VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 99.0%

Time: 24.0s

Alternatives: 7

Speedup: 3.3×

• 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: 99.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_] := N[(N[(N[Log[N[(1.0 / N[Tanh[N[(f * N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $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. expm1-log1p-u 6.3%

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

   2. expm1-udef 6.3%

      \[\leadsto -4 \cdot \frac{\log \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{e^{0.25 \cdot \left(f \cdot \pi\right)} - e^{-0.25 \cdot \left(f \cdot \pi\right)}}\right)} - 1\right)}}{\pi} \]

4. Applied egg-rr 98.9%

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

   1. expm1-def 98.9%

      \[\leadsto -4 \cdot \frac{\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}\right)\right)\right)}}{\pi} \]

   2. expm1-log1p 98.9%

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

   3. *-commutative 98.9%

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

   4. associate-*l* 98.9%

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

   5. *-commutative 98.9%

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

6. Simplified 98.9%

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

7. Final simplification 98.9%

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

Alternative 2: 95.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_] := (-N[Abs[N[(4.0 * N[(N[Log[N[(N[(4.0 / Pi), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision] / Pi), $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 96.0%

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

   1. *-commutative 96.0%

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

   2. associate-/r* 96.0%

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

   3. distribute-rgt-out-- 96.0%

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

   4. metadata-eval 96.0%

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

4. Simplified 96.0%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{\frac{2}{\pi \cdot 0.5}}{f}\right)} \]
5. Step-by-step derivation

   1. diff-log 96.1%

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

   2. add-log-exp 78.1%

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

   3. *-commutative 78.1%

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

   4. diff-log 78.0%

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

   5. exp-to-pow 78.0%

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

   6. associate-/l/ 78.0%

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

   7. *-commutative 78.0%

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

   8. associate-/r/ 78.0%

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

6. Applied egg-rr 78.0%

   \[\leadsto -\color{blue}{\log \left({\left(\frac{2}{\left(\pi \cdot 0.5\right) \cdot f}\right)}^{\left(\frac{1}{\pi} \cdot 4\right)}\right)} \]
7. Step-by-step derivation

   1. log-pow 96.0%

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

   2. associate-*l/ 96.0%

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

   3. metadata-eval 96.0%

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

   4. metadata-eval 96.0%

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

   5. distribute-rgt-out-- 96.0%

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

   6. *-commutative 96.0%

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

   7. distribute-rgt-out-- 96.0%

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

   8. metadata-eval 96.0%

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

8. Simplified 96.0%

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

10. Applied egg-rr 96.3%

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

    1. unpow2 96.3%

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

    2. rem-sqrt-square 96.3%

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

    3. associate-*r* 96.3%

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

12. Simplified 96.3%

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

    1. frac-2neg 96.3%

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

    2. div-inv 96.1%

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

    3. *-commutative 96.1%

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

    4. neg-log 96.1%

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

    5. associate-*r* 96.1%

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

    6. *-commutative 96.1%

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

    7. metadata-eval 96.1%

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

    8. div-inv 96.1%

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

    9. clear-num 96.1%

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

    10. distribute-rgt-neg-in 96.1%

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

    11. metadata-eval 96.1%

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

    12. metadata-eval 96.1%

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

    13. div-inv 96.1%

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

    14. clear-num 96.1%

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

    15. *-commutative 96.1%

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

    16. associate-*l/ 96.3%

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

14. Applied egg-rr 96.3%

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

15. Final simplification 96.3%

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

Alternative 3: 99.0% 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 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. clear-num 6.3%

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

   2. log-rec 6.3%

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

   3. +-commutative 6.3%

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

   4. tanh-undef 98.9%

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

   5. associate-*r* 98.9%

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

4. Applied egg-rr 98.9%

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

5. Final simplification 98.9%

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

Alternative 4: 95.7% accurate, 3.3× speedup

Math

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

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(4.0 / Float64(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[(4.0 / N[(f * Pi), $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 96.0%

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

   1. *-commutative 96.0%

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

   2. associate-/r* 96.0%

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

   3. distribute-rgt-out-- 96.0%

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

   4. metadata-eval 96.0%

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

4. Simplified 96.0%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{\frac{2}{\pi \cdot 0.5}}{f}\right)} \]
5. Step-by-step derivation

   1. diff-log 96.1%

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

   2. add-cbrt-cube 95.7%

      \[\leadsto -\color{blue}{\sqrt[3]{\left(\left(\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f\right)\right) \cdot \left(\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f\right)\right)\right) \cdot \left(\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f\right)\right)}} \]

   3. pow3 95.7%

      \[\leadsto -\sqrt[3]{\color{blue}{{\left(\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f\right)\right)}^{3}}} \]

6. Applied egg-rr 95.7%

   \[\leadsto -\color{blue}{\sqrt[3]{{\left(\frac{\log \left(\frac{2}{\left(\pi \cdot 0.5\right) \cdot f}\right)}{\pi \cdot 0.25}\right)}^{3}}} \]
7. Step-by-step derivation

   1. rem-cbrt-cube 96.2%

      \[\leadsto -\color{blue}{\frac{\log \left(\frac{2}{\left(\pi \cdot 0.5\right) \cdot f}\right)}{\pi \cdot 0.25}} \]

   2. clear-num 96.1%

      \[\leadsto -\color{blue}{\frac{1}{\frac{\pi \cdot 0.25}{\log \left(\frac{2}{\left(\pi \cdot 0.5\right) \cdot f}\right)}}} \]

   3. associate-/r/ 96.0%

      \[\leadsto -\color{blue}{\frac{1}{\pi \cdot 0.25} \cdot \log \left(\frac{2}{\left(\pi \cdot 0.5\right) \cdot f}\right)} \]

   4. metadata-eval 96.0%

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

   5. div-inv 96.0%

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

   6. clear-num 96.0%

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

   7. associate-/r* 96.0%

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

   8. log-div 96.1%

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

   9. *-commutative 96.1%

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

   10. associate-/r* 96.1%

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

   11. metadata-eval 96.1%

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

   12. clear-num 96.1%

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

   13. diff-log 96.0%

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

   14. clear-num 96.0%

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

   15. associate-/l/ 96.0%

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

   16. *-commutative 96.0%

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

8. Applied egg-rr 96.0%

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

9. Final simplification 96.0%

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

Alternative 5: 95.7% accurate, 3.3× speedup

Math

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

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) / Float64(pi / log(Float64(4.0 / Float64(f * pi)))))
end

MATLAB

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

Wolfram

code[f_] := N[((-4.0) / N[(Pi / N[Log[N[(4.0 / N[(f * Pi), $MachinePrecision]), $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 96.0%

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

   1. *-commutative 96.0%

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

   2. associate-/r* 96.0%

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

   3. distribute-rgt-out-- 96.0%

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

   4. metadata-eval 96.0%

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

4. Simplified 96.0%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{\frac{2}{\pi \cdot 0.5}}{f}\right)} \]
5. Step-by-step derivation

   1. diff-log 96.1%

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

   2. add-log-exp 78.1%

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

   3. *-commutative 78.1%

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

   4. diff-log 78.0%

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

   5. exp-to-pow 78.0%

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

   6. associate-/l/ 78.0%

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

   7. *-commutative 78.0%

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

   8. associate-/r/ 78.0%

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

6. Applied egg-rr 78.0%

   \[\leadsto -\color{blue}{\log \left({\left(\frac{2}{\left(\pi \cdot 0.5\right) \cdot f}\right)}^{\left(\frac{1}{\pi} \cdot 4\right)}\right)} \]
7. Step-by-step derivation

   1. log-pow 96.0%

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

   2. associate-*l/ 96.0%

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

   3. metadata-eval 96.0%

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

   4. metadata-eval 96.0%

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

   5. distribute-rgt-out-- 96.0%

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

   6. *-commutative 96.0%

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

   7. distribute-rgt-out-- 96.0%

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

   8. metadata-eval 96.0%

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

8. Simplified 96.0%

   \[\leadsto -\color{blue}{\frac{4}{\pi} \cdot \log \left(\frac{2}{f \cdot \left(\pi \cdot 0.5\right)}\right)} \]
9. Step-by-step derivation

   1. associate-*l/ 96.2%

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

   2. *-commutative 96.2%

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

   3. associate-/l* 96.1%

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

   4. associate-/r* 96.1%

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

   5. log-div 96.1%

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

   6. *-commutative 96.1%

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

   7. associate-/r* 96.1%

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

   8. metadata-eval 96.1%

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

   9. clear-num 96.1%

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

   10. diff-log 96.1%

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

   11. clear-num 96.1%

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

   12. associate-/l/ 96.1%

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

   13. *-commutative 96.1%

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

10. Applied egg-rr 96.1%

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

11. Final simplification 96.1%

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

Alternative 6: 95.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_] := N[((-N[Log[N[(f * N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(Pi * -0.25), $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 96.0%

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

   1. *-commutative 96.0%

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

   2. associate-/r* 96.0%

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

   3. distribute-rgt-out-- 96.0%

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

   4. metadata-eval 96.0%

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

4. Simplified 96.0%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{\frac{2}{\pi \cdot 0.5}}{f}\right)} \]
5. Step-by-step derivation

   1. diff-log 96.1%

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

   2. add-log-exp 78.1%

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

   3. *-commutative 78.1%

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

   4. diff-log 78.0%

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

   5. exp-to-pow 78.0%

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

   6. associate-/l/ 78.0%

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

   7. *-commutative 78.0%

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

   8. associate-/r/ 78.0%

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

6. Applied egg-rr 78.0%

   \[\leadsto -\color{blue}{\log \left({\left(\frac{2}{\left(\pi \cdot 0.5\right) \cdot f}\right)}^{\left(\frac{1}{\pi} \cdot 4\right)}\right)} \]
7. Step-by-step derivation

   1. log-pow 96.0%

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

   2. associate-*l/ 96.0%

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

   3. metadata-eval 96.0%

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

   4. metadata-eval 96.0%

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

   5. distribute-rgt-out-- 96.0%

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

   6. *-commutative 96.0%

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

   7. distribute-rgt-out-- 96.0%

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

   8. metadata-eval 96.0%

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

8. Simplified 96.0%

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

9. Taylor expanded in f around 0 96.2%

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

    1. *-commutative 96.2%

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

    2. *-commutative 96.2%

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

    3. cancel-sign-sub 96.2%

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

    4. *-commutative 96.2%

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

    5. log-rec 96.2%

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

11. Simplified 96.2%

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

12. Final simplification 96.2%

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

Alternative 7: 3.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \frac{\log 0}{\pi} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_] := N[(4.0 * N[(N[Log[0.0], $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. expm1-log1p-u 6.3%

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

   2. expm1-udef 6.3%

      \[\leadsto -4 \cdot \frac{\log \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{e^{0.25 \cdot \left(f \cdot \pi\right)} - e^{-0.25 \cdot \left(f \cdot \pi\right)}}\right)} - 1\right)}}{\pi} \]

4. Applied egg-rr 98.9%

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

   1. expm1-def 98.9%

      \[\leadsto -4 \cdot \frac{\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}\right)\right)\right)}}{\pi} \]

   2. expm1-log1p 98.9%

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

   3. *-commutative 98.9%

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

   4. associate-*l* 98.9%

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

   5. *-commutative 98.9%

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

6. Simplified 98.9%

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

   1. log-div 98.9%

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

   2. div-sub 98.9%

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

   3. metadata-eval 98.9%

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

   4. *-commutative 98.9%

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

   5. associate-*l* 98.9%

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

8. Applied egg-rr 98.9%

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

   1. div-sub 98.9%

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

   2. neg-sub0 98.9%

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

   3. rem-exp-log 98.9%

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

   4. rem-exp-log 98.9%

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

   5. associate-*r* 98.9%

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

10. Simplified 98.9%

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

    1. *-commutative 98.9%

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

    2. tanh-def-a 6.3%

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

    3. div-sub 6.3%

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

    4. exp-prod 6.2%

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

    5. cosh-undef 6.2%

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

    6. *-commutative 6.2%

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

    7. associate-*l* 6.2%

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

12. Applied egg-rr 3.1%

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

    1. +-inverses 3.1%

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

14. Simplified 3.1%

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

15. Final simplification 3.1%

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

Reproduce

herbie shell --seed 2023315 
(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)))))))))