VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.1% → 99.0%

Time: 20.2s

Alternatives: 6

Speedup: 4.9×

• could not determine a ground truth

  1. f = 1275492679.4975288

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.4%

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

   1. associate-*l/ 7.4%

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

4. Applied egg-rr 98.8%

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

   1. neg-mul-1 98.8%

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

   2. *-commutative 98.8%

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

   3. times-frac 98.8%

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

   4. metadata-eval 98.8%

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

   5. *-commutative 98.8%

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

6. Simplified 98.8%

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

   1. metadata-eval 98.8%

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

   2. times-frac 98.8%

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

   3. *-commutative 98.8%

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

   4. neg-mul-1 98.8%

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

   5. *-commutative 98.8%

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

   6. associate-/r* 98.8%

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

   7. distribute-neg-frac 98.8%

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

   8. distribute-frac-neg2 98.8%

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

   9. *-commutative 98.8%

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

   10. frac-2neg 98.8%

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

8. Applied egg-rr 98.8%

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

   1. add-sqr-sqrt 1.2%

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

   2. sqrt-unprod 2.7%

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

   3. sqr-neg 2.7%

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

   4. *-commutative 2.7%

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

   5. *-commutative 2.7%

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

   6. sqrt-unprod 2.7%

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

   7. add-sqr-sqrt 2.7%

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

   8. neg-sub0 2.7%

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

   9. metadata-eval 2.7%

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

   10. *-commutative 2.7%

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

10. Applied egg-rr 98.8%

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

Alternative 2: 99.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.4%

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

   1. associate-*l/ 7.4%

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

4. Applied egg-rr 98.8%

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

   1. neg-mul-1 98.8%

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

   2. *-commutative 98.8%

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

   3. times-frac 98.8%

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

   4. metadata-eval 98.8%

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

   5. *-commutative 98.8%

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

6. Simplified 98.8%

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

   1. metadata-eval 98.8%

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

   2. times-frac 98.8%

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

   3. *-commutative 98.8%

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

   4. neg-mul-1 98.8%

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

   5. *-commutative 98.8%

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

   6. associate-/r* 98.8%

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

   7. distribute-neg-frac 98.8%

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

   8. distribute-frac-neg2 98.8%

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

   9. *-commutative 98.8%

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

   10. frac-2neg 98.8%

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

8. Applied egg-rr 98.8%

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

Alternative 3: 95.8% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.4%

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

   1. associate-*l/ 7.4%

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

4. Applied egg-rr 98.8%

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

   1. neg-mul-1 98.8%

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

   2. *-commutative 98.8%

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

   3. times-frac 98.8%

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

   4. metadata-eval 98.8%

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

   5. *-commutative 98.8%

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

6. Simplified 98.8%

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

   1. metadata-eval 98.8%

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

   2. times-frac 98.8%

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

   3. *-commutative 98.8%

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

   4. neg-mul-1 98.8%

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

   5. *-commutative 98.8%

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

   6. associate-/r* 98.8%

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

   7. distribute-neg-frac 98.8%

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

   8. distribute-frac-neg2 98.8%

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

   9. *-commutative 98.8%

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

   10. frac-2neg 98.8%

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

8. Applied egg-rr 98.8%

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

9. Taylor expanded in f around 0 96.2%

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

    1. *-commutative 96.2%

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

    2. associate-*r* 96.2%

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

    3. *-commutative 96.2%

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

11. Simplified 96.2%

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

12. Final simplification 96.2%

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

Alternative 4: 95.7% accurate, 4.9× speedup

Math

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

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

Derivation

1. Initial program 7.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in f around 0 95.7%

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

Alternative 5: 3.1% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \log 0 \end{array} \]

FPCore

(FPCore (f) :precision binary64 (log 0.0))

C

double code(double f) {
	return log(0.0);
}

Fortran

real(8) function code(f)
    real(8), intent (in) :: f
    code = log(0.0d0)
end function

Java

public static double code(double f) {
	return Math.log(0.0);
}

Python

def code(f):
	return math.log(0.0)

Julia

function code(f)
	return log(0.0)
end

MATLAB

function tmp = code(f)
	tmp = log(0.0);
end

Wolfram

code[f_] := N[Log[0.0], $MachinePrecision]

Derivation

1. Initial program 7.4%

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

   1. associate-*l/ 7.4%

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

4. Applied egg-rr 98.8%

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

   1. neg-mul-1 98.8%

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

   2. *-commutative 98.8%

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

   3. times-frac 98.8%

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

   4. metadata-eval 98.8%

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

   5. *-commutative 98.8%

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

6. Simplified 98.8%

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

   1. metadata-eval 98.8%

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

   2. times-frac 98.8%

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

   3. *-commutative 98.8%

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

   4. neg-mul-1 98.8%

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

   5. *-commutative 98.8%

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

   6. associate-/r* 98.8%

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

   7. distribute-neg-frac 98.8%

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

   8. distribute-frac-neg2 98.8%

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

   9. *-commutative 98.8%

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

   10. frac-2neg 98.8%

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

8. Applied egg-rr 98.8%

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

10. Applied egg-rr 3.1%

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

    1. +-inverses 3.1%

       \[\leadsto \frac{\frac{\log \color{blue}{0}}{\pi}}{0.25} \]

12. Simplified 3.1%

    \[\leadsto \frac{\frac{\log \color{blue}{0}}{\pi}}{0.25} \]
13. Step-by-step derivation

    1. *-un-lft-identity 3.1%

       \[\leadsto \color{blue}{1 \cdot \frac{\frac{\log 0}{\pi}}{0.25}} \]

    2. div-inv 3.1%

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

    3. metadata-eval 3.1%

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

14. Applied egg-rr 3.1%

    \[\leadsto \color{blue}{1 \cdot \left(\frac{\log 0}{\pi} \cdot 4\right)} \]
15. Step-by-step derivation

    1. *-lft-identity 3.1%

       \[\leadsto \color{blue}{\frac{\log 0}{\pi} \cdot 4} \]

    2. log1p-expm1 3.1%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\log 0}{\pi} \cdot 4\right)\right)} \]

    3. log1p-define 3.1%

       \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{\log 0}{\pi} \cdot 4\right)\right)} \]

    4. rem-exp-log 3.1%

       \[\leadsto \log \color{blue}{\left(e^{\log \left(1 + \mathsf{expm1}\left(\frac{\log 0}{\pi} \cdot 4\right)\right)}\right)} \]

    5. log1p-define 3.1%

       \[\leadsto \log \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\log 0}{\pi} \cdot 4\right)\right)}}\right) \]

    6. log1p-expm1 3.1%

       \[\leadsto \log \left(e^{\color{blue}{\frac{\log 0}{\pi} \cdot 4}}\right) \]

    7. exp-prod 3.1%

       \[\leadsto \log \color{blue}{\left({\left(e^{\frac{\log 0}{\pi}}\right)}^{4}\right)} \]

    8. *-rgt-identity 3.1%

       \[\leadsto \log \left({\left(e^{\frac{\color{blue}{\log 0 \cdot 1}}{\pi}}\right)}^{4}\right) \]

    9. associate-*r/ 3.1%

       \[\leadsto \log \left({\left(e^{\color{blue}{\log 0 \cdot \frac{1}{\pi}}}\right)}^{4}\right) \]

    10. exp-to-pow 3.1%

        \[\leadsto \log \left({\color{blue}{\left({0}^{\left(\frac{1}{\pi}\right)}\right)}}^{4}\right) \]

    11. pow-base-0 3.1%

        \[\leadsto \log \left({\color{blue}{0}}^{4}\right) \]

    12. metadata-eval 3.1%

        \[\leadsto \log \color{blue}{0} \]

16. Simplified 3.1%

    \[\leadsto \color{blue}{\log 0} \]
17. Add Preprocessing

Alternative 6: 5.0% accurate, 106.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0}{\pi}}{0.25} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (/ 0.0 PI) 0.25))

C

double code(double f) {
	return (0.0 / ((double) M_PI)) / 0.25;
}

Java

public static double code(double f) {
	return (0.0 / Math.PI) / 0.25;
}

Python

def code(f):
	return (0.0 / math.pi) / 0.25

Julia

function code(f)
	return Float64(Float64(0.0 / pi) / 0.25)
end

MATLAB

function tmp = code(f)
	tmp = (0.0 / pi) / 0.25;
end

Wolfram

code[f_] := N[(N[(0.0 / Pi), $MachinePrecision] / 0.25), $MachinePrecision]

Derivation

1. Initial program 7.4%

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

   1. associate-*l/ 7.4%

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

4. Applied egg-rr 98.8%

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

   1. neg-mul-1 98.8%

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

   2. *-commutative 98.8%

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

   3. times-frac 98.8%

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

   4. metadata-eval 98.8%

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

   5. *-commutative 98.8%

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

6. Simplified 98.8%

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

   1. metadata-eval 98.8%

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

   2. times-frac 98.8%

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

   3. *-commutative 98.8%

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

   4. neg-mul-1 98.8%

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

   5. *-commutative 98.8%

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

   6. associate-/r* 98.8%

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

   7. distribute-neg-frac 98.8%

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

   8. distribute-frac-neg2 98.8%

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

   9. *-commutative 98.8%

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

   10. frac-2neg 98.8%

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

8. Applied egg-rr 98.8%

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

   1. add-exp-log 98.8%

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

   2. add-sqr-sqrt 1.2%

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

   3. sqrt-unprod 2.7%

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

   4. sqr-neg 2.7%

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

   5. log-rec 2.7%

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

   6. log-rec 2.7%

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

   7. sqrt-unprod 2.7%

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

   8. add-sqr-sqrt 2.7%

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

   9. add-exp-log 2.7%

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

   10. add-sqr-sqrt 2.7%

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

10. Applied egg-rr 4.2%

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

    1. *-inverses 4.2%

       \[\leadsto \frac{\frac{\log \color{blue}{1}}{\pi}}{0.25} \]

12. Simplified 4.2%

    \[\leadsto \frac{\frac{\log \color{blue}{1}}{\pi}}{0.25} \]

13. Final simplification 4.2%

    \[\leadsto \frac{\frac{0}{\pi}}{0.25} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024177 
(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)))))))))