VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.1% → 99.0%

Time: 29.3s

Alternatives: 8

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: 7.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.7%

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

3. Taylor expanded in f around inf 8.7%

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

   1. *-un-lft-identity 8.7%

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

5. Applied egg-rr 99.1%

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

   1. associate-*r/ 99.1%

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

   2. *-lft-identity 99.1%

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

   3. *-commutative 99.1%

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

   4. *-commutative 99.1%

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

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

Alternative 2: 95.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.7%

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

3. Taylor expanded in f around 0 95.8%

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

   1. distribute-rgt-out-- 95.8%

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

   2. metadata-eval 95.8%

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

5. Simplified 95.8%

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

6. Taylor expanded in f around 0 95.8%

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

   1. *-commutative 95.8%

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

8. Simplified 95.8%

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

   1. add-sqr-sqrt 95.5%

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

   2. sqrt-unprod 95.9%

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

   3. pow2 95.9%

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

   4. clear-num 95.9%

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

10. Applied egg-rr 95.9%

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

    1. unpow2 95.9%

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

    2. rem-sqrt-square 95.9%

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

    3. associate-*l/ 96.0%

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

    4. *-lft-identity 96.0%

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

    5. times-frac 96.0%

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

    6. metadata-eval 96.0%

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

    7. *-commutative 96.0%

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

12. Simplified 96.0%

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

13. Final simplification 96.0%

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

Alternative 3: 95.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.7%

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

3. Taylor expanded in f around 0 95.8%

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

   1. distribute-rgt-out-- 95.8%

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

   2. metadata-eval 95.8%

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

5. Simplified 95.8%

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

6. Taylor expanded in f around 0 95.8%

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

   1. *-commutative 95.8%

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

8. Simplified 95.8%

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

   1. associate-*l/ 95.9%

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

   2. div-inv 95.9%

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

   3. metadata-eval 95.9%

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

   4. *-un-lft-identity 95.9%

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

10. Applied egg-rr 95.9%

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

    1. add-sqr-sqrt 95.4%

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

    2. sqrt-unprod 96.0%

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

    3. pow2 96.0%

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

12. Applied egg-rr 96.0%

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

    1. unpow2 96.0%

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

    2. rem-sqrt-square 96.0%

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

    3. associate-/r* 96.0%

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

14. Simplified 96.0%

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

15. Final simplification 96.0%

    \[\leadsto \frac{\left|\log \left(\frac{\frac{4}{\pi}}{f}\right)\right|}{\pi \cdot \left(-0.25\right)} \]
16. Add Preprocessing

Alternative 4: 95.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.7%

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

3. Simplified 98.9%

   \[\leadsto \color{blue}{\log \left(\frac{1}{\mathsf{expm1}\left(\left(\pi \cdot f\right) \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Taylor expanded in f around 0 96.0%

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

   1. mul-1-neg 96.0%

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

   2. unsub-neg 96.0%

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

7. Simplified 96.0%

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

8. Final simplification 96.0%

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

Alternative 5: 95.7% accurate, 4.8× speedup

Math

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

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(Float64(4.0 / pi) / f)) * Float64(4.0 / Float64(-pi)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.7%

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

3. Taylor expanded in f around 0 95.8%

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

   1. distribute-rgt-out-- 95.8%

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

   2. metadata-eval 95.8%

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

5. Simplified 95.8%

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

6. Taylor expanded in f around 0 95.8%

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

   1. *-commutative 95.8%

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

8. Simplified 95.8%

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

   1. associate-*l/ 95.9%

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

   2. div-inv 95.9%

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

   3. metadata-eval 95.9%

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

   4. *-un-lft-identity 95.9%

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

10. Applied egg-rr 95.9%

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

    1. div-inv 95.8%

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

    2. inv-pow 95.8%

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

    3. *-commutative 95.8%

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

    4. unpow-prod-down 95.8%

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

    5. metadata-eval 95.8%

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

    6. inv-pow 95.8%

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

12. Applied egg-rr 95.8%

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

    1. associate-/r* 95.8%

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

    2. associate-*r/ 95.8%

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

    3. metadata-eval 95.8%

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

14. Simplified 95.8%

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

15. Final simplification 95.8%

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

Alternative 6: 95.9% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.7%

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

3. Taylor expanded in f around 0 95.8%

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

   1. distribute-rgt-out-- 95.8%

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

   2. metadata-eval 95.8%

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

5. Simplified 95.8%

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

6. Taylor expanded in f around 0 95.8%

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

   1. *-commutative 95.8%

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

8. Simplified 95.8%

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

   1. associate-*l/ 95.9%

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

   2. div-inv 95.9%

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

   3. metadata-eval 95.9%

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

   4. *-un-lft-identity 95.9%

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

10. Applied egg-rr 95.9%

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

11. Final simplification 95.9%

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

Alternative 7: 95.9% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.7%

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

3. Taylor expanded in f around 0 95.8%

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

   1. distribute-rgt-out-- 95.8%

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

   2. metadata-eval 95.8%

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

5. Simplified 95.8%

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

6. Taylor expanded in f around 0 95.8%

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

   1. *-commutative 95.8%

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

8. Simplified 95.8%

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

   1. associate-*l/ 95.9%

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

   2. div-inv 95.9%

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

   3. metadata-eval 95.9%

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

   4. *-un-lft-identity 95.9%

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

10. Applied egg-rr 95.9%

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

    1. *-un-lft-identity 95.9%

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

    2. log-prod 95.9%

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

    3. metadata-eval 95.9%

       \[\leadsto -\frac{\color{blue}{0} + \log \left(\frac{4}{\pi \cdot f}\right)}{\pi \cdot 0.25} \]

12. Applied egg-rr 95.9%

    \[\leadsto -\frac{\color{blue}{0 + \log \left(\frac{4}{\pi \cdot f}\right)}}{\pi \cdot 0.25} \]
13. Step-by-step derivation

    1. +-lft-identity 95.9%

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

    2. associate-/r* 96.0%

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

14. Simplified 96.0%

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

15. Final simplification 96.0%

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

Alternative 8: 1.6% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\log \left(f \cdot \left(\pi \cdot 0.041666666666666664\right)\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (* -4.0 (/ (log (* f (* PI 0.041666666666666664))) PI)))

C

double code(double f) {
	return -4.0 * (log((f * (((double) M_PI) * 0.041666666666666664))) / ((double) M_PI));
}

Java

public static double code(double f) {
	return -4.0 * (Math.log((f * (Math.PI * 0.041666666666666664))) / Math.PI);
}

Python

def code(f):
	return -4.0 * (math.log((f * (math.pi * 0.041666666666666664))) / math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(log(Float64(f * Float64(pi * 0.041666666666666664))) / pi))
end

MATLAB

function tmp = code(f)
	tmp = -4.0 * (log((f * (pi * 0.041666666666666664))) / pi);
end

Wolfram

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

Derivation

1. Initial program 8.7%

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

3. Simplified 98.9%

   \[\leadsto \color{blue}{\log \left(\frac{1}{\mathsf{expm1}\left(\left(\pi \cdot f\right) \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Taylor expanded in f around 0 96.7%

   \[\leadsto \log \left(\color{blue}{\frac{f \cdot \left(-1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right) \cdot \frac{-4}{\pi} \]

6. Taylor expanded in f around inf 1.6%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(-1 \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right) + -1 \cdot \log \left(\frac{1}{f}\right)}{\pi}} \]
7. Step-by-step derivation

   1. +-commutative 1.6%

      \[\leadsto -4 \cdot \frac{\color{blue}{-1 \cdot \log \left(\frac{1}{f}\right) + \log \left(-1 \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)}}{\pi} \]

   2. mul-1-neg 1.6%

      \[\leadsto -4 \cdot \frac{\color{blue}{\left(-\log \left(\frac{1}{f}\right)\right)} + \log \left(-1 \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)}{\pi} \]

   3. log-rec 1.6%

      \[\leadsto -4 \cdot \frac{\left(-\color{blue}{\left(-\log f\right)}\right) + \log \left(-1 \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)}{\pi} \]

   4. remove-double-neg 1.6%

      \[\leadsto -4 \cdot \frac{\color{blue}{\log f} + \log \left(-1 \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)}{\pi} \]

   5. mul-1-neg 1.6%

      \[\leadsto -4 \cdot \frac{\log f + \log \color{blue}{\left(-\left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)}}{\pi} \]

   6. distribute-rgt-out 1.6%

      \[\leadsto -4 \cdot \frac{\log f + \log \left(-\color{blue}{\pi \cdot \left(-0.125 + 0.08333333333333333\right)}\right)}{\pi} \]

   7. metadata-eval 1.6%

      \[\leadsto -4 \cdot \frac{\log f + \log \left(-\pi \cdot \color{blue}{-0.041666666666666664}\right)}{\pi} \]

   8. distribute-rgt-neg-in 1.6%

      \[\leadsto -4 \cdot \frac{\log f + \log \color{blue}{\left(\pi \cdot \left(--0.041666666666666664\right)\right)}}{\pi} \]

   9. metadata-eval 1.6%

      \[\leadsto -4 \cdot \frac{\log f + \log \left(\pi \cdot \color{blue}{0.041666666666666664}\right)}{\pi} \]

8. Simplified 1.6%

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

   1. *-un-lft-identity 1.6%

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

   2. sum-log 1.6%

      \[\leadsto -4 \cdot \left(1 \cdot \frac{\color{blue}{\log \left(f \cdot \left(\pi \cdot 0.041666666666666664\right)\right)}}{\pi}\right) \]

10. Applied egg-rr 1.6%

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

    1. *-lft-identity 1.6%

       \[\leadsto -4 \cdot \color{blue}{\frac{\log \left(f \cdot \left(\pi \cdot 0.041666666666666664\right)\right)}{\pi}} \]

12. Simplified 1.6%

    \[\leadsto -4 \cdot \color{blue}{\frac{\log \left(f \cdot \left(\pi \cdot 0.041666666666666664\right)\right)}{\pi}} \]

13. Final simplification 1.6%

    \[\leadsto -4 \cdot \frac{\log \left(f \cdot \left(\pi \cdot 0.041666666666666664\right)\right)}{\pi} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024056 
(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)))))))))