VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 99.0%

Time: 19.0s

Alternatives: 8

Speedup: 4.9×

• 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: 6.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{-4 \cdot \mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot 0.5\right)\right)} + \left(-1 + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (*
   -4.0
   (log1p
    (+
     (/ 1.0 (expm1 (* PI (* f 0.5))))
     (+ -1.0 (/ -1.0 (expm1 (* PI (* f -0.5))))))))
  PI))

C

double code(double f) {
	return (-4.0 * log1p(((1.0 / expm1((((double) M_PI) * (f * 0.5)))) + (-1.0 + (-1.0 / expm1((((double) M_PI) * (f * -0.5)))))))) / ((double) M_PI);
}

Java

public static double code(double f) {
	return (-4.0 * Math.log1p(((1.0 / Math.expm1((Math.PI * (f * 0.5)))) + (-1.0 + (-1.0 / Math.expm1((Math.PI * (f * -0.5)))))))) / Math.PI;
}

Python

def code(f):
	return (-4.0 * math.log1p(((1.0 / math.expm1((math.pi * (f * 0.5)))) + (-1.0 + (-1.0 / math.expm1((math.pi * (f * -0.5)))))))) / math.pi

Julia

function code(f)
	return Float64(Float64(-4.0 * log1p(Float64(Float64(1.0 / expm1(Float64(pi * Float64(f * 0.5)))) + Float64(-1.0 + Float64(-1.0 / expm1(Float64(pi * Float64(f * -0.5)))))))) / pi)
end

Wolfram

code[f_] := N[(N[(-4.0 * N[Log[1 + N[(N[(1.0 / N[(Exp[N[(Pi * N[(f * 0.5), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] + N[(-1.0 + N[(-1.0 / N[(Exp[N[(Pi * N[(f * -0.5), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 7.9%

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

3. Simplified 99.0%

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

5. Taylor expanded in f around inf 7.1%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{1}{e^{0.5 \cdot \left(f \cdot \pi\right)} - 1} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi}} \]
6. Step-by-step derivation

   1. associate-*r/ 7.1%

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

7. Simplified 99.1%

   \[\leadsto \color{blue}{\frac{-4 \cdot \log \left(\frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot 0.5\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\pi}} \]
8. Step-by-step derivation

   1. log1p-expm1-u 99.1%

      \[\leadsto \frac{-4 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot 0.5\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)\right)\right)}}{\pi} \]

   2. expm1-undefine 99.1%

      \[\leadsto \frac{-4 \cdot \mathsf{log1p}\left(\color{blue}{e^{\log \left(\frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot 0.5\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)} - 1}\right)}{\pi} \]

   3. add-exp-log 99.1%

      \[\leadsto \frac{-4 \cdot \mathsf{log1p}\left(\color{blue}{\left(\frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot 0.5\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)} - 1\right)}{\pi} \]

9. Applied egg-rr 99.1%

   \[\leadsto \frac{-4 \cdot \color{blue}{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot 0.5\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right) - 1\right)}}{\pi} \]
10. Step-by-step derivation

    1. associate--l+ 99.2%

       \[\leadsto \frac{-4 \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot 0.5\right)\right)} + \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)} - 1\right)}\right)}{\pi} \]

11. Simplified 99.2%

    \[\leadsto \frac{-4 \cdot \color{blue}{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot 0.5\right)\right)} + \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)} - 1\right)\right)}}{\pi} \]

12. Final simplification 99.2%

    \[\leadsto \frac{-4 \cdot \mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot 0.5\right)\right)} + \left(-1 + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)\right)}{\pi} \]
13. Add Preprocessing

Alternative 2: 99.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \frac{\log \left(\frac{1}{\tanh \left(\pi \cdot \left(f \cdot 0.25\right)\right)}\right)}{-\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (* 4.0 (/ (log (/ 1.0 (tanh (* PI (* f 0.25))))) (- PI))))

C

double code(double f) {
	return 4.0 * (log((1.0 / tanh((((double) M_PI) * (f * 0.25))))) / -((double) M_PI));
}

Java

public static double code(double f) {
	return 4.0 * (Math.log((1.0 / Math.tanh((Math.PI * (f * 0.25))))) / -Math.PI);
}

Python

def code(f):
	return 4.0 * (math.log((1.0 / math.tanh((math.pi * (f * 0.25))))) / -math.pi)

Julia

function code(f)
	return Float64(4.0 * Float64(log(Float64(1.0 / tanh(Float64(pi * Float64(f * 0.25))))) / Float64(-pi)))
end

MATLAB

function tmp = code(f)
	tmp = 4.0 * (log((1.0 / tanh((pi * (f * 0.25))))) / -pi);
end

Wolfram

code[f_] := N[(4.0 * N[(N[Log[N[(1.0 / N[Tanh[N[(Pi * N[(f * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-Pi)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.9%

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

3. Taylor expanded in f around inf 7.9%

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

   1. *-un-lft-identity 7.9%

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

   2. clear-num 7.9%

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

   3. +-commutative 7.9%

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

   4. tanh-undef 99.1%

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

   5. associate-*r* 99.1%

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

5. Applied egg-rr 99.1%

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

   1. *-lft-identity 99.1%

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

   2. *-commutative 99.1%

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

   3. *-commutative 99.1%

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

   \[\leadsto 4 \cdot \frac{\log \left(\frac{1}{\tanh \left(\pi \cdot \left(f \cdot 0.25\right)\right)}\right)}{-\pi} \]
9. Add Preprocessing

Alternative 3: 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 7.9%

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

3. Taylor expanded in f around inf 7.9%

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

   1. *-un-lft-identity 7.9%

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

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 4: 98.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.9%

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

3. Taylor expanded in f around inf 7.9%

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

   1. clear-num 7.9%

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

   2. inv-pow 7.9%

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

5. Applied egg-rr 99.0%

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

   1. unpow-1 99.0%

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

   2. distribute-frac-neg2 99.0%

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

   3. distribute-neg-frac 99.0%

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

   4. *-commutative 99.0%

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

   5. *-commutative 99.0%

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

7. Simplified 99.0%

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

   1. neg-sub0 99.0%

      \[\leadsto \color{blue}{0 - 4 \cdot \frac{1}{\frac{-\pi}{\log \tanh \left(\pi \cdot \left(f \cdot 0.25\right)\right)}}} \]

   2. un-div-inv 99.0%

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

9. Applied egg-rr 99.0%

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

    1. neg-sub0 99.0%

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

    2. distribute-neg-frac2 99.0%

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

    3. distribute-frac-neg 99.0%

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

    4. remove-double-neg 99.0%

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

11. Simplified 99.0%

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

Alternative 5: 96.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{-4 \cdot \mathsf{log1p}\left(\frac{4 \cdot \frac{1}{\pi} - f \cdot \left(f \cdot \left(\left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right) - \left(\pi \cdot -0.08333333333333333 + \pi \cdot 0.125\right)\right) - -1\right)}{f}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (*
   -4.0
   (log1p
    (/
     (-
      (* 4.0 (/ 1.0 PI))
      (*
       f
       (-
        (*
         f
         (-
          (+ (* PI -0.125) (* PI 0.08333333333333333))
          (+ (* PI -0.08333333333333333) (* PI 0.125))))
        -1.0)))
     f)))
  PI))

C

double code(double f) {
	return (-4.0 * log1p((((4.0 * (1.0 / ((double) M_PI))) - (f * ((f * (((((double) M_PI) * -0.125) + (((double) M_PI) * 0.08333333333333333)) - ((((double) M_PI) * -0.08333333333333333) + (((double) M_PI) * 0.125)))) - -1.0))) / f))) / ((double) M_PI);
}

Java

public static double code(double f) {
	return (-4.0 * Math.log1p((((4.0 * (1.0 / Math.PI)) - (f * ((f * (((Math.PI * -0.125) + (Math.PI * 0.08333333333333333)) - ((Math.PI * -0.08333333333333333) + (Math.PI * 0.125)))) - -1.0))) / f))) / Math.PI;
}

Python

def code(f):
	return (-4.0 * math.log1p((((4.0 * (1.0 / math.pi)) - (f * ((f * (((math.pi * -0.125) + (math.pi * 0.08333333333333333)) - ((math.pi * -0.08333333333333333) + (math.pi * 0.125)))) - -1.0))) / f))) / math.pi

Julia

function code(f)
	return Float64(Float64(-4.0 * log1p(Float64(Float64(Float64(4.0 * Float64(1.0 / pi)) - Float64(f * Float64(Float64(f * Float64(Float64(Float64(pi * -0.125) + Float64(pi * 0.08333333333333333)) - Float64(Float64(pi * -0.08333333333333333) + Float64(pi * 0.125)))) - -1.0))) / f))) / pi)
end

Wolfram

code[f_] := N[(N[(-4.0 * N[Log[1 + N[(N[(N[(4.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision] - N[(f * N[(N[(f * N[(N[(N[(Pi * -0.125), $MachinePrecision] + N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision] - N[(N[(Pi * -0.08333333333333333), $MachinePrecision] + N[(Pi * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 7.9%

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

3. Simplified 99.0%

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

5. Taylor expanded in f around inf 7.1%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{1}{e^{0.5 \cdot \left(f \cdot \pi\right)} - 1} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi}} \]
6. Step-by-step derivation

   1. associate-*r/ 7.1%

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

7. Simplified 99.1%

   \[\leadsto \color{blue}{\frac{-4 \cdot \log \left(\frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot 0.5\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\pi}} \]
8. Step-by-step derivation

   1. log1p-expm1-u 99.1%

      \[\leadsto \frac{-4 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot 0.5\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)\right)\right)}}{\pi} \]

   2. expm1-undefine 99.1%

      \[\leadsto \frac{-4 \cdot \mathsf{log1p}\left(\color{blue}{e^{\log \left(\frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot 0.5\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)} - 1}\right)}{\pi} \]

   3. add-exp-log 99.1%

      \[\leadsto \frac{-4 \cdot \mathsf{log1p}\left(\color{blue}{\left(\frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot 0.5\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)} - 1\right)}{\pi} \]

9. Applied egg-rr 99.1%

   \[\leadsto \frac{-4 \cdot \color{blue}{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot 0.5\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right) - 1\right)}}{\pi} \]
10. Step-by-step derivation

    1. associate--l+ 99.2%

       \[\leadsto \frac{-4 \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot 0.5\right)\right)} + \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)} - 1\right)}\right)}{\pi} \]

11. Simplified 99.2%

    \[\leadsto \frac{-4 \cdot \color{blue}{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot 0.5\right)\right)} + \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)} - 1\right)\right)}}{\pi} \]

12. Taylor expanded in f around 0 96.4%

    \[\leadsto \frac{-4 \cdot \mathsf{log1p}\left(\color{blue}{\frac{f \cdot \left(f \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right) - 1\right) + 4 \cdot \frac{1}{\pi}}{f}}\right)}{\pi} \]

13. Final simplification 96.4%

    \[\leadsto \frac{-4 \cdot \mathsf{log1p}\left(\frac{4 \cdot \frac{1}{\pi} - f \cdot \left(f \cdot \left(\left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right) - \left(\pi \cdot -0.08333333333333333 + \pi \cdot 0.125\right)\right) - -1\right)}{f}\right)}{\pi} \]
14. Add Preprocessing

Alternative 6: 95.7% accurate, 4.9× speedup

Math

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

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(Float64(-4.0 * 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[(N[(-4.0 * N[Log[N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 7.9%

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

3. Simplified 99.0%

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

5. Taylor expanded in f around 0 95.5%

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

   1. associate-/r* 95.5%

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

7. Simplified 95.5%

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

   1. associate-*r/ 95.6%

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

   2. associate-/l/ 95.6%

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

9. Applied egg-rr 95.6%

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

10. Final simplification 95.6%

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

Alternative 7: 95.5% accurate, 4.9× speedup

Math

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

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(Float64(4.0 / 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[(N[(4.0 / f), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.9%

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

3. Simplified 99.0%

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

5. Taylor expanded in f around 0 95.5%

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

   1. associate-/r* 95.5%

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

7. Simplified 95.5%

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

Alternative 8: 5.4% accurate, 76.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-16}{\pi \cdot f}}{\pi} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (/ -16.0 (* PI f)) PI))

C

double code(double f) {
	return (-16.0 / (((double) M_PI) * f)) / ((double) M_PI);
}

Java

public static double code(double f) {
	return (-16.0 / (Math.PI * f)) / Math.PI;
}

Python

def code(f):
	return (-16.0 / (math.pi * f)) / math.pi

Julia

function code(f)
	return Float64(Float64(-16.0 / Float64(pi * f)) / pi)
end

MATLAB

function tmp = code(f)
	tmp = (-16.0 / (pi * f)) / pi;
end

Wolfram

code[f_] := N[(N[(-16.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 7.9%

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

3. Simplified 99.0%

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

5. Taylor expanded in f around inf 7.1%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{1}{e^{0.5 \cdot \left(f \cdot \pi\right)} - 1} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi}} \]
6. Step-by-step derivation

   1. associate-*r/ 7.1%

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

7. Simplified 99.1%

   \[\leadsto \color{blue}{\frac{-4 \cdot \log \left(\frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot 0.5\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\pi}} \]
8. Step-by-step derivation

   1. log1p-expm1-u 99.1%

      \[\leadsto \frac{-4 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot 0.5\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)\right)\right)}}{\pi} \]

   2. expm1-undefine 99.1%

      \[\leadsto \frac{-4 \cdot \mathsf{log1p}\left(\color{blue}{e^{\log \left(\frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot 0.5\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)} - 1}\right)}{\pi} \]

   3. add-exp-log 99.1%

      \[\leadsto \frac{-4 \cdot \mathsf{log1p}\left(\color{blue}{\left(\frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot 0.5\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)} - 1\right)}{\pi} \]

9. Applied egg-rr 99.1%

   \[\leadsto \frac{-4 \cdot \color{blue}{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot 0.5\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right) - 1\right)}}{\pi} \]
10. Step-by-step derivation

    1. associate--l+ 99.2%

       \[\leadsto \frac{-4 \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot 0.5\right)\right)} + \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)} - 1\right)}\right)}{\pi} \]

11. Simplified 99.2%

    \[\leadsto \frac{-4 \cdot \color{blue}{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot 0.5\right)\right)} + \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)} - 1\right)\right)}}{\pi} \]

12. Taylor expanded in f around 0 94.5%

    \[\leadsto \frac{-4 \cdot \mathsf{log1p}\left(\color{blue}{\frac{4}{f \cdot \pi}}\right)}{\pi} \]
13. Step-by-step derivation

    1. *-commutative 94.5%

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

14. Simplified 94.5%

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

15. Taylor expanded in f around inf 5.6%

    \[\leadsto \frac{\color{blue}{\frac{-16}{f \cdot \pi}}}{\pi} \]
16. Step-by-step derivation

    1. *-commutative 5.6%

       \[\leadsto \frac{\frac{-16}{\color{blue}{\pi \cdot f}}}{\pi} \]

17. Simplified 5.6%

    \[\leadsto \frac{\color{blue}{\frac{-16}{\pi \cdot f}}}{\pi} \]
18. Add Preprocessing

Reproduce

herbie shell --seed 2024157 
(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)))))))))