VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.2% → 96.5%

Time: 26.7s

Alternatives: 4

Speedup: 3.3×

• could not determine a ground truth

  1. f = -2.879952693614454e+115

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: 7.2% 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: 96.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(\pi \cdot \left(\pi \cdot 0.08333333333333333\right)\right)}{\pi}, f \cdot f, 0\right)\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (fma
  (/ (- (log (/ 4.0 PI)) (log f)) PI)
  -4.0
  (* -2.0 (fma (/ (* 0.5 (* PI (* PI 0.08333333333333333))) PI) (* f f) 0.0))))

C

double code(double f) {
	return fma(((log((4.0 / ((double) M_PI))) - log(f)) / ((double) M_PI)), -4.0, (-2.0 * fma(((0.5 * (((double) M_PI) * (((double) M_PI) * 0.08333333333333333))) / ((double) M_PI)), (f * f), 0.0)));
}

Julia

function code(f)
	return fma(Float64(Float64(log(Float64(4.0 / pi)) - log(f)) / pi), -4.0, Float64(-2.0 * fma(Float64(Float64(0.5 * Float64(pi * Float64(pi * 0.08333333333333333))) / pi), Float64(f * f), 0.0)))
end

Wolfram

code[f_] := N[(N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] * -4.0 + N[(-2.0 * N[(N[(N[(0.5 * N[(Pi * N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] * N[(f * f), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.2%

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

   1. distribute-lft-neg-in 6.2%

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

   2. *-commutative 6.2%

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

   3. associate-/r/ 6.2%

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

   4. associate-*l/ 6.2%

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

   5. metadata-eval 6.2%

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

   6. distribute-neg-frac 6.2%

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

3. Simplified 6.1%

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

4. Taylor expanded in f around 0 97.1%

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

6. Simplified 97.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, 1 \cdot \frac{\pi}{0.5}, \frac{-2}{\frac{{\pi}^{2}}{{\pi}^{3}} \cdot 48}\right), 0\right)}{\pi}, f \cdot f, 0\right)\right)} \]
7. Step-by-step derivation

   1. *-un-lft-identity 97.1%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \color{blue}{\frac{\pi}{0.5}}, \frac{-2}{\frac{{\pi}^{2}}{{\pi}^{3}} \cdot 48}\right), 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

   2. fma-udef 97.1%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{0.0625 \cdot \frac{\pi}{0.5} + \frac{-2}{\frac{{\pi}^{2}}{{\pi}^{3}} \cdot 48}}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

   3. div-inv 97.1%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, 0.0625 \cdot \color{blue}{\left(\pi \cdot \frac{1}{0.5}\right)} + \frac{-2}{\frac{{\pi}^{2}}{{\pi}^{3}} \cdot 48}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

   4. metadata-eval 97.1%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, 0.0625 \cdot \left(\pi \cdot \color{blue}{2}\right) + \frac{-2}{\frac{{\pi}^{2}}{{\pi}^{3}} \cdot 48}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

   5. pow-div 97.1%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, 0.0625 \cdot \left(\pi \cdot 2\right) + \frac{-2}{\color{blue}{{\pi}^{\left(2 - 3\right)}} \cdot 48}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

   6. metadata-eval 97.1%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, 0.0625 \cdot \left(\pi \cdot 2\right) + \frac{-2}{{\pi}^{\color{blue}{-1}} \cdot 48}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

8. Applied egg-rr 97.1%

   \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{0.0625 \cdot \left(\pi \cdot 2\right) + \frac{-2}{{\pi}^{-1} \cdot 48}}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]
9. Step-by-step derivation

   1. *-commutative 97.1%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{\left(\pi \cdot 2\right) \cdot 0.0625} + \frac{-2}{{\pi}^{-1} \cdot 48}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

   2. associate-*l* 97.1%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{\pi \cdot \left(2 \cdot 0.0625\right)} + \frac{-2}{{\pi}^{-1} \cdot 48}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

   3. metadata-eval 97.1%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \pi \cdot \color{blue}{0.125} + \frac{-2}{{\pi}^{-1} \cdot 48}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

   4. unpow-1 97.1%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \pi \cdot 0.125 + \frac{-2}{\color{blue}{\frac{1}{\pi}} \cdot 48}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

   5. associate-*l/ 97.1%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \pi \cdot 0.125 + \frac{-2}{\color{blue}{\frac{1 \cdot 48}{\pi}}}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

   6. metadata-eval 97.1%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \pi \cdot 0.125 + \frac{-2}{\frac{\color{blue}{48}}{\pi}}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

10. Simplified 97.1%

    \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{\pi \cdot 0.125 + \frac{-2}{\frac{48}{\pi}}}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]
11. Step-by-step derivation

    1. fma-udef 97.1%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\color{blue}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.125 + \frac{-2}{\frac{48}{\pi}}\right) + 0}}{\pi}, f \cdot f, 0\right)\right) \]

    2. fma-def 97.1%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\left(\pi \cdot 0.5\right) \cdot \color{blue}{\mathsf{fma}\left(\pi, 0.125, \frac{-2}{\frac{48}{\pi}}\right)} + 0}{\pi}, f \cdot f, 0\right)\right) \]

    3. associate-/r/ 97.1%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(\pi, 0.125, \color{blue}{\frac{-2}{48} \cdot \pi}\right) + 0}{\pi}, f \cdot f, 0\right)\right) \]

    4. metadata-eval 97.1%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(\pi, 0.125, \color{blue}{-0.041666666666666664} \cdot \pi\right) + 0}{\pi}, f \cdot f, 0\right)\right) \]

12. Applied egg-rr 97.1%

    \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\color{blue}{\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(\pi, 0.125, -0.041666666666666664 \cdot \pi\right) + 0}}{\pi}, f \cdot f, 0\right)\right) \]
13. Step-by-step derivation

    1. +-rgt-identity 97.1%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\color{blue}{\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(\pi, 0.125, -0.041666666666666664 \cdot \pi\right)}}{\pi}, f \cdot f, 0\right)\right) \]

    2. *-commutative 97.1%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\color{blue}{\left(0.5 \cdot \pi\right)} \cdot \mathsf{fma}\left(\pi, 0.125, -0.041666666666666664 \cdot \pi\right)}{\pi}, f \cdot f, 0\right)\right) \]

    3. associate-*l* 97.1%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\color{blue}{0.5 \cdot \left(\pi \cdot \mathsf{fma}\left(\pi, 0.125, -0.041666666666666664 \cdot \pi\right)\right)}}{\pi}, f \cdot f, 0\right)\right) \]

    4. fma-udef 97.1%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(\pi \cdot \color{blue}{\left(\pi \cdot 0.125 + -0.041666666666666664 \cdot \pi\right)}\right)}{\pi}, f \cdot f, 0\right)\right) \]

    5. *-commutative 97.1%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(\pi \cdot \left(\pi \cdot 0.125 + \color{blue}{\pi \cdot -0.041666666666666664}\right)\right)}{\pi}, f \cdot f, 0\right)\right) \]

    6. distribute-lft-out 97.1%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(\pi \cdot \color{blue}{\left(\pi \cdot \left(0.125 + -0.041666666666666664\right)\right)}\right)}{\pi}, f \cdot f, 0\right)\right) \]

    7. metadata-eval 97.1%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(\pi \cdot \left(\pi \cdot \color{blue}{0.08333333333333333}\right)\right)}{\pi}, f \cdot f, 0\right)\right) \]

14. Simplified 97.1%

    \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\color{blue}{0.5 \cdot \left(\pi \cdot \left(\pi \cdot 0.08333333333333333\right)\right)}}{\pi}, f \cdot f, 0\right)\right) \]
15. Step-by-step derivation

    1. associate-/r* 97.1%

       \[\leadsto \mathsf{fma}\left(\frac{\log \color{blue}{\left(\frac{\frac{2}{\pi}}{0.5}\right)} - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(\pi \cdot \left(\pi \cdot 0.08333333333333333\right)\right)}{\pi}, f \cdot f, 0\right)\right) \]

    2. log-div 97.1%

       \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\log \left(\frac{2}{\pi}\right) - \log 0.5\right)} - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(\pi \cdot \left(\pi \cdot 0.08333333333333333\right)\right)}{\pi}, f \cdot f, 0\right)\right) \]

16. Applied egg-rr 97.1%

    \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\log \left(\frac{2}{\pi}\right) - \log 0.5\right)} - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(\pi \cdot \left(\pi \cdot 0.08333333333333333\right)\right)}{\pi}, f \cdot f, 0\right)\right) \]
17. Step-by-step derivation

    1. log-div 97.1%

       \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\log \left(\frac{\frac{2}{\pi}}{0.5}\right)} - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(\pi \cdot \left(\pi \cdot 0.08333333333333333\right)\right)}{\pi}, f \cdot f, 0\right)\right) \]

    2. associate-/r* 97.1%

       \[\leadsto \mathsf{fma}\left(\frac{\log \color{blue}{\left(\frac{2}{\pi \cdot 0.5}\right)} - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(\pi \cdot \left(\pi \cdot 0.08333333333333333\right)\right)}{\pi}, f \cdot f, 0\right)\right) \]

    3. *-commutative 97.1%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\color{blue}{0.5 \cdot \pi}}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(\pi \cdot \left(\pi \cdot 0.08333333333333333\right)\right)}{\pi}, f \cdot f, 0\right)\right) \]

    4. associate-/r* 97.1%

       \[\leadsto \mathsf{fma}\left(\frac{\log \color{blue}{\left(\frac{\frac{2}{0.5}}{\pi}\right)} - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(\pi \cdot \left(\pi \cdot 0.08333333333333333\right)\right)}{\pi}, f \cdot f, 0\right)\right) \]

    5. metadata-eval 97.1%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{\color{blue}{4}}{\pi}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(\pi \cdot \left(\pi \cdot 0.08333333333333333\right)\right)}{\pi}, f \cdot f, 0\right)\right) \]

18. Simplified 97.1%

    \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\log \left(\frac{4}{\pi}\right)} - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(\pi \cdot \left(\pi \cdot 0.08333333333333333\right)\right)}{\pi}, f \cdot f, 0\right)\right) \]

19. Final simplification 97.1%

    \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(\pi \cdot \left(\pi \cdot 0.08333333333333333\right)\right)}{\pi}, f \cdot f, 0\right)\right) \]

Alternative 2: 95.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\mathsf{log1p}\left(\frac{\frac{4}{\pi}}{f} + -1\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (* -4.0 (/ (log1p (+ (/ (/ 4.0 PI) f) -1.0)) PI)))

C

double code(double f) {
	return -4.0 * (log1p((((4.0 / ((double) M_PI)) / f) + -1.0)) / ((double) M_PI));
}

Java

public static double code(double f) {
	return -4.0 * (Math.log1p((((4.0 / Math.PI) / f) + -1.0)) / Math.PI);
}

Python

def code(f):
	return -4.0 * (math.log1p((((4.0 / math.pi) / f) + -1.0)) / math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(log1p(Float64(Float64(Float64(4.0 / pi) / f) + -1.0)) / pi))
end

Wolfram

code[f_] := N[(-4.0 * N[(N[Log[1 + N[(N[(N[(4.0 / Pi), $MachinePrecision] / f), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.2%

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

   1. distribute-lft-neg-in 6.2%

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

   2. *-commutative 6.2%

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

   3. associate-/r/ 6.2%

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

   4. associate-*l/ 6.2%

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

   5. metadata-eval 6.2%

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

   6. distribute-neg-frac 6.2%

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

3. Simplified 6.1%

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

4. Taylor expanded in f around 0 97.0%

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

   1. *-commutative 97.0%

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

   2. mul-1-neg 97.0%

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

   3. unsub-neg 97.0%

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

   4. distribute-rgt-out-- 97.0%

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

   5. metadata-eval 97.0%

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

   6. associate-/r* 97.0%

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

6. Simplified 97.0%

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

7. Taylor expanded in f around 0 97.0%

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

   1. log1p-expm1-u 97.0%

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

   2. expm1-udef 97.0%

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

   3. diff-log 97.0%

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

   4. add-exp-log 97.0%

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

9. Applied egg-rr 97.0%

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

10. Final simplification 97.0%

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

Alternative 3: 95.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.2%

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

   1. distribute-lft-neg-in 6.2%

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

   2. *-commutative 6.2%

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

   3. associate-/r/ 6.2%

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

   4. associate-*l/ 6.2%

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

   5. metadata-eval 6.2%

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

   6. distribute-neg-frac 6.2%

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

3. Simplified 6.1%

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

4. Taylor expanded in f around 0 97.0%

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

   1. *-commutative 97.0%

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

   2. mul-1-neg 97.0%

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

   3. unsub-neg 97.0%

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

   4. distribute-rgt-out-- 97.0%

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

   5. metadata-eval 97.0%

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

   6. associate-/r* 97.0%

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

6. Simplified 97.0%

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

7. Taylor expanded in f around 0 97.0%

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

   1. sub-neg 97.0%

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

   2. log-rec 97.0%

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

   3. +-commutative 97.0%

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

   4. log-rec 97.0%

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

   5. neg-mul-1 97.0%

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

   6. neg-mul-1 97.0%

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

   7. log-rec 97.0%

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

   8. +-commutative 97.0%

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

   9. log-rec 97.0%

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

   10. sub-neg 97.0%

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

   11. log-div 97.0%

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

   12. associate--l- 96.9%

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

   13. log-prod 96.9%

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

   14. log-div 97.0%

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

   15. *-commutative 97.0%

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

9. Simplified 97.0%

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

10. Final simplification 97.0%

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

Alternative 4: 1.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \left(-\frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi}\right) \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* 4.0 (- (/ (log 7.62939453125e-6) PI))))

C

double code(double f) {
	return 4.0 * -(log(7.62939453125e-6) / ((double) M_PI));
}

Java

public static double code(double f) {
	return 4.0 * -(Math.log(7.62939453125e-6) / Math.PI);
}

Python

def code(f):
	return 4.0 * -(math.log(7.62939453125e-6) / math.pi)

Julia

function code(f)
	return Float64(4.0 * Float64(-Float64(log(7.62939453125e-6) / pi)))
end

MATLAB

function tmp = code(f)
	tmp = 4.0 * -(log(7.62939453125e-6) / pi);
end

Wolfram

code[f_] := N[(4.0 * (-N[(N[Log[7.62939453125e-6], $MachinePrecision] / Pi), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 6.2%

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

3. Applied egg-rr 1.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{262144}}\right) \]

4. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi}} \]

5. Final simplification 1.6%

   \[\leadsto 4 \cdot \left(-\frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi}\right) \]

Reproduce

herbie shell --seed 2023240 
(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)))))))))