VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.6% → 96.4%

Time: 55.9s

Alternatives: 5

Speedup: 3.3×

• could not determine a ground truth

  1. f = -2.281083592197991e+145

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: 6.6% 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: 96.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\pi}{4}} \cdot \left(\log f - \mathsf{fma}\left(\pi \cdot \left(\pi \cdot 0.041666666666666664\right), \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{4}{\pi}\right)\right)\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (/ 1.0 (/ PI 4.0))
  (-
   (log f)
   (fma (* PI (* PI 0.041666666666666664)) (* (* f f) 0.5) (log (/ 4.0 PI))))))

C

double code(double f) {
	return (1.0 / (((double) M_PI) / 4.0)) * (log(f) - fma((((double) M_PI) * (((double) M_PI) * 0.041666666666666664)), ((f * f) * 0.5), log((4.0 / ((double) M_PI)))));
}

Julia

function code(f)
	return Float64(Float64(1.0 / Float64(pi / 4.0)) * Float64(log(f) - fma(Float64(pi * Float64(pi * 0.041666666666666664)), Float64(Float64(f * f) * 0.5), log(Float64(4.0 / pi)))))
end

Wolfram

code[f_] := N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[(N[Log[f], $MachinePrecision] - N[(N[(Pi * N[(Pi * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] * N[(N[(f * f), $MachinePrecision] * 0.5), $MachinePrecision] + N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.9%

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

2. Taylor expanded in f around 0 96.2%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(0.5 \cdot \left(\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right)\right) \cdot {f}^{2}\right) + \left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + \left(0.5 \cdot \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)\right) + -1 \cdot \log f\right)\right)\right)} \]

4. Simplified 96.2%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, 1 \cdot \frac{\pi}{0.5}, \left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332\right) \cdot -2\right), 0\right), \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right)} \]
5. Step-by-step derivation

   1. fma-udef 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(0.0625, 1 \cdot \frac{\pi}{0.5}, \left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332\right) \cdot -2\right) + 0}, \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

   2. *-un-lft-identity 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(0.0625, \color{blue}{\frac{\pi}{0.5}}, \left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332\right) \cdot -2\right) + 0, \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

   3. div-inv 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(0.0625, \color{blue}{\pi \cdot \frac{1}{0.5}}, \left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332\right) \cdot -2\right) + 0, \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

   4. metadata-eval 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(0.0625, \pi \cdot \color{blue}{2}, \left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332\right) \cdot -2\right) + 0, \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

   5. pow-div 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(0.0625, \pi \cdot 2, \left(\color{blue}{{\pi}^{\left(3 - 2\right)}} \cdot 0.020833333333333332\right) \cdot -2\right) + 0, \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

   6. metadata-eval 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(0.0625, \pi \cdot 2, \left({\pi}^{\color{blue}{1}} \cdot 0.020833333333333332\right) \cdot -2\right) + 0, \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

   7. pow1 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(0.0625, \pi \cdot 2, \left(\color{blue}{\pi} \cdot 0.020833333333333332\right) \cdot -2\right) + 0, \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

   8. associate-*l* 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(0.0625, \pi \cdot 2, \color{blue}{\pi \cdot \left(0.020833333333333332 \cdot -2\right)}\right) + 0, \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

   9. metadata-eval 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(0.0625, \pi \cdot 2, \pi \cdot \color{blue}{-0.041666666666666664}\right) + 0, \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

6. Applied egg-rr 96.2%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(0.0625, \pi \cdot 2, \pi \cdot -0.041666666666666664\right) + 0}, \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]
7. Step-by-step derivation

   1. +-rgt-identity 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(0.0625, \pi \cdot 2, \pi \cdot -0.041666666666666664\right)}, \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

   2. fma-def 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(0.0625 \cdot \left(\pi \cdot 2\right) + \pi \cdot -0.041666666666666664\right)}, \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

   3. +-commutative 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(\pi \cdot -0.041666666666666664 + 0.0625 \cdot \left(\pi \cdot 2\right)\right)}, \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

   4. fma-def 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\left(\pi \cdot 0.5\right) \cdot \color{blue}{\mathsf{fma}\left(\pi, -0.041666666666666664, 0.0625 \cdot \left(\pi \cdot 2\right)\right)}, \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

   5. *-commutative 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(\pi, -0.041666666666666664, \color{blue}{\left(\pi \cdot 2\right) \cdot 0.0625}\right), \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

   6. associate-*l* 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(\pi, -0.041666666666666664, \color{blue}{\pi \cdot \left(2 \cdot 0.0625\right)}\right), \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

   7. metadata-eval 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(\pi, -0.041666666666666664, \pi \cdot \color{blue}{0.125}\right), \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

8. Simplified 96.2%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(\pi, -0.041666666666666664, \pi \cdot 0.125\right)}, \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]
9. Step-by-step derivation

   1. pow1 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\color{blue}{{\left(\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(\pi, -0.041666666666666664, \pi \cdot 0.125\right)\right)}^{1}}, \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

   2. associate-*l* 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left({\color{blue}{\left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\pi, -0.041666666666666664, \pi \cdot 0.125\right)\right)\right)}}^{1}, \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

10. Applied egg-rr 96.2%

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\color{blue}{{\left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\pi, -0.041666666666666664, \pi \cdot 0.125\right)\right)\right)}^{1}}, \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]
11. Step-by-step derivation

    1. unpow1 96.2%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\color{blue}{\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\pi, -0.041666666666666664, \pi \cdot 0.125\right)\right)}, \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

    2. *-commutative 96.2%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\pi \cdot \color{blue}{\left(\mathsf{fma}\left(\pi, -0.041666666666666664, \pi \cdot 0.125\right) \cdot 0.5\right)}, \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

    3. fma-def 96.2%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\pi \cdot \left(\color{blue}{\left(\pi \cdot -0.041666666666666664 + \pi \cdot 0.125\right)} \cdot 0.5\right), \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

    4. distribute-lft-out 96.2%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\pi \cdot \left(\color{blue}{\left(\pi \cdot \left(-0.041666666666666664 + 0.125\right)\right)} \cdot 0.5\right), \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

    5. associate-*l* 96.2%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\pi \cdot \color{blue}{\left(\pi \cdot \left(\left(-0.041666666666666664 + 0.125\right) \cdot 0.5\right)\right)}, \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

    6. metadata-eval 96.2%

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

    7. metadata-eval 96.2%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\pi \cdot \left(\pi \cdot \color{blue}{0.041666666666666664}\right), \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

12. Simplified 96.2%

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\color{blue}{\pi \cdot \left(\pi \cdot 0.041666666666666664\right)}, \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]
13. Step-by-step derivation

    1. log-div 96.2%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\pi \cdot \left(\pi \cdot 0.041666666666666664\right), \left(f \cdot f\right) \cdot 0.5, \color{blue}{\log \left(\frac{2}{\pi}\right) - \log 0.5}\right) - \log f\right) \]

14. Applied egg-rr 96.2%

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\pi \cdot \left(\pi \cdot 0.041666666666666664\right), \left(f \cdot f\right) \cdot 0.5, \color{blue}{\log \left(\frac{2}{\pi}\right) - \log 0.5}\right) - \log f\right) \]
15. Step-by-step derivation

    1. log-div 96.2%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\pi \cdot \left(\pi \cdot 0.041666666666666664\right), \left(f \cdot f\right) \cdot 0.5, \color{blue}{\log \left(\frac{\frac{2}{\pi}}{0.5}\right)}\right) - \log f\right) \]

    2. associate-/r* 96.2%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\pi \cdot \left(\pi \cdot 0.041666666666666664\right), \left(f \cdot f\right) \cdot 0.5, \log \color{blue}{\left(\frac{2}{\pi \cdot 0.5}\right)}\right) - \log f\right) \]

    3. *-commutative 96.2%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\pi \cdot \left(\pi \cdot 0.041666666666666664\right), \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{2}{\color{blue}{0.5 \cdot \pi}}\right)\right) - \log f\right) \]

    4. associate-/r* 96.2%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\pi \cdot \left(\pi \cdot 0.041666666666666664\right), \left(f \cdot f\right) \cdot 0.5, \log \color{blue}{\left(\frac{\frac{2}{0.5}}{\pi}\right)}\right) - \log f\right) \]

    5. metadata-eval 96.2%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\pi \cdot \left(\pi \cdot 0.041666666666666664\right), \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{\color{blue}{4}}{\pi}\right)\right) - \log f\right) \]

16. Simplified 96.2%

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left(\pi \cdot \left(\pi \cdot 0.041666666666666664\right), \left(f \cdot f\right) \cdot 0.5, \color{blue}{\log \left(\frac{4}{\pi}\right)}\right) - \log f\right) \]

17. Final simplification 96.2%

    \[\leadsto \frac{1}{\frac{\pi}{4}} \cdot \left(\log f - \mathsf{fma}\left(\pi \cdot \left(\pi \cdot 0.041666666666666664\right), \left(f \cdot f\right) \cdot 0.5, \log \left(\frac{4}{\pi}\right)\right)\right) \]

Alternative 2: 96.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \left(\pi \cdot 2\right) + \pi \cdot -0.041666666666666664, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (fma
    f
    (+ (* 0.0625 (* PI 2.0)) (* PI -0.041666666666666664))
    (/ (/ (/ 2.0 PI) 0.5) f)))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(fma(f, ((0.0625 * (((double) M_PI) * 2.0)) + (((double) M_PI) * -0.041666666666666664)), (((2.0 / ((double) M_PI)) / 0.5) / f))) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(log(fma(f, Float64(Float64(0.0625 * Float64(pi * 2.0)) + Float64(pi * -0.041666666666666664)), Float64(Float64(Float64(2.0 / pi) / 0.5) / f))) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

code[f_] := N[(N[Log[N[(f * N[(N[(0.0625 * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision] + N[(Pi * -0.041666666666666664), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(2.0 / Pi), $MachinePrecision] / 0.5), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.9%

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

2. Taylor expanded in f around 0 95.8%

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

4. Simplified 95.8%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, 1 \cdot \frac{\pi}{0.5}, \left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332\right) \cdot -2\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right)} \]
5. Step-by-step derivation

   1. *-un-lft-identity 95.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \color{blue}{\frac{\pi}{0.5}}, \left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332\right) \cdot -2\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   2. fma-udef 95.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{0.0625 \cdot \frac{\pi}{0.5} + \left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332\right) \cdot -2}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   3. div-inv 95.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \color{blue}{\left(\pi \cdot \frac{1}{0.5}\right)} + \left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332\right) \cdot -2, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   4. metadata-eval 95.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \left(\pi \cdot \color{blue}{2}\right) + \left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332\right) \cdot -2, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   5. pow-div 95.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \left(\pi \cdot 2\right) + \left(\color{blue}{{\pi}^{\left(3 - 2\right)}} \cdot 0.020833333333333332\right) \cdot -2, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   6. metadata-eval 95.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \left(\pi \cdot 2\right) + \left({\pi}^{\color{blue}{1}} \cdot 0.020833333333333332\right) \cdot -2, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   7. pow1 95.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \left(\pi \cdot 2\right) + \left(\color{blue}{\pi} \cdot 0.020833333333333332\right) \cdot -2, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   8. associate-*l* 95.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \left(\pi \cdot 2\right) + \color{blue}{\pi \cdot \left(0.020833333333333332 \cdot -2\right)}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   9. metadata-eval 95.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \left(\pi \cdot 2\right) + \pi \cdot \color{blue}{-0.041666666666666664}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

6. Applied egg-rr 95.8%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{0.0625 \cdot \left(\pi \cdot 2\right) + \pi \cdot -0.041666666666666664}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

7. Final simplification 95.8%

   \[\leadsto \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \left(\pi \cdot 2\right) + \pi \cdot -0.041666666666666664, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}} \]

Alternative 3: 96.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.9%

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

   1. *-commutative 7.9%

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

   2. distribute-rgt-neg-in 7.9%

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

   3. associate-/r/ 7.9%

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

   4. associate-*l/ 7.9%

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

   5. metadata-eval 7.9%

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

   6. distribute-neg-frac 7.9%

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

3. Simplified 7.9%

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

4. Taylor expanded in f around 0 95.6%

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

   1. associate-*r/ 95.6%

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

   2. associate-/l* 95.4%

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

   3. mul-1-neg 95.4%

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

   4. unsub-neg 95.4%

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

   5. distribute-rgt-out-- 95.4%

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

   6. metadata-eval 95.4%

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

   7. associate-/r* 95.4%

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

6. Simplified 95.4%

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

7. Taylor expanded in f around 0 95.6%

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

8. Final simplification 95.6%

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

Alternative 4: 95.8% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.9%

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

   1. *-commutative 7.9%

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

   2. distribute-rgt-neg-in 7.9%

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

   3. associate-/r/ 7.9%

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

   4. associate-*l/ 7.9%

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

   5. metadata-eval 7.9%

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

   6. distribute-neg-frac 7.9%

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

3. Simplified 7.9%

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

4. Taylor expanded in f around 0 95.6%

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

   1. associate-*r/ 95.6%

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

   2. associate-/l* 95.4%

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

   3. mul-1-neg 95.4%

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

   4. unsub-neg 95.4%

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

   5. distribute-rgt-out-- 95.4%

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

   6. metadata-eval 95.4%

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

   7. associate-/r* 95.4%

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

6. Simplified 95.4%

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

7. Taylor expanded in f around 0 95.6%

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

   1. associate-*r/ 95.6%

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

   2. *-commutative 95.6%

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

   3. metadata-eval 95.6%

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

   4. associate-/r* 95.6%

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

   5. *-commutative 95.6%

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

   6. log-div 95.1%

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

   7. associate-/r* 95.1%

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

   8. associate-*r/ 95.0%

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

   9. *-commutative 95.0%

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

   10. associate-/r* 95.0%

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

   11. *-commutative 95.0%

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

   12. associate-/r* 95.0%

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

   13. metadata-eval 95.0%

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

   14. associate-/r* 95.0%

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

   15. *-commutative 95.0%

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

9. Simplified 95.0%

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

10. Taylor expanded in f around 0 95.6%

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

    1. mul-1-neg 95.6%

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

    2. log-rec 95.2%

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

    3. +-commutative 95.2%

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

    4. log-rec 95.6%

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

    5. unsub-neg 95.6%

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

    6. log-div 95.1%

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

    7. associate-/r* 95.1%

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

    8. associate-/l/ 95.1%

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

12. Simplified 95.1%

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

13. Final simplification 95.1%

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

Alternative 5: 1.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ -\frac{4 \cdot \log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (- (/ (* 4.0 (log 7.62939453125e-6)) PI)))

C

double code(double f) {
	return -((4.0 * log(7.62939453125e-6)) / ((double) M_PI));
}

Java

public static double code(double f) {
	return -((4.0 * Math.log(7.62939453125e-6)) / Math.PI);
}

Python

def code(f):
	return -((4.0 * math.log(7.62939453125e-6)) / math.pi)

Julia

function code(f)
	return Float64(-Float64(Float64(4.0 * log(7.62939453125e-6)) / pi))
end

MATLAB

function tmp = code(f)
	tmp = -((4.0 * log(7.62939453125e-6)) / pi);
end

Wolfram

code[f_] := (-N[(N[(4.0 * N[Log[7.62939453125e-6], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision])

Derivation

1. Initial program 7.9%

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

3. Applied egg-rr 1.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{262144}}\right) \]

4. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi}} \]
5. Step-by-step derivation

   1. associate-*r/ 1.6%

      \[\leadsto -\color{blue}{\frac{4 \cdot \log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi}} \]

6. Simplified 1.6%

   \[\leadsto -\color{blue}{\frac{4 \cdot \log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi}} \]

7. Final simplification 1.6%

   \[\leadsto -\frac{4 \cdot \log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi} \]

Reproduce

herbie shell --seed 2023279 
(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)))))))))