VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 99.1%

Time: 30.2s

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: 6.8% 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: 99.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\frac{1}{\tanh \left(\pi \cdot \left(f \cdot 0.25\right)\right)}\right)}{\pi} \cdot \left(-4\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (* (/ (log (/ 1.0 (tanh (* PI (* f 0.25))))) PI) (- 4.0)))

C

double code(double f) {
	return (log((1.0 / tanh((((double) M_PI) * (f * 0.25))))) / ((double) M_PI)) * -4.0;
}

Java

public static double code(double f) {
	return (Math.log((1.0 / Math.tanh((Math.PI * (f * 0.25))))) / Math.PI) * -4.0;
}

Python

def code(f):
	return (math.log((1.0 / math.tanh((math.pi * (f * 0.25))))) / math.pi) * -4.0

Julia

function code(f)
	return Float64(Float64(log(Float64(1.0 / tanh(Float64(pi * Float64(f * 0.25))))) / pi) * Float64(-4.0))
end

MATLAB

function tmp = code(f)
	tmp = (log((1.0 / tanh((pi * (f * 0.25))))) / pi) * -4.0;
end

Wolfram

code[f_] := N[(N[(N[Log[N[(1.0 / N[Tanh[N[(Pi * N[(f * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * (-4.0)), $MachinePrecision]

Derivation

1. Initial program 5.5%

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

3. Taylor expanded in f around inf 5.5%

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

   1. *-un-lft-identity 5.5%

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

   2. clear-num 5.5%

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

   3. +-commutative 5.5%

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

   4. tanh-undef 99.5%

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

   5. associate-*r* 99.5%

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

5. Applied egg-rr 99.5%

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

   1. *-lft-identity 99.5%

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

   2. *-commutative 99.5%

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

   3. *-commutative 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

   \[\leadsto \frac{\log \left(\frac{1}{\tanh \left(\pi \cdot \left(f \cdot 0.25\right)\right)}\right)}{\pi} \cdot \left(-4\right) \]
9. Add Preprocessing

Alternative 2: 98.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.5%

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

3. Simplified 99.3%

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

5. Taylor expanded in f around 0 98.8%

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

   1. associate-*r/ 98.8%

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

   2. mul-1-neg 98.8%

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

   3. unsub-neg 98.8%

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

7. Simplified 98.8%

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

   1. expm1-log1p-u 97.5%

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

   2. diff-log 97.5%

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

9. Applied egg-rr 97.5%

   \[\leadsto \frac{-4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\frac{\frac{4}{\pi}}{f}\right)\right)\right)}}{\pi} \]
10. Step-by-step derivation

    1. expm1-log1p-u 98.8%

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

    2. associate-/l/ 98.8%

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

    3. metadata-eval 98.8%

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

    4. frac-times 98.4%

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

    5. div-inv 98.4%

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

    6. add-sqr-sqrt 97.9%

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

    7. sqrt-unprod 98.5%

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

    8. pow2 98.5%

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

    9. associate-/l/ 98.8%

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

11. Applied egg-rr 98.8%

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

    1. unpow2 98.8%

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

    2. rem-sqrt-square 98.8%

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

    3. associate-/r* 98.8%

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

13. Simplified 98.8%

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

14. Final simplification 98.8%

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

Alternative 3: 99.2% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.5%

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

3. Taylor expanded in f around inf 5.5%

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

   1. *-un-lft-identity 5.5%

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

5. Applied egg-rr 99.5%

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

   1. associate-*r/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. *-commutative 99.5%

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

   4. *-commutative 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 4: 98.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \frac{-4 \cdot \left(-1 - \left(-1 - \log \left(\frac{4}{\pi \cdot f}\right)\right)\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/ (* -4.0 (- -1.0 (- -1.0 (log (/ 4.0 (* PI f)))))) PI))

C

double code(double f) {
	return (-4.0 * (-1.0 - (-1.0 - log((4.0 / (((double) M_PI) * f)))))) / ((double) M_PI);
}

Java

public static double code(double f) {
	return (-4.0 * (-1.0 - (-1.0 - Math.log((4.0 / (Math.PI * f)))))) / Math.PI;
}

Python

def code(f):
	return (-4.0 * (-1.0 - (-1.0 - math.log((4.0 / (math.pi * f)))))) / math.pi

Julia

function code(f)
	return Float64(Float64(-4.0 * Float64(-1.0 - Float64(-1.0 - log(Float64(4.0 / Float64(pi * f)))))) / pi)
end

MATLAB

function tmp = code(f)
	tmp = (-4.0 * (-1.0 - (-1.0 - log((4.0 / (pi * f)))))) / pi;
end

Wolfram

code[f_] := N[(N[(-4.0 * N[(-1.0 - N[(-1.0 - N[Log[N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 5.5%

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

3. Simplified 99.3%

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

5. Taylor expanded in f around 0 98.8%

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

   1. associate-*r/ 98.8%

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

   2. mul-1-neg 98.8%

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

   3. unsub-neg 98.8%

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

7. Simplified 98.8%

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

   1. expm1-log1p-u 97.5%

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

   2. diff-log 97.5%

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

9. Applied egg-rr 97.5%

   \[\leadsto \frac{-4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\frac{\frac{4}{\pi}}{f}\right)\right)\right)}}{\pi} \]
10. Step-by-step derivation

    1. expm1-undefine 97.5%

       \[\leadsto \frac{-4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log \left(\frac{\frac{4}{\pi}}{f}\right)\right)} - 1\right)}}{\pi} \]

    2. log1p-undefine 97.5%

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

    3. rem-exp-log 98.8%

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

    4. associate-/l/ 98.8%

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

    5. metadata-eval 98.8%

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

    6. frac-times 98.4%

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

    7. div-inv 98.4%

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

    8. associate-/l/ 98.8%

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

11. Applied egg-rr 98.8%

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

12. Final simplification 98.8%

    \[\leadsto \frac{-4 \cdot \left(-1 - \left(-1 - \log \left(\frac{4}{\pi \cdot f}\right)\right)\right)}{\pi} \]
13. Add Preprocessing

Alternative 5: 97.9% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.5%

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

3. Simplified 99.3%

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

5. Taylor expanded in f around 0 98.8%

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

   1. associate-*r/ 98.8%

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

   2. mul-1-neg 98.8%

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

   3. unsub-neg 98.8%

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

7. Simplified 98.8%

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

   1. *-un-lft-identity 98.8%

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

   2. diff-log 98.8%

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

9. Applied egg-rr 98.8%

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

    1. *-lft-identity 98.8%

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

    2. associate-/l/ 98.8%

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

    3. associate-/r* 98.4%

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

11. Simplified 98.4%

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

    1. div-inv 98.3%

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

    2. *-commutative 98.3%

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

    3. associate-*l* 98.3%

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

    4. associate-/l/ 98.6%

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

    5. metadata-eval 98.6%

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

    6. distribute-lft-neg-in 98.6%

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

    7. div-inv 98.6%

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

    8. distribute-neg-frac 98.6%

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

    9. metadata-eval 98.6%

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

13. Applied egg-rr 98.6%

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

14. Final simplification 98.6%

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

Alternative 6: 97.9% accurate, 4.9× 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(-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 5.5%

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

3. Simplified 99.3%

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

5. Taylor expanded in f around 0 98.8%

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

   1. associate-*r/ 98.8%

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

   2. mul-1-neg 98.8%

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

   3. unsub-neg 98.8%

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

7. Simplified 98.8%

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

   1. *-un-lft-identity 98.8%

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

   2. diff-log 98.8%

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

9. Applied egg-rr 98.8%

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

    1. *-lft-identity 98.8%

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

    2. associate-/l/ 98.8%

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

    3. associate-/r* 98.4%

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

11. Simplified 98.4%

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

    1. clear-num 98.3%

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

    2. inv-pow 98.3%

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

    3. *-un-lft-identity 98.3%

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

    4. times-frac 98.3%

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

    5. metadata-eval 98.3%

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

    6. associate-/l/ 98.7%

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

13. Applied egg-rr 98.7%

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

    1. unpow-1 98.7%

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

    2. associate-/r* 98.7%

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

    3. metadata-eval 98.7%

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

    4. associate-/r* 98.7%

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

15. Simplified 98.7%

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

16. Final simplification 98.7%

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

Alternative 7: 98.0% accurate, 4.9× speedup

Math

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

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

Derivation

1. Initial program 5.5%

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

3. Simplified 99.3%

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

5. Taylor expanded in f around 0 98.8%

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

   1. associate-*r/ 98.8%

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

   2. mul-1-neg 98.8%

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

   3. unsub-neg 98.8%

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

7. Simplified 98.8%

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

   1. diff-log 98.8%

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

9. Applied egg-rr 98.8%

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

10. Final simplification 98.8%

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

Reproduce

herbie shell --seed 2024076 
(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)))))))))