VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.0% → 96.1%

Time: 29.8s

Alternatives: 5

Speedup: 3.3×

• could not determine a ground truth

  1. f = -9.627577009425023e+291

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: 7.0% 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: 96.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.4%

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

2. Taylor expanded in f around 0 96.4%

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

   1. *-commutative 96.4%

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

   2. associate-*l/ 96.4%

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

   3. mul-1-neg 96.4%

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

   4. unsub-neg 96.4%

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

   5. distribute-rgt-out-- 96.4%

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

   6. metadata-eval 96.4%

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

4. Simplified 96.4%

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

5. Taylor expanded in f around 0 96.4%

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

6. Final simplification 96.4%

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

Alternative 2: 96.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.4%

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

2. Taylor expanded in f around 0 96.4%

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

   1. *-commutative 96.4%

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

   2. associate-*l/ 96.4%

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

   3. mul-1-neg 96.4%

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

   4. unsub-neg 96.4%

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

   5. distribute-rgt-out-- 96.4%

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

   6. metadata-eval 96.4%

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

4. Simplified 96.4%

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

5. Taylor expanded in f around 0 96.4%

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

7. Simplified 95.8%

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

8. Taylor expanded in f around 0 96.3%

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

   1. mul-1-neg 96.3%

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

   2. sub-neg 96.3%

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

10. Simplified 96.3%

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

11. Final simplification 96.3%

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

Alternative 3: 95.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.4%

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

2. Taylor expanded in f around 0 96.4%

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

   1. *-commutative 96.4%

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

   2. associate-*l/ 96.4%

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

   3. mul-1-neg 96.4%

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

   4. unsub-neg 96.4%

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

   5. distribute-rgt-out-- 96.4%

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

   6. metadata-eval 96.4%

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

4. Simplified 96.4%

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

5. Taylor expanded in f around 0 96.4%

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

7. Simplified 95.8%

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

8. Final simplification 95.8%

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

Alternative 4: 96.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.4%

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

2. Taylor expanded in f around 0 96.4%

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

   1. *-commutative 96.4%

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

   2. associate-*l/ 96.4%

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

   3. mul-1-neg 96.4%

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

   4. unsub-neg 96.4%

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

   5. distribute-rgt-out-- 96.4%

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

   6. metadata-eval 96.4%

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

4. Simplified 96.4%

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

5. Taylor expanded in f around 0 96.4%

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

   1. log-div 96.4%

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

   2. associate--l- 96.2%

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

   3. log-prod 96.2%

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

   4. *-commutative 96.2%

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

   5. log-div 96.0%

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

   6. associate-/r* 96.0%

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

7. Simplified 96.0%

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

8. Final simplification 96.0%

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

Alternative 5: 1.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{-4}{\frac{\pi}{\log \left( 7.62939453125 \cdot 10^{-6} \right)}} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (- 4.0) (/ PI (log 7.62939453125e-6))))

C

double code(double f) {
	return -4.0 / (((double) M_PI) / log(7.62939453125e-6));
}

Java

public static double code(double f) {
	return -4.0 / (Math.PI / Math.log(7.62939453125e-6));
}

Python

def code(f):
	return -4.0 / (math.pi / math.log(7.62939453125e-6))

Julia

function code(f)
	return Float64(Float64(-4.0) / Float64(pi / log(7.62939453125e-6)))
end

MATLAB

function tmp = code(f)
	tmp = -4.0 / (pi / log(7.62939453125e-6));
end

Wolfram

code[f_] := N[((-4.0) / N[(Pi / N[Log[7.62939453125e-6], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.4%

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

3. Applied egg-rr 1.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{262144}}\right) \]

4. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi}} \]
5. Step-by-step derivation

   1. associate-*r/ 1.6%

      \[\leadsto -\color{blue}{\frac{4 \cdot \log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi}} \]

   2. associate-/l* 1.6%

      \[\leadsto -\color{blue}{\frac{4}{\frac{\pi}{\log \left( 7.62939453125 \cdot 10^{-6} \right)}}} \]

6. Simplified 1.6%

   \[\leadsto -\color{blue}{\frac{4}{\frac{\pi}{\log \left( 7.62939453125 \cdot 10^{-6} \right)}}} \]

7. Final simplification 1.6%

   \[\leadsto \frac{-4}{\frac{\pi}{\log \left( 7.62939453125 \cdot 10^{-6} \right)}} \]

Reproduce

herbie shell --seed 2023292 
(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)))))))))