (*
    (log
     (fma f (* PI 0.08333333333333333) (* (/ (pow t_0 2.0) f) (/ t_0 0.5))))
    (/ -1.0 (/ PI 4.0)))))

C

double code(double f) {
	double t_0 = cbrt((2.0 / ((double) M_PI)));
	return log(fma(f, (((double) M_PI) * 0.08333333333333333), ((pow(t_0, 2.0) / f) * (t_0 / 0.5)))) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	t_0 = cbrt(Float64(2.0 / pi))
	return Float64(log(fma(f, Float64(pi * 0.08333333333333333), Float64(Float64((t_0 ^ 2.0) / f) * Float64(t_0 / 0.5)))) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

code[f_] := Block[{t$95$0 = N[Power[N[(2.0 / Pi), $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[Log[N[(f * N[(Pi * 0.08333333333333333), $MachinePrecision] + N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] / f), $MachinePrecision] * N[(t$95$0 / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 5.4%

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

3. Taylor expanded in f around 0 96.7%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + \left(0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + \left(f \cdot \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right) + 2 \cdot \frac{1}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}\right)\right)\right)} \]

5. Simplified 96.7%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332, -2, 0.0625 \cdot \frac{\pi}{0.5}\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right)} \]
6. Step-by-step derivation

   1. fma-udef 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332\right) \cdot -2 + 0.0625 \cdot \frac{\pi}{0.5}}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   2. pow-div 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left(\color{blue}{{\pi}^{\left(3 - 2\right)}} \cdot 0.020833333333333332\right) \cdot -2 + 0.0625 \cdot \frac{\pi}{0.5}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   3. metadata-eval 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left({\pi}^{\color{blue}{1}} \cdot 0.020833333333333332\right) \cdot -2 + 0.0625 \cdot \frac{\pi}{0.5}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   4. pow1 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left(\color{blue}{\pi} \cdot 0.020833333333333332\right) \cdot -2 + 0.0625 \cdot \frac{\pi}{0.5}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   5. associate-*r/ 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left(\pi \cdot 0.020833333333333332\right) \cdot -2 + \color{blue}{\frac{0.0625 \cdot \pi}{0.5}}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   6. div-inv 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left(\pi \cdot 0.020833333333333332\right) \cdot -2 + \color{blue}{\left(0.0625 \cdot \pi\right) \cdot \frac{1}{0.5}}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   7. metadata-eval 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left(\pi \cdot 0.020833333333333332\right) \cdot -2 + \left(0.0625 \cdot \pi\right) \cdot \color{blue}{2}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

7. Applied egg-rr 96.7%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\left(\pi \cdot 0.020833333333333332\right) \cdot -2 + \left(0.0625 \cdot \pi\right) \cdot 2}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]
8. Step-by-step derivation

   1. associate-*l* 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(0.020833333333333332 \cdot -2\right)} + \left(0.0625 \cdot \pi\right) \cdot 2, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   2. metadata-eval 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot \color{blue}{-0.041666666666666664} + \left(0.0625 \cdot \pi\right) \cdot 2, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   3. *-commutative 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot -0.041666666666666664 + \color{blue}{2 \cdot \left(0.0625 \cdot \pi\right)}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   4. *-commutative 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot -0.041666666666666664 + 2 \cdot \color{blue}{\left(\pi \cdot 0.0625\right)}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

9. Simplified 96.7%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\pi \cdot -0.041666666666666664 + 2 \cdot \left(\pi \cdot 0.0625\right)}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]
10. Step-by-step derivation

    1. associate-/l/ 96.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot -0.041666666666666664 + 2 \cdot \left(\pi \cdot 0.0625\right), \color{blue}{\frac{\frac{2}{\pi}}{f \cdot 0.5}}\right)\right) \]

    2. add-cube-cbrt 96.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot -0.041666666666666664 + 2 \cdot \left(\pi \cdot 0.0625\right), \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\pi}} \cdot \sqrt[3]{\frac{2}{\pi}}\right) \cdot \sqrt[3]{\frac{2}{\pi}}}}{f \cdot 0.5}\right)\right) \]

    3. times-frac 96.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot -0.041666666666666664 + 2 \cdot \left(\pi \cdot 0.0625\right), \color{blue}{\frac{\sqrt[3]{\frac{2}{\pi}} \cdot \sqrt[3]{\frac{2}{\pi}}}{f} \cdot \frac{\sqrt[3]{\frac{2}{\pi}}}{0.5}}\right)\right) \]

    4. pow2 96.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot -0.041666666666666664 + 2 \cdot \left(\pi \cdot 0.0625\right), \frac{\color{blue}{{\left(\sqrt[3]{\frac{2}{\pi}}\right)}^{2}}}{f} \cdot \frac{\sqrt[3]{\frac{2}{\pi}}}{0.5}\right)\right) \]

11. Applied egg-rr 96.7%

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot -0.041666666666666664 + 2 \cdot \left(\pi \cdot 0.0625\right), \color{blue}{\frac{{\left(\sqrt[3]{\frac{2}{\pi}}\right)}^{2}}{f} \cdot \frac{\sqrt[3]{\frac{2}{\pi}}}{0.5}}\right)\right) \]
12. Step-by-step derivation

    1. expm1-log1p-u 96.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot -0.041666666666666664 + 2 \cdot \left(\pi \cdot 0.0625\right)\right)\right)}, \frac{{\left(\sqrt[3]{\frac{2}{\pi}}\right)}^{2}}{f} \cdot \frac{\sqrt[3]{\frac{2}{\pi}}}{0.5}\right)\right) \]

    2. expm1-udef 96.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{e^{\mathsf{log1p}\left(\pi \cdot -0.041666666666666664 + 2 \cdot \left(\pi \cdot 0.0625\right)\right)} - 1}, \frac{{\left(\sqrt[3]{\frac{2}{\pi}}\right)}^{2}}{f} \cdot \frac{\sqrt[3]{\frac{2}{\pi}}}{0.5}\right)\right) \]

    3. fma-def 96.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\pi, -0.041666666666666664, 2 \cdot \left(\pi \cdot 0.0625\right)\right)}\right)} - 1, \frac{{\left(\sqrt[3]{\frac{2}{\pi}}\right)}^{2}}{f} \cdot \frac{\sqrt[3]{\frac{2}{\pi}}}{0.5}\right)\right) \]

    4. *-commutative 96.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, e^{\mathsf{log1p}\left(\mathsf{fma}\left(\pi, -0.041666666666666664, \color{blue}{\left(\pi \cdot 0.0625\right) \cdot 2}\right)\right)} - 1, \frac{{\left(\sqrt[3]{\frac{2}{\pi}}\right)}^{2}}{f} \cdot \frac{\sqrt[3]{\frac{2}{\pi}}}{0.5}\right)\right) \]

    5. associate-*l* 96.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, e^{\mathsf{log1p}\left(\mathsf{fma}\left(\pi, -0.041666666666666664, \color{blue}{\pi \cdot \left(0.0625 \cdot 2\right)}\right)\right)} - 1, \frac{{\left(\sqrt[3]{\frac{2}{\pi}}\right)}^{2}}{f} \cdot \frac{\sqrt[3]{\frac{2}{\pi}}}{0.5}\right)\right) \]

    6. metadata-eval 96.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, e^{\mathsf{log1p}\left(\mathsf{fma}\left(\pi, -0.041666666666666664, \pi \cdot \color{blue}{0.125}\right)\right)} - 1, \frac{{\left(\sqrt[3]{\frac{2}{\pi}}\right)}^{2}}{f} \cdot \frac{\sqrt[3]{\frac{2}{\pi}}}{0.5}\right)\right) \]

13. Applied egg-rr 96.7%

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\pi, -0.041666666666666664, \pi \cdot 0.125\right)\right)} - 1}, \frac{{\left(\sqrt[3]{\frac{2}{\pi}}\right)}^{2}}{f} \cdot \frac{\sqrt[3]{\frac{2}{\pi}}}{0.5}\right)\right) \]
14. Step-by-step derivation

    1. expm1-def 96.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, -0.041666666666666664, \pi \cdot 0.125\right)\right)\right)}, \frac{{\left(\sqrt[3]{\frac{2}{\pi}}\right)}^{2}}{f} \cdot \frac{\sqrt[3]{\frac{2}{\pi}}}{0.5}\right)\right) \]

    2. expm1-log1p 96.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\mathsf{fma}\left(\pi, -0.041666666666666664, \pi \cdot 0.125\right)}, \frac{{\left(\sqrt[3]{\frac{2}{\pi}}\right)}^{2}}{f} \cdot \frac{\sqrt[3]{\frac{2}{\pi}}}{0.5}\right)\right) \]

    3. fma-udef 96.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\pi \cdot -0.041666666666666664 + \pi \cdot 0.125}, \frac{{\left(\sqrt[3]{\frac{2}{\pi}}\right)}^{2}}{f} \cdot \frac{\sqrt[3]{\frac{2}{\pi}}}{0.5}\right)\right) \]

    4. distribute-lft-out 96.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(-0.041666666666666664 + 0.125\right)}, \frac{{\left(\sqrt[3]{\frac{2}{\pi}}\right)}^{2}}{f} \cdot \frac{\sqrt[3]{\frac{2}{\pi}}}{0.5}\right)\right) \]

    5. metadata-eval 96.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.08333333333333333}, \frac{{\left(\sqrt[3]{\frac{2}{\pi}}\right)}^{2}}{f} \cdot \frac{\sqrt[3]{\frac{2}{\pi}}}{0.5}\right)\right) \]

15. Simplified 96.7%

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\pi \cdot 0.08333333333333333}, \frac{{\left(\sqrt[3]{\frac{2}{\pi}}\right)}^{2}}{f} \cdot \frac{\sqrt[3]{\frac{2}{\pi}}}{0.5}\right)\right) \]

16. Final simplification 96.7%

    \[\leadsto \log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{{\left(\sqrt[3]{\frac{2}{\pi}}\right)}^{2}}{f} \cdot \frac{\sqrt[3]{\frac{2}{\pi}}}{0.5}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}} \]
17. Add Preprocessing

Alternative 2: 96.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \log \left(\mathsf{fma}\left(f, \pi \cdot -0.041666666666666664 + 2 \cdot \left(\pi \cdot 0.0625\right), \frac{\frac{4}{\pi \cdot \sqrt{f}}}{\sqrt{f}}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (fma
    f
    (+ (* PI -0.041666666666666664) (* 2.0 (* PI 0.0625)))
    (/ (/ 4.0 (* PI (sqrt f))) (sqrt f))))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(fma(f, ((((double) M_PI) * -0.041666666666666664) + (2.0 * (((double) M_PI) * 0.0625))), ((4.0 / (((double) M_PI) * sqrt(f))) / sqrt(f)))) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(log(fma(f, Float64(Float64(pi * -0.041666666666666664) + Float64(2.0 * Float64(pi * 0.0625))), Float64(Float64(4.0 / Float64(pi * sqrt(f))) / sqrt(f)))) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

code[f_] := N[(N[Log[N[(f * N[(N[(Pi * -0.041666666666666664), $MachinePrecision] + N[(2.0 * N[(Pi * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 / N[(Pi * N[Sqrt[f], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[f], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.4%

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

3. Taylor expanded in f around 0 96.7%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + \left(0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + \left(f \cdot \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right) + 2 \cdot \frac{1}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}\right)\right)\right)} \]

5. Simplified 96.7%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332, -2, 0.0625 \cdot \frac{\pi}{0.5}\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right)} \]
6. Step-by-step derivation

   1. fma-udef 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332\right) \cdot -2 + 0.0625 \cdot \frac{\pi}{0.5}}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   2. pow-div 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left(\color{blue}{{\pi}^{\left(3 - 2\right)}} \cdot 0.020833333333333332\right) \cdot -2 + 0.0625 \cdot \frac{\pi}{0.5}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   3. metadata-eval 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left({\pi}^{\color{blue}{1}} \cdot 0.020833333333333332\right) \cdot -2 + 0.0625 \cdot \frac{\pi}{0.5}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   4. pow1 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left(\color{blue}{\pi} \cdot 0.020833333333333332\right) \cdot -2 + 0.0625 \cdot \frac{\pi}{0.5}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   5. associate-*r/ 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left(\pi \cdot 0.020833333333333332\right) \cdot -2 + \color{blue}{\frac{0.0625 \cdot \pi}{0.5}}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   6. div-inv 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left(\pi \cdot 0.020833333333333332\right) \cdot -2 + \color{blue}{\left(0.0625 \cdot \pi\right) \cdot \frac{1}{0.5}}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   7. metadata-eval 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left(\pi \cdot 0.020833333333333332\right) \cdot -2 + \left(0.0625 \cdot \pi\right) \cdot \color{blue}{2}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

7. Applied egg-rr 96.7%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\left(\pi \cdot 0.020833333333333332\right) \cdot -2 + \left(0.0625 \cdot \pi\right) \cdot 2}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]
8. Step-by-step derivation

   1. associate-*l* 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(0.020833333333333332 \cdot -2\right)} + \left(0.0625 \cdot \pi\right) \cdot 2, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   2. metadata-eval 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot \color{blue}{-0.041666666666666664} + \left(0.0625 \cdot \pi\right) \cdot 2, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   3. *-commutative 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot -0.041666666666666664 + \color{blue}{2 \cdot \left(0.0625 \cdot \pi\right)}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   4. *-commutative 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot -0.041666666666666664 + 2 \cdot \color{blue}{\left(\pi \cdot 0.0625\right)}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

9. Simplified 96.7%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\pi \cdot -0.041666666666666664 + 2 \cdot \left(\pi \cdot 0.0625\right)}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]
10. Step-by-step derivation

    1. associate-/l/ 96.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot -0.041666666666666664 + 2 \cdot \left(\pi \cdot 0.0625\right), \frac{\color{blue}{\frac{2}{0.5 \cdot \pi}}}{f}\right)\right) \]

    2. associate-/r* 96.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot -0.041666666666666664 + 2 \cdot \left(\pi \cdot 0.0625\right), \frac{\color{blue}{\frac{\frac{2}{0.5}}{\pi}}}{f}\right)\right) \]

    3. metadata-eval 96.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot -0.041666666666666664 + 2 \cdot \left(\pi \cdot 0.0625\right), \frac{\frac{\color{blue}{4}}{\pi}}{f}\right)\right) \]

    4. *-un-lft-identity 96.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot -0.041666666666666664 + 2 \cdot \left(\pi \cdot 0.0625\right), \frac{\color{blue}{1 \cdot \frac{4}{\pi}}}{f}\right)\right) \]

    5. add-sqr-sqrt 96.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot -0.041666666666666664 + 2 \cdot \left(\pi \cdot 0.0625\right), \frac{1 \cdot \frac{4}{\pi}}{\color{blue}{\sqrt{f} \cdot \sqrt{f}}}\right)\right) \]

    6. times-frac 96.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot -0.041666666666666664 + 2 \cdot \left(\pi \cdot 0.0625\right), \color{blue}{\frac{1}{\sqrt{f}} \cdot \frac{\frac{4}{\pi}}{\sqrt{f}}}\right)\right) \]

11. Applied egg-rr 96.7%

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot -0.041666666666666664 + 2 \cdot \left(\pi \cdot 0.0625\right), \color{blue}{\frac{1}{\sqrt{f}} \cdot \frac{\frac{4}{\pi}}{\sqrt{f}}}\right)\right) \]
12. Step-by-step derivation

    1. associate-*l/ 96.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot -0.041666666666666664 + 2 \cdot \left(\pi \cdot 0.0625\right), \color{blue}{\frac{1 \cdot \frac{\frac{4}{\pi}}{\sqrt{f}}}{\sqrt{f}}}\right)\right) \]

    2. *-lft-identity 96.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot -0.041666666666666664 + 2 \cdot \left(\pi \cdot 0.0625\right), \frac{\color{blue}{\frac{\frac{4}{\pi}}{\sqrt{f}}}}{\sqrt{f}}\right)\right) \]

    3. associate-/l/ 96.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot -0.041666666666666664 + 2 \cdot \left(\pi \cdot 0.0625\right), \frac{\color{blue}{\frac{4}{\sqrt{f} \cdot \pi}}}{\sqrt{f}}\right)\right) \]

13. Simplified 96.7%

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot -0.041666666666666664 + 2 \cdot \left(\pi \cdot 0.0625\right), \color{blue}{\frac{\frac{4}{\sqrt{f} \cdot \pi}}{\sqrt{f}}}\right)\right) \]

14. Final simplification 96.7%

    \[\leadsto \log \left(\mathsf{fma}\left(f, \pi \cdot -0.041666666666666664 + 2 \cdot \left(\pi \cdot 0.0625\right), \frac{\frac{4}{\pi \cdot \sqrt{f}}}{\sqrt{f}}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}} \]
15. Add Preprocessing

Alternative 3: 96.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log (fma f (* PI 0.08333333333333333) (/ (/ (/ 2.0 PI) 0.5) f)))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(fma(f, (((double) M_PI) * 0.08333333333333333), (((2.0 / ((double) M_PI)) / 0.5) / f))) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(log(fma(f, Float64(pi * 0.08333333333333333), Float64(Float64(Float64(2.0 / pi) / 0.5) / f))) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

code[f_] := N[(N[Log[N[(f * N[(Pi * 0.08333333333333333), $MachinePrecision] + N[(N[(N[(2.0 / Pi), $MachinePrecision] / 0.5), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.4%

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

3. Taylor expanded in f around 0 96.7%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + \left(0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + \left(f \cdot \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right) + 2 \cdot \frac{1}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}\right)\right)\right)} \]

5. Simplified 96.7%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332, -2, 0.0625 \cdot \frac{\pi}{0.5}\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right)} \]
6. Step-by-step derivation

   1. add-log-exp 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\log \left(e^{\mathsf{fma}\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332, -2, 0.0625 \cdot \frac{\pi}{0.5}\right)}\right)}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   2. pow-div 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \log \left(e^{\mathsf{fma}\left(\color{blue}{{\pi}^{\left(3 - 2\right)}} \cdot 0.020833333333333332, -2, 0.0625 \cdot \frac{\pi}{0.5}\right)}\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   3. metadata-eval 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \log \left(e^{\mathsf{fma}\left({\pi}^{\color{blue}{1}} \cdot 0.020833333333333332, -2, 0.0625 \cdot \frac{\pi}{0.5}\right)}\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   4. pow1 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \log \left(e^{\mathsf{fma}\left(\color{blue}{\pi} \cdot 0.020833333333333332, -2, 0.0625 \cdot \frac{\pi}{0.5}\right)}\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   5. associate-*r/ 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \log \left(e^{\mathsf{fma}\left(\pi \cdot 0.020833333333333332, -2, \color{blue}{\frac{0.0625 \cdot \pi}{0.5}}\right)}\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   6. div-inv 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \log \left(e^{\mathsf{fma}\left(\pi \cdot 0.020833333333333332, -2, \color{blue}{\left(0.0625 \cdot \pi\right) \cdot \frac{1}{0.5}}\right)}\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   7. metadata-eval 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \log \left(e^{\mathsf{fma}\left(\pi \cdot 0.020833333333333332, -2, \left(0.0625 \cdot \pi\right) \cdot \color{blue}{2}\right)}\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

7. Applied egg-rr 96.7%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\log \left(e^{\mathsf{fma}\left(\pi \cdot 0.020833333333333332, -2, \left(0.0625 \cdot \pi\right) \cdot 2\right)}\right)}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]
8. Step-by-step derivation

   1. rem-log-exp 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\mathsf{fma}\left(\pi \cdot 0.020833333333333332, -2, \left(0.0625 \cdot \pi\right) \cdot 2\right)}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   2. fma-udef 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\left(\pi \cdot 0.020833333333333332\right) \cdot -2 + \left(0.0625 \cdot \pi\right) \cdot 2}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   3. associate-*l* 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(0.020833333333333332 \cdot -2\right)} + \left(0.0625 \cdot \pi\right) \cdot 2, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   4. metadata-eval 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot \color{blue}{-0.041666666666666664} + \left(0.0625 \cdot \pi\right) \cdot 2, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   5. *-commutative 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot -0.041666666666666664 + \color{blue}{\left(\pi \cdot 0.0625\right)} \cdot 2, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   6. associate-*l* 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot -0.041666666666666664 + \color{blue}{\pi \cdot \left(0.0625 \cdot 2\right)}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   7. metadata-eval 96.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot -0.041666666666666664 + \pi \cdot \color{blue}{0.125}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

9. Applied egg-rr 96.7%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\pi \cdot -0.041666666666666664 + \pi \cdot 0.125}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]
10. Step-by-step derivation

    1. distribute-lft-out 96.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(-0.041666666666666664 + 0.125\right)}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

    2. metadata-eval 96.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.08333333333333333}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

11. Simplified 96.7%

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\pi \cdot 0.08333333333333333}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

12. Final simplification 96.7%

    \[\leadsto \log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}} \]
13. Add Preprocessing

Alternative 4: 95.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.4%

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

3. Taylor expanded in f around 0 96.5%

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

   1. mul-1-neg 96.5%

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

   2. unsub-neg 96.5%

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

   3. distribute-rgt-out-- 96.5%

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

   4. metadata-eval 96.5%

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

   5. associate-/r* 96.5%

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

5. Simplified 96.5%

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

   1. add-sqr-sqrt 96.0%

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

   2. sqrt-unprod 96.6%

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

   3. pow2 96.6%

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

   4. diff-log 96.6%

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

   5. associate-/l/ 96.6%

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

   6. *-commutative 96.6%

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

7. Applied egg-rr 96.6%

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

   1. unpow2 96.6%

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

   2. rem-sqrt-square 96.6%

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

   3. log-div 96.6%

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

   4. *-commutative 96.6%

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

   5. associate-/r* 96.6%

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

   6. metadata-eval 96.6%

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

   7. metadata-eval 96.6%

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

   8. associate-*r/ 96.6%

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

   9. log-div 96.6%

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

   10. associate-*l/ 96.6%

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

   11. associate-*r/ 96.6%

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

   12. associate-*l/ 96.6%

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

   13. *-commutative 96.6%

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

   14. associate-/l/ 96.6%

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

   15. associate-*r/ 96.6%

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

   16. metadata-eval 96.6%

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

9. Simplified 96.6%

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

10. Final simplification 96.6%

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

Alternative 5: 95.6% accurate, 4.8× 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 5.4%

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

3. Taylor expanded in f around 0 96.5%

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

   1. mul-1-neg 96.5%

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

   2. unsub-neg 96.5%

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

   3. distribute-rgt-out-- 96.5%

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

   4. metadata-eval 96.5%

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

   5. associate-/r* 96.5%

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

5. Simplified 96.5%

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

6. Taylor expanded in f around 0 96.5%

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

   1. associate-*l/ 96.5%

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

   2. *-un-lft-identity 96.5%

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

   3. frac-2neg 96.5%

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

8. Applied egg-rr 96.5%

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

   1. *-commutative 96.5%

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

   2. associate-*l* 96.5%

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

10. Simplified 96.5%

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

    1. *-commutative 96.5%

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

    2. associate-*r* 96.5%

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

    3. *-commutative 96.5%

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

    4. metadata-eval 96.5%

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

    5. div-inv 96.5%

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

    6. clear-num 96.5%

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

    7. *-commutative 96.5%

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

    8. associate-/r* 96.5%

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

    9. neg-log 96.5%

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

    10. diff-log 96.5%

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

    11. neg-mul-1 96.5%

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

    12. associate-*l/ 96.5%

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

12. Applied egg-rr 96.4%

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

13. Final simplification 96.4%

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

Alternative 6: 95.8% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.4%

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

3. Taylor expanded in f around 0 96.5%

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

   1. mul-1-neg 96.5%

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

   2. unsub-neg 96.5%

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

   3. distribute-rgt-out-- 96.5%

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

   4. metadata-eval 96.5%

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

   5. associate-/r* 96.5%

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

5. Simplified 96.5%

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

6. Taylor expanded in f around 0 96.5%

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

   1. associate-*l/ 96.5%

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

   2. *-un-lft-identity 96.5%

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

   3. frac-2neg 96.5%

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

8. Applied egg-rr 96.5%

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

   1. *-commutative 96.5%

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

   2. associate-*l* 96.5%

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

10. Simplified 96.5%

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

11. Final simplification 96.5%

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

Alternative 7: 1.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \log 0.125 \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (log 0.125) (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(0.125) * (-1.0 / (((double) M_PI) / 4.0));
}

Java

public static double code(double f) {
	return Math.log(0.125) * (-1.0 / (Math.PI / 4.0));
}

Python

def code(f):
	return math.log(0.125) * (-1.0 / (math.pi / 4.0))

Julia

function code(f)
	return Float64(log(0.125) * Float64(-1.0 / Float64(pi / 4.0)))
end

MATLAB

function tmp = code(f)
	tmp = log(0.125) * (-1.0 / (pi / 4.0));
end

Wolfram

code[f_] := N[(N[Log[0.125], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.4%

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

4. Applied egg-rr 1.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{16}}\right) \]

5. Taylor expanded in f around 0 1.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\log 0.125} \]

6. Final simplification 1.6%

   \[\leadsto \log 0.125 \cdot \frac{-1}{\frac{\pi}{4}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(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)))))))))