VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 96.6%

Time: 31.5s

Alternatives: 3

Speedup: 3.3×

• could not determine a ground truth

  1. f = -2.0480390492834746e+252

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: 96.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. distribute-lft-neg-in 6.9%

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

   2. *-commutative 6.9%

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

3. Simplified 6.9%

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

4. Taylor expanded in f around inf 6.9%

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

   1. associate-*r* 6.9%

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

   2. *-commutative 6.9%

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

   3. exp-prod 6.5%

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

   4. *-commutative 6.5%

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

   5. *-commutative 6.5%

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

   6. *-commutative 6.5%

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

   7. associate-*l* 6.5%

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

   8. exp-prod 6.5%

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

6. Simplified 6.5%

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

7. Taylor expanded in f around 0 96.3%

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

9. Simplified 96.3%

   \[\leadsto \log \color{blue}{\left(0 + \mathsf{fma}\left(f, 0.125 \cdot \frac{{\pi}^{2}}{\pi} + -2 \cdot \frac{{\pi}^{3} \cdot 0.005208333333333333}{0.25 \cdot {\pi}^{2}}, \frac{\frac{2}{f}}{\pi \cdot 0.5}\right)\right)} \cdot \frac{-4}{\pi} \]

10. Taylor expanded in f around 0 96.4%

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

    1. fma-def 96.4%

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

    2. mul-1-neg 96.4%

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

    3. sub-neg 96.4%

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

    4. log-div 96.5%

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

    5. associate-/r* 96.5%

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

    6. *-commutative 96.5%

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

    7. associate-/r* 96.5%

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

    8. mul-1-neg 96.5%

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

    9. distribute-rgt-neg-in 96.5%

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

    10. distribute-rgt-out 96.5%

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

    11. distribute-rgt-neg-in 96.5%

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

    12. metadata-eval 96.5%

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

    13. metadata-eval 96.5%

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

12. Simplified 96.5%

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

13. Final simplification 96.5%

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

Alternative 2: 96.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. distribute-lft-neg-in 6.9%

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

   2. *-commutative 6.9%

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

3. Simplified 6.9%

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

4. Taylor expanded in f around 0 95.7%

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

   1. associate-/r* 95.7%

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

   2. distribute-rgt-out-- 95.7%

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

   3. metadata-eval 95.7%

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

6. Simplified 95.7%

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

7. Taylor expanded in f around 0 95.7%

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

   1. *-commutative 95.7%

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

9. Simplified 95.7%

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

10. Final simplification 95.7%

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

Alternative 3: 96.2% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. distribute-lft-neg-in 6.9%

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

   2. *-commutative 6.9%

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

3. Simplified 6.9%

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

4. Taylor expanded in f around 0 95.7%

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

   1. associate-/r* 95.7%

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

   2. distribute-rgt-out-- 95.7%

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

   3. metadata-eval 95.7%

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

6. Simplified 95.7%

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

7. Taylor expanded in f around 0 95.7%

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

   1. *-commutative 95.7%

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

9. Simplified 95.7%

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

    1. associate-*r/ 95.9%

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

11. Applied egg-rr 95.9%

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

12. Final simplification 95.9%

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

Reproduce

herbie shell --seed 2023319 
(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)))))))))