VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 99.0%

Time: 19.1s

Alternatives: 7

Speedup: 4.9×

• could not determine a ground truth

  1. f = 2.00000000000000007e154

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: 6.9% 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

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

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 9.1%

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

   1. add-cube-cbrt 9.1%

      \[\leadsto -\color{blue}{\left(\sqrt[3]{\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)} \cdot \sqrt[3]{\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)}\right) \cdot \sqrt[3]{\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)}} \]

   2. pow3 9.1%

      \[\leadsto -\color{blue}{{\left(\sqrt[3]{\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)}\right)}^{3}} \]

4. Applied egg-rr 97.4%

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

   1. rem-cube-cbrt 98.8%

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

6. Applied egg-rr 98.8%

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

Alternative 2: 98.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.1%

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

   1. add-exp-log 9.1%

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

   2. associate-*l/ 9.1%

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

4. Applied egg-rr 97.6%

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

   1. rem-exp-log 98.8%

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

   2. distribute-neg-frac 98.8%

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

   3. *-commutative 98.8%

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

   4. remove-double-neg 98.8%

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

   5. associate-/r* 98.8%

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

   6. div-inv 98.8%

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

   7. clear-num 98.7%

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

   8. metadata-eval 98.7%

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

   9. associate-*l/ 98.7%

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

   10. metadata-eval 98.7%

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

   11. *-commutative 98.7%

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

   12. associate-*l* 98.7%

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

   13. *-commutative 98.7%

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

6. Applied egg-rr 98.7%

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

Alternative 3: 96.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.1%

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

3. Simplified 98.7%

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

5. Taylor expanded in f around 0 96.0%

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

   1. add-sqr-sqrt 95.6%

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

   2. sqrt-unprod 96.1%

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

   3. pow2 96.1%

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

   4. associate-/r* 96.1%

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

7. Applied egg-rr 96.1%

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

   1. unpow2 96.1%

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

   2. rem-sqrt-square 96.1%

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

   3. associate-/r* 96.1%

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

9. Simplified 96.1%

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

Alternative 4: 96.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.1%

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

3. Simplified 98.7%

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

5. Taylor expanded in f around 0 96.1%

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

   1. mul-1-neg 96.1%

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

   2. unsub-neg 96.1%

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

7. Simplified 96.1%

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

Alternative 5: 96.0% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.1%

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

3. Simplified 98.7%

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

5. Taylor expanded in f around 0 96.0%

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

   1. associate-*r/ 96.1%

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

   2. associate-/r* 96.1%

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

7. Applied egg-rr 96.1%

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

   1. clear-num 96.1%

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

   2. log-rec 96.1%

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

   3. div-inv 96.1%

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

   4. clear-num 96.1%

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

   5. div-inv 96.1%

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

   6. metadata-eval 96.1%

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

9. Applied egg-rr 96.1%

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

10. Final simplification 96.1%

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

Alternative 6: 95.9% accurate, 4.9× speedup

Math

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

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 * 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 9.1%

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

3. Simplified 98.7%

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

5. Taylor expanded in f around 0 96.0%

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

   1. associate-*r/ 96.1%

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

   2. associate-/r* 96.1%

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

7. Applied egg-rr 96.1%

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

8. Final simplification 96.1%

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

Alternative 7: 95.9% 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 9.1%

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

3. Simplified 98.7%

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

5. Taylor expanded in f around 0 96.0%

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

Reproduce

herbie shell --seed 2024136 
(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)))))))))