VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.6% → 99.1%

Time: 9.1s

Alternatives: 4

Speedup: 4.9×

• could not determine a ground truth

  1. f = 1481668238.5496356

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: 6.6% 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: 99.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.6%

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

3. Applied rewrites 98.8%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. lift-tanh.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. mult-flip N/A

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

   9. metadata-eval N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. sqrt-div N/A

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

   13. lower-*.f64 N/A

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

5. Applied rewrites 99.0%

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

7. Applied rewrites 99.1%

   \[\leadsto -\frac{4}{\sqrt{\pi}} \cdot \color{blue}{\log \left({\tanh \left(\left(f \cdot \pi\right) \cdot 0.25\right)}^{\left(-1 \cdot \frac{1}{\sqrt{\pi}}\right)}\right)} \]
8. Add Preprocessing

Alternative 2: 99.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.6%

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

3. Applied rewrites 98.8%

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

4. Taylor expanded in f around inf

   \[\leadsto \color{blue}{4 \cdot \frac{\log \left(\frac{e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)} - \frac{1}{e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)}}}{e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)} + \frac{1}{e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)}}}\right)}{\mathsf{PI}\left(\right)}} \]
5. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

6. Applied rewrites 99.0%

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

Alternative 3: 95.8% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.6%

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

3. Applied rewrites 98.8%

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

4. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

6. Applied rewrites 95.8%

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

   1. lift-log.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

      \[\leadsto \frac{\log \left(\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot f\right)}{\pi} \cdot 4 \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot f\right)}{\pi} \cdot 4 \]

   5. log-prod N/A

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

   6. lower-+.f64 N/A

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

   7. lower-log.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lower-log.f64 95.8

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

8. Applied rewrites 95.8%

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

Alternative 4: 95.8% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.6%

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

3. Applied rewrites 98.8%

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

4. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

6. Applied rewrites 95.8%

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

Reproduce

herbie shell --seed 2025120 
(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)))))))))