VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.6% → 99.0%

Time: 9.4s

Alternatives: 3

Speedup: 6.1×

• could not determine a ground truth

  1. f = 1090480515.5390706

Specification

Math

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

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

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

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, 4.1× speedup

Math

\[\frac{\log \tanh \left(0.7853981633974483 \cdot f\right) \cdot 4}{\pi} \]

FPCore

(FPCore (f)
 :precision binary64
 (/ (* (log (tanh (* 0.7853981633974483 f))) 4.0) PI))

C

double code(double f) {
	return (log(tanh((0.7853981633974483 * f))) * 4.0) / ((double) M_PI);
}

Java

public static double code(double f) {
	return (Math.log(Math.tanh((0.7853981633974483 * f))) * 4.0) / Math.PI;
}

Python

def code(f):
	return (math.log(math.tanh((0.7853981633974483 * f))) * 4.0) / math.pi

Julia

function code(f)
	return Float64(Float64(log(tanh(Float64(0.7853981633974483 * f))) * 4.0) / pi)
end

MATLAB

function tmp = code(f)
	tmp = (log(tanh((0.7853981633974483 * f))) * 4.0) / pi;
end

Wolfram

code[f_] := N[(N[(N[Log[N[Tanh[N[(0.7853981633974483 * f), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 4.0), $MachinePrecision] / Pi), $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 97.3%

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

   1. lift-/.f64 N/A

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

   2. lift-sinh.f64 N/A

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

   3. sinh-def N/A

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

   4. sinh-undef-rev N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. mult-flip N/A

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

   11. lift-PI.f64 N/A

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

   12. sinh-undef N/A

       \[\leadsto \frac{\log \left(\frac{\frac{\color{blue}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{4} \cdot f\right)}}}{2}}{\cosh \left(\left(f \cdot \pi\right) \cdot \frac{-1}{4}\right)}\right) \cdot 4}{\pi} \]

   13. associate-/l/ N/A

       \[\leadsto \frac{\log \color{blue}{\left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{4} \cdot f\right)}}{2 \cdot \cosh \left(\left(f \cdot \pi\right) \cdot \frac{-1}{4}\right)}\right)} \cdot 4}{\pi} \]

5. Applied rewrites 99.0%

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

6. Evaluated real constant 99.0%

   \[\leadsto \frac{\log \tanh \left(\color{blue}{0.7853981633974483} \cdot f\right) \cdot 4}{\pi} \]
7. Add Preprocessing

Alternative 2: 96.3% accurate, 6.1× speedup

Math

\[\frac{\left(\log f + -0.24156447527049044\right) \cdot 4}{\pi} \]

FPCore

(FPCore (f)
 :precision binary64
 (/ (* (+ (log f) -0.24156447527049044) 4.0) PI))

C

double code(double f) {
	return ((log(f) + -0.24156447527049044) * 4.0) / ((double) M_PI);
}

Java

public static double code(double f) {
	return ((Math.log(f) + -0.24156447527049044) * 4.0) / Math.PI;
}

Python

def code(f):
	return ((math.log(f) + -0.24156447527049044) * 4.0) / math.pi

Julia

function code(f)
	return Float64(Float64(Float64(log(f) + -0.24156447527049044) * 4.0) / pi)
end

MATLAB

function tmp = code(f)
	tmp = ((log(f) + -0.24156447527049044) * 4.0) / pi;
end

Wolfram

code[f_] := N[(N[(N[(N[Log[f], $MachinePrecision] + -0.24156447527049044), $MachinePrecision] * 4.0), $MachinePrecision] / Pi), $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 97.3%

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

4. Taylor expanded in f around 0

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

   1. lower-+.f64 N/A

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

   2. lower-log.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-PI.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-PI.f64 96.3%

      \[\leadsto \frac{\left(\log f + \log \left(0.5 \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)\right) \cdot 4}{\pi} \]

6. Applied rewrites 96.3%

   \[\leadsto \frac{\color{blue}{\left(\log f + \log \left(0.5 \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)\right)} \cdot 4}{\pi} \]

7. Evaluated real constant 96.3%

   \[\leadsto \frac{\left(\log f + -0.24156447527049044\right) \cdot 4}{\pi} \]
8. Add Preprocessing

Alternative 3: 96.2% accurate, 6.1× speedup

Math

\[\frac{4}{\pi} \cdot \left(\log f - 0.24156447527049044\right) \]

FPCore

(FPCore (f) :precision binary64 (* (/ 4.0 PI) (- (log f) 0.24156447527049044)))

C

double code(double f) {
	return (4.0 / ((double) M_PI)) * (log(f) - 0.24156447527049044);
}

Java

public static double code(double f) {
	return (4.0 / Math.PI) * (Math.log(f) - 0.24156447527049044);
}

Python

def code(f):
	return (4.0 / math.pi) * (math.log(f) - 0.24156447527049044)

Julia

function code(f)
	return Float64(Float64(4.0 / pi) * Float64(log(f) - 0.24156447527049044))
end

MATLAB

function tmp = code(f)
	tmp = (4.0 / pi) * (log(f) - 0.24156447527049044);
end

Wolfram

code[f_] := N[(N[(4.0 / Pi), $MachinePrecision] * N[(N[Log[f], $MachinePrecision] - 0.24156447527049044), $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 97.3%

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

4. Taylor expanded in f around 0

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

   1. lower-+.f64 N/A

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

   2. lower-log.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-PI.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-PI.f64 96.3%

      \[\leadsto \frac{\left(\log f + \log \left(0.5 \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)\right) \cdot 4}{\pi} \]

6. Applied rewrites 96.3%

   \[\leadsto \frac{\color{blue}{\left(\log f + \log \left(0.5 \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)\right)} \cdot 4}{\pi} \]

7. Evaluated real constant 96.3%

   \[\leadsto \frac{\left(\log f + -0.24156447527049044\right) \cdot 4}{\pi} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(\log f + \frac{-8703277446513041}{36028797018963968}\right) \cdot 4}{\pi}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\log f + \frac{-8703277446513041}{36028797018963968}\right) \cdot 4}}{\pi} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{\left(\log f + \frac{-8703277446513041}{36028797018963968}\right) \cdot \frac{4}{\pi}} \]

   4. div-flip-rev N/A

      \[\leadsto \left(\log f + \frac{-8703277446513041}{36028797018963968}\right) \cdot \color{blue}{\frac{1}{\frac{\pi}{4}}} \]

   5. lift-/.f64 N/A

      \[\leadsto \left(\log f + \frac{-8703277446513041}{36028797018963968}\right) \cdot \frac{1}{\color{blue}{\frac{\pi}{4}}} \]

   6. lift-/.f64 N/A

      \[\leadsto \left(\log f + \frac{-8703277446513041}{36028797018963968}\right) \cdot \color{blue}{\frac{1}{\frac{\pi}{4}}} \]

   7. *-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\pi}{4}} \cdot \left(\log f + \frac{-8703277446513041}{36028797018963968}\right)} \]

   8. lower-*.f64 96.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\pi}{4}} \cdot \left(\log f + -0.24156447527049044\right)} \]

   9. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\pi}{4}}} \cdot \left(\log f + \frac{-8703277446513041}{36028797018963968}\right) \]

   10. lift-/.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{\pi}{4}}} \cdot \left(\log f + \frac{-8703277446513041}{36028797018963968}\right) \]

   11. div-flip-rev N/A

       \[\leadsto \color{blue}{\frac{4}{\pi}} \cdot \left(\log f + \frac{-8703277446513041}{36028797018963968}\right) \]

   12. lower-/.f64 96.2%

       \[\leadsto \color{blue}{\frac{4}{\pi}} \cdot \left(\log f + -0.24156447527049044\right) \]

9. Applied rewrites 96.2%

   \[\leadsto \color{blue}{\frac{4}{\pi} \cdot \left(\log f - 0.24156447527049044\right)} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025207 
(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)))))))))