VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.4% → 96.7%

Time: 1.1min

Alternatives: 6

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: 7.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ t_1 := e^{t\_0}\\ t_2 := e^{-t\_0}\\ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{t\_1 + t\_2}{t\_1 - t\_2}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (/ PI 4.0) f)) (t_1 (exp t_0)) (t_2 (exp (- t_0))))
   (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ t_1 t_2) (- t_1 t_2)))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) / 4.0) * f;
	double t_1 = exp(t_0);
	double t_2 = exp(-t_0);
	return -((1.0 / (((double) M_PI) / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
}

Java

public static double code(double f) {
	double t_0 = (Math.PI / 4.0) * f;
	double t_1 = Math.exp(t_0);
	double t_2 = Math.exp(-t_0);
	return -((1.0 / (Math.PI / 4.0)) * Math.log(((t_1 + t_2) / (t_1 - t_2))));
}

Python

def code(f):
	t_0 = (math.pi / 4.0) * f
	t_1 = math.exp(t_0)
	t_2 = math.exp(-t_0)
	return -((1.0 / (math.pi / 4.0)) * math.log(((t_1 + t_2) / (t_1 - t_2))))

Julia

function code(f)
	t_0 = Float64(Float64(pi / 4.0) * f)
	t_1 = exp(t_0)
	t_2 = exp(Float64(-t_0))
	return Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(t_1 + t_2) / Float64(t_1 - t_2)))))
end

MATLAB

function tmp = code(f)
	t_0 = (pi / 4.0) * f;
	t_1 = exp(t_0);
	t_2 = exp(-t_0);
	tmp = -((1.0 / (pi / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]}, Block[{t$95$1 = N[Exp[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-t$95$0)], $MachinePrecision]}, (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(t$95$1 + t$95$2), $MachinePrecision] / N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Alternative 1: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\pi \cdot \left(\pi \cdot 0.041666666666666664\right)}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (fma
  (pow f 2.0)
  (* -2.0 (/ (* PI (* PI 0.041666666666666664)) PI))
  (fma (/ (- (log (/ 4.0 PI)) (log f)) PI) -4.0 0.0)))

C

double code(double f) {
	return fma(pow(f, 2.0), (-2.0 * ((((double) M_PI) * (((double) M_PI) * 0.041666666666666664)) / ((double) M_PI))), fma(((log((4.0 / ((double) M_PI))) - log(f)) / ((double) M_PI)), -4.0, 0.0));
}

Julia

function code(f)
	return fma((f ^ 2.0), Float64(-2.0 * Float64(Float64(pi * Float64(pi * 0.041666666666666664)) / pi)), fma(Float64(Float64(log(Float64(4.0 / pi)) - log(f)) / pi), -4.0, 0.0))
end

Wolfram

code[f_] := N[(N[Power[f, 2.0], $MachinePrecision] * N[(-2.0 * N[(N[(Pi * N[(Pi * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] * -4.0 + 0.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.1%

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

   1. *-commutative 7.1%

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

   2. distribute-rgt-neg-in 7.1%

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

3. Simplified 7.1%

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

5. Taylor expanded in f around 0 96.6%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} + \left(-2 \cdot \frac{f \cdot \left(\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)}{\pi} + -2 \cdot \frac{{f}^{2} \cdot \left(-0.25 \cdot \left({\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2} \cdot {\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}\right) + \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right) \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)}{\pi}\right)} \]

7. Simplified 96.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\mathsf{fma}\left(\pi, \mathsf{fma}\left(0.0625, \pi \cdot 2, \left(0.005208333333333333 \cdot \left(2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot -2\right) \cdot 0.5, 0\right)}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right)} \]
8. Step-by-step derivation

   1. fma-undefine 96.6%

      \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\color{blue}{\pi \cdot \left(\mathsf{fma}\left(0.0625, \pi \cdot 2, \left(0.005208333333333333 \cdot \left(2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot -2\right) \cdot 0.5\right) + 0}}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

   2. associate-*r* 96.6%

      \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\pi \cdot \left(\mathsf{fma}\left(0.0625, \pi \cdot 2, \color{blue}{\left(\left(0.005208333333333333 \cdot 2\right) \cdot \left(\pi \cdot 2\right)\right)} \cdot -2\right) \cdot 0.5\right) + 0}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

   3. metadata-eval 96.6%

      \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\pi \cdot \left(\mathsf{fma}\left(0.0625, \pi \cdot 2, \left(\color{blue}{0.010416666666666666} \cdot \left(\pi \cdot 2\right)\right) \cdot -2\right) \cdot 0.5\right) + 0}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

9. Applied egg-rr 96.6%

   \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\color{blue}{\pi \cdot \left(\mathsf{fma}\left(0.0625, \pi \cdot 2, \left(0.010416666666666666 \cdot \left(\pi \cdot 2\right)\right) \cdot -2\right) \cdot 0.5\right) + 0}}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]
10. Step-by-step derivation

    1. +-rgt-identity 96.6%

       \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\color{blue}{\pi \cdot \left(\mathsf{fma}\left(0.0625, \pi \cdot 2, \left(0.010416666666666666 \cdot \left(\pi \cdot 2\right)\right) \cdot -2\right) \cdot 0.5\right)}}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

    2. *-commutative 96.6%

       \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\color{blue}{\left(\mathsf{fma}\left(0.0625, \pi \cdot 2, \left(0.010416666666666666 \cdot \left(\pi \cdot 2\right)\right) \cdot -2\right) \cdot 0.5\right) \cdot \pi}}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

    3. associate-*l* 96.6%

       \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\color{blue}{\mathsf{fma}\left(0.0625, \pi \cdot 2, \left(0.010416666666666666 \cdot \left(\pi \cdot 2\right)\right) \cdot -2\right) \cdot \left(0.5 \cdot \pi\right)}}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

    4. fma-define 96.6%

       \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\color{blue}{\left(0.0625 \cdot \left(\pi \cdot 2\right) + \left(0.010416666666666666 \cdot \left(\pi \cdot 2\right)\right) \cdot -2\right)} \cdot \left(0.5 \cdot \pi\right)}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

    5. +-commutative 96.6%

       \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\color{blue}{\left(\left(0.010416666666666666 \cdot \left(\pi \cdot 2\right)\right) \cdot -2 + 0.0625 \cdot \left(\pi \cdot 2\right)\right)} \cdot \left(0.5 \cdot \pi\right)}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

    6. associate-*l* 96.6%

       \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\left(\color{blue}{0.010416666666666666 \cdot \left(\left(\pi \cdot 2\right) \cdot -2\right)} + 0.0625 \cdot \left(\pi \cdot 2\right)\right) \cdot \left(0.5 \cdot \pi\right)}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

    7. fma-define 96.6%

       \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\color{blue}{\mathsf{fma}\left(0.010416666666666666, \left(\pi \cdot 2\right) \cdot -2, 0.0625 \cdot \left(\pi \cdot 2\right)\right)} \cdot \left(0.5 \cdot \pi\right)}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

    8. associate-*l* 96.6%

       \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\mathsf{fma}\left(0.010416666666666666, \color{blue}{\pi \cdot \left(2 \cdot -2\right)}, 0.0625 \cdot \left(\pi \cdot 2\right)\right) \cdot \left(0.5 \cdot \pi\right)}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

    9. metadata-eval 96.6%

       \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\mathsf{fma}\left(0.010416666666666666, \pi \cdot \color{blue}{-4}, 0.0625 \cdot \left(\pi \cdot 2\right)\right) \cdot \left(0.5 \cdot \pi\right)}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

    10. *-commutative 96.6%

        \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\mathsf{fma}\left(0.010416666666666666, \pi \cdot -4, \color{blue}{\left(\pi \cdot 2\right) \cdot 0.0625}\right) \cdot \left(0.5 \cdot \pi\right)}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

    11. associate-*l* 96.6%

        \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\mathsf{fma}\left(0.010416666666666666, \pi \cdot -4, \color{blue}{\pi \cdot \left(2 \cdot 0.0625\right)}\right) \cdot \left(0.5 \cdot \pi\right)}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

    12. metadata-eval 96.6%

        \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\mathsf{fma}\left(0.010416666666666666, \pi \cdot -4, \pi \cdot \color{blue}{0.125}\right) \cdot \left(0.5 \cdot \pi\right)}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

    13. *-commutative 96.6%

        \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\mathsf{fma}\left(0.010416666666666666, \pi \cdot -4, \pi \cdot 0.125\right) \cdot \color{blue}{\left(\pi \cdot 0.5\right)}}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

11. Simplified 96.6%

    \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\color{blue}{\mathsf{fma}\left(0.010416666666666666, \pi \cdot -4, \pi \cdot 0.125\right) \cdot \left(\pi \cdot 0.5\right)}}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]
12. Step-by-step derivation

    1. *-commutative 96.6%

       \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\color{blue}{\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(0.010416666666666666, \pi \cdot -4, \pi \cdot 0.125\right)}}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

    2. fma-undefine 96.6%

       \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(0.010416666666666666 \cdot \left(\pi \cdot -4\right) + \pi \cdot 0.125\right)}}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

    3. distribute-lft-in 96.6%

       \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\color{blue}{\left(\pi \cdot 0.5\right) \cdot \left(0.010416666666666666 \cdot \left(\pi \cdot -4\right)\right) + \left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.125\right)}}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

    4. *-commutative 96.6%

       \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\left(\pi \cdot 0.5\right) \cdot \left(0.010416666666666666 \cdot \color{blue}{\left(-4 \cdot \pi\right)}\right) + \left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.125\right)}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

    5. associate-*r* 96.6%

       \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(\left(0.010416666666666666 \cdot -4\right) \cdot \pi\right)} + \left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.125\right)}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

    6. metadata-eval 96.6%

       \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\left(\pi \cdot 0.5\right) \cdot \left(\color{blue}{-0.041666666666666664} \cdot \pi\right) + \left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.125\right)}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

13. Applied egg-rr 96.6%

    \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\color{blue}{\left(\pi \cdot 0.5\right) \cdot \left(-0.041666666666666664 \cdot \pi\right) + \left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.125\right)}}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]
14. Step-by-step derivation

    1. distribute-lft-out 96.6%

       \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\color{blue}{\left(\pi \cdot 0.5\right) \cdot \left(-0.041666666666666664 \cdot \pi + \pi \cdot 0.125\right)}}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

    2. *-commutative 96.6%

       \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\left(\pi \cdot 0.5\right) \cdot \left(\color{blue}{\pi \cdot -0.041666666666666664} + \pi \cdot 0.125\right)}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

    3. distribute-lft-out 96.6%

       \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(\pi \cdot \left(-0.041666666666666664 + 0.125\right)\right)}}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

    4. metadata-eval 96.6%

       \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot \color{blue}{0.08333333333333333}\right)}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

15. Simplified 96.6%

    \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\color{blue}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.08333333333333333\right)}}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]
16. Step-by-step derivation

    1. pow1 96.6%

       \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\color{blue}{{\left(\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.08333333333333333\right)\right)}^{1}}}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

    2. associate-*l* 96.6%

       \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{{\color{blue}{\left(\pi \cdot \left(0.5 \cdot \left(\pi \cdot 0.08333333333333333\right)\right)\right)}}^{1}}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

    3. *-commutative 96.6%

       \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{{\left(\pi \cdot \left(0.5 \cdot \color{blue}{\left(0.08333333333333333 \cdot \pi\right)}\right)\right)}^{1}}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

    4. associate-*r* 96.6%

       \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{{\left(\pi \cdot \color{blue}{\left(\left(0.5 \cdot 0.08333333333333333\right) \cdot \pi\right)}\right)}^{1}}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

    5. metadata-eval 96.6%

       \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{{\left(\pi \cdot \left(\color{blue}{0.041666666666666664} \cdot \pi\right)\right)}^{1}}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

17. Applied egg-rr 96.6%

    \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\color{blue}{{\left(\pi \cdot \left(0.041666666666666664 \cdot \pi\right)\right)}^{1}}}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]
18. Step-by-step derivation

    1. unpow1 96.6%

       \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\color{blue}{\pi \cdot \left(0.041666666666666664 \cdot \pi\right)}}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

    2. *-commutative 96.6%

       \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\pi \cdot \color{blue}{\left(\pi \cdot 0.041666666666666664\right)}}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

19. Simplified 96.6%

    \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\color{blue}{\pi \cdot \left(\pi \cdot 0.041666666666666664\right)}}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]

20. Final simplification 96.6%

    \[\leadsto \mathsf{fma}\left({f}^{2}, -2 \cdot \frac{\pi \cdot \left(\pi \cdot 0.041666666666666664\right)}{\pi}, \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, 0\right)\right) \]
21. Add Preprocessing

Alternative 2: 96.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, 2 \cdot \pi, -2 \cdot \left(0.005208333333333333 \cdot \left(2 \cdot \left(2 \cdot \pi\right)\right)\right)\right), \frac{\frac{4}{\pi}}{f}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (fma
    f
    (fma
     0.0625
     (* 2.0 PI)
     (* -2.0 (* 0.005208333333333333 (* 2.0 (* 2.0 PI)))))
    (/ (/ 4.0 PI) f)))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(fma(f, fma(0.0625, (2.0 * ((double) M_PI)), (-2.0 * (0.005208333333333333 * (2.0 * (2.0 * ((double) M_PI)))))), ((4.0 / ((double) M_PI)) / f))) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(log(fma(f, fma(0.0625, Float64(2.0 * pi), Float64(-2.0 * Float64(0.005208333333333333 * Float64(2.0 * Float64(2.0 * pi))))), Float64(Float64(4.0 / pi) / f))) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

code[f_] := N[(N[Log[N[(f * N[(0.0625 * N[(2.0 * Pi), $MachinePrecision] + N[(-2.0 * N[(0.005208333333333333 * N[(2.0 * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 / Pi), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.1%

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

3. Taylor expanded in f around 0 96.5%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + \left(0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + \left(f \cdot \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right) + 2 \cdot \frac{1}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}\right)\right)\right)} \]

5. Simplified 96.5%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \pi \cdot 2, \left(0.005208333333333333 \cdot \left(2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot -2\right), \frac{\frac{4}{\pi}}{f}\right)\right)} \]

6. Final simplification 96.5%

   \[\leadsto \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, 2 \cdot \pi, -2 \cdot \left(0.005208333333333333 \cdot \left(2 \cdot \left(2 \cdot \pi\right)\right)\right)\right), \frac{\frac{4}{\pi}}{f}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}} \]
7. Add Preprocessing

Alternative 3: 96.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

3. Taylor expanded in f around 0 95.9%

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

   1. associate-/l/ 95.9%

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

   2. distribute-rgt-out-- 95.9%

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

   3. *-commutative 95.9%

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

   4. associate-/r* 95.9%

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

   5. metadata-eval 95.9%

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

   6. metadata-eval 95.9%

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

5. Simplified 95.9%

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

6. Taylor expanded in f around inf 96.1%

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

7. Final simplification 96.1%

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

Alternative 4: 96.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{-4 \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/ (* -4.0 (- (log (/ 2.0 (* PI 0.5))) (log f))) PI))

C

double code(double f) {
	return (-4.0 * (log((2.0 / (((double) M_PI) * 0.5))) - log(f))) / ((double) M_PI);
}

Java

public static double code(double f) {
	return (-4.0 * (Math.log((2.0 / (Math.PI * 0.5))) - Math.log(f))) / Math.PI;
}

Python

def code(f):
	return (-4.0 * (math.log((2.0 / (math.pi * 0.5))) - math.log(f))) / math.pi

Julia

function code(f)
	return Float64(Float64(-4.0 * Float64(log(Float64(2.0 / Float64(pi * 0.5))) - log(f))) / pi)
end

MATLAB

function tmp = code(f)
	tmp = (-4.0 * (log((2.0 / (pi * 0.5))) - log(f))) / pi;
end

Wolfram

code[f_] := N[(N[(-4.0 * N[(N[Log[N[(2.0 / N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 7.1%

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

   1. *-commutative 7.1%

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

   2. distribute-rgt-neg-in 7.1%

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

3. Simplified 7.1%

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

5. Taylor expanded in f around 0 96.1%

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

   1. *-commutative 96.1%

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

   2. associate-*l/ 96.1%

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

   3. mul-1-neg 96.1%

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

   4. unsub-neg 96.1%

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

   5. distribute-rgt-out-- 96.1%

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

   6. metadata-eval 96.1%

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

7. Simplified 96.1%

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

8. Final simplification 96.1%

   \[\leadsto \frac{-4 \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f\right)}{\pi} \]
9. Add Preprocessing

Alternative 5: 96.1% accurate, 4.9× speedup

Math

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

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

   1. *-commutative 7.1%

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

   2. distribute-rgt-neg-in 7.1%

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

3. Simplified 7.1%

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

5. Taylor expanded in f around 0 96.1%

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

   1. *-commutative 96.1%

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

   2. associate-*l/ 96.1%

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

   3. mul-1-neg 96.1%

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

   4. unsub-neg 96.1%

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

   5. distribute-rgt-out-- 96.1%

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

   6. metadata-eval 96.1%

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

7. Simplified 96.1%

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

8. Taylor expanded in f around 0 96.1%

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

   1. div-sub 96.0%

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

   2. remove-double-neg 96.0%

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

   3. mul-1-neg 96.0%

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

   4. log-rec 96.0%

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

   5. div-sub 96.1%

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

   6. div-sub 96.0%

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

   7. log-rec 96.0%

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

   8. mul-1-neg 96.0%

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

   9. remove-double-neg 96.0%

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

   10. div-sub 96.1%

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

10. Simplified 96.1%

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

11. Final simplification 96.1%

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

Alternative 6: 1.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \log 0.5}{-\pi} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (* 4.0 (log 0.5)) (- PI)))

C

double code(double f) {
	return (4.0 * log(0.5)) / -((double) M_PI);
}

Java

public static double code(double f) {
	return (4.0 * Math.log(0.5)) / -Math.PI;
}

Python

def code(f):
	return (4.0 * math.log(0.5)) / -math.pi

Julia

function code(f)
	return Float64(Float64(4.0 * log(0.5)) / Float64(-pi))
end

MATLAB

function tmp = code(f)
	tmp = (4.0 * log(0.5)) / -pi;
end

Wolfram

code[f_] := N[(N[(4.0 * N[Log[0.5], $MachinePrecision]), $MachinePrecision] / (-Pi)), $MachinePrecision]

Derivation

1. Initial program 7.1%

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

4. Applied egg-rr 1.6%

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

5. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log 0.5}{\pi}} \]
6. Step-by-step derivation

   1. associate-*r/ 1.6%

      \[\leadsto -\color{blue}{\frac{4 \cdot \log 0.5}{\pi}} \]

7. Simplified 1.6%

   \[\leadsto -\color{blue}{\frac{4 \cdot \log 0.5}{\pi}} \]

8. Final simplification 1.6%

   \[\leadsto \frac{4 \cdot \log 0.5}{-\pi} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024052 
(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)))))))))