VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 98.9%

Time: 31.4s

Alternatives: 6

Speedup: 3.3×

• could not determine a ground truth

  1. f = -5.362937640208645e+226

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: 6.8% 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: 98.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.3%

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

2. Taylor expanded in f around inf 6.3%

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

   1. *-un-lft-identity 6.3%

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

   2. log-prod 6.3%

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

   3. metadata-eval 6.3%

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

   4. clear-num 6.3%

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

   5. log-rec 6.3%

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

4. Applied egg-rr 98.9%

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

   1. +-lft-identity 98.9%

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

   2. associate-*r* 98.9%

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

   3. *-commutative 98.9%

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

6. Simplified 98.9%

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

7. Final simplification 98.9%

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

Alternative 2: 95.7% accurate, 2.5× speedup

Math

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

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

Derivation

1. Initial program 6.3%

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

2. Taylor expanded in f around 0 96.1%

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

   1. *-commutative 96.1%

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

   2. associate-/r* 96.1%

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

   3. distribute-rgt-out-- 96.1%

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

   4. metadata-eval 96.1%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\frac{2}{\pi \cdot \color{blue}{0.5}}}{f}\right) \]

4. Simplified 96.1%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{\frac{2}{\pi \cdot 0.5}}{f}\right)} \]

5. Taylor expanded in f around 0 96.2%

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

   1. neg-mul-1 96.2%

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

   2. sub-neg 96.2%

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

7. Simplified 96.2%

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

8. Final simplification 96.2%

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

Alternative 3: 95.5% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.3%

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

2. Taylor expanded in f around 0 96.1%

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

   1. *-commutative 96.1%

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

   2. associate-/r* 96.1%

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

   3. distribute-rgt-out-- 96.1%

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

   4. metadata-eval 96.1%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\frac{2}{\pi \cdot \color{blue}{0.5}}}{f}\right) \]

4. Simplified 96.1%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{\frac{2}{\pi \cdot 0.5}}{f}\right)} \]

5. Taylor expanded in f around 0 96.1%

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

   1. associate-/r* 96.1%

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

7. Simplified 96.1%

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

8. Taylor expanded in f around 0 96.2%

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

   1. associate-*r/ 96.2%

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

   2. mul-1-neg 96.2%

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

   3. unsub-neg 96.2%

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

   4. log-div 96.2%

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

   5. associate-/r* 96.2%

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

   6. associate-*l/ 96.1%

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

   7. *-commutative 96.1%

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

10. Simplified 96.1%

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

11. Final simplification 96.1%

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

Alternative 4: 95.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.3%

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

2. Taylor expanded in f around 0 96.1%

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

   1. *-commutative 96.1%

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

   2. associate-/r* 96.1%

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

   3. distribute-rgt-out-- 96.1%

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

   4. metadata-eval 96.1%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\frac{2}{\pi \cdot \color{blue}{0.5}}}{f}\right) \]

4. Simplified 96.1%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{\frac{2}{\pi \cdot 0.5}}{f}\right)} \]

5. Taylor expanded in f around 0 96.1%

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

   1. associate-/r* 96.1%

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

7. Simplified 96.1%

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

8. Taylor expanded in f around 0 96.2%

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

   1. associate-*r/ 96.2%

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

   2. mul-1-neg 96.2%

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

   3. unsub-neg 96.2%

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

   4. log-div 96.2%

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

   5. associate-/r* 96.2%

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

   6. associate-*l/ 96.1%

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

   7. *-commutative 96.1%

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

10. Simplified 96.1%

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

    1. associate-*l/ 96.2%

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

    2. *-commutative 96.2%

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

    3. associate-/l* 96.1%

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

    4. associate-/r* 96.1%

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

12. Applied egg-rr 96.1%

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

13. Final simplification 96.1%

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

Alternative 5: 95.7% accurate, 3.3× speedup

Math

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

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 * Float64(-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 6.3%

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

2. Taylor expanded in f around 0 96.1%

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

   1. *-commutative 96.1%

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

   2. associate-/r* 96.1%

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

   3. distribute-rgt-out-- 96.1%

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

   4. metadata-eval 96.1%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\frac{2}{\pi \cdot \color{blue}{0.5}}}{f}\right) \]

4. Simplified 96.1%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{\frac{2}{\pi \cdot 0.5}}{f}\right)} \]

5. Taylor expanded in f around 0 96.1%

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

   1. associate-/r* 96.1%

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

7. Simplified 96.1%

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

   1. clear-num 96.1%

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

   2. *-commutative 96.1%

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

   3. associate-*r/ 96.2%

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

   4. associate-/l/ 96.2%

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

9. Applied egg-rr 96.2%

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

10. Final simplification 96.2%

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

Alternative 6: 0.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log 0}{\pi} \cdot \left(-4\right) \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (/ (log 0.0) PI) (- 4.0)))

C

double code(double f) {
	return (log(0.0) / ((double) M_PI)) * -4.0;
}

Java

public static double code(double f) {
	return (Math.log(0.0) / Math.PI) * -4.0;
}

Python

def code(f):
	return (math.log(0.0) / math.pi) * -4.0

Julia

function code(f)
	return Float64(Float64(log(0.0) / pi) * Float64(-4.0))
end

MATLAB

function tmp = code(f)
	tmp = (log(0.0) / pi) * -4.0;
end

Wolfram

code[f_] := N[(N[(N[Log[0.0], $MachinePrecision] / Pi), $MachinePrecision] * (-4.0)), $MachinePrecision]

Derivation

1. Initial program 6.3%

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

2. Taylor expanded in f around inf 6.3%

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

   1. *-un-lft-identity 6.3%

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

   2. log-prod 6.3%

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

   3. metadata-eval 6.3%

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

   4. clear-num 6.3%

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

   5. log-rec 6.3%

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

4. Applied egg-rr 98.9%

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

   1. +-lft-identity 98.9%

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

   2. associate-*r* 98.9%

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

   3. *-commutative 98.9%

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

6. Simplified 98.9%

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

8. Applied egg-rr 1.6%

   \[\leadsto -4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\log \tanh \left(\pi \cdot \left(f \cdot 0.25\right)\right)}{\pi}\right)} - 1\right)} \]
9. Step-by-step derivation

   1. expm1-def 1.6%

      \[\leadsto -4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \tanh \left(\pi \cdot \left(f \cdot 0.25\right)\right)}{\pi}\right)\right)} \]

   2. expm1-log1p 3.1%

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

   3. *-commutative 3.1%

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

   4. associate-*l* 3.1%

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

10. Simplified 3.1%

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

12. Applied egg-rr 0.6%

    \[\leadsto -4 \cdot \frac{\log \color{blue}{\left(\frac{{\left(e^{f}\right)}^{\left(0.25 \cdot \pi\right)}}{2 \cdot \cosh \left(f \cdot \left(0.25 \cdot \pi\right)\right)} - \frac{{\left(e^{f}\right)}^{\left(0.25 \cdot \pi\right)}}{2 \cdot \cosh \left(f \cdot \left(0.25 \cdot \pi\right)\right)}\right)}}{\pi} \]
13. Step-by-step derivation

    1. +-inverses 0.7%

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

14. Simplified 0.7%

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

15. Final simplification 0.7%

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

Reproduce

herbie shell --seed 2023300 
(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)))))))))