VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 96.2%

Time: 31.1s

Alternatives: 9

Speedup: 3.3×

• could not determine a ground truth

  1. f = -3.0087855136929642e+287

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: 96.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(\left(\log \left(\frac{4}{\pi}\right) + 0.020833333333333332 \cdot \left({f}^{2} \cdot {\pi}^{2}\right)\right) - \log f\right) \cdot \frac{-4}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (-
   (+ (log (/ 4.0 PI)) (* 0.020833333333333332 (* (pow f 2.0) (pow PI 2.0))))
   (log f))
  (/ -4.0 PI)))

C

double code(double f) {
	return ((log((4.0 / ((double) M_PI))) + (0.020833333333333332 * (pow(f, 2.0) * pow(((double) M_PI), 2.0)))) - log(f)) * (-4.0 / ((double) M_PI));
}

Java

public static double code(double f) {
	return ((Math.log((4.0 / Math.PI)) + (0.020833333333333332 * (Math.pow(f, 2.0) * Math.pow(Math.PI, 2.0)))) - Math.log(f)) * (-4.0 / Math.PI);
}

Python

def code(f):
	return ((math.log((4.0 / math.pi)) + (0.020833333333333332 * (math.pow(f, 2.0) * math.pow(math.pi, 2.0)))) - math.log(f)) * (-4.0 / math.pi)

Julia

function code(f)
	return Float64(Float64(Float64(log(Float64(4.0 / pi)) + Float64(0.020833333333333332 * Float64((f ^ 2.0) * (pi ^ 2.0)))) - log(f)) * Float64(-4.0 / pi))
end

MATLAB

function tmp = code(f)
	tmp = ((log((4.0 / pi)) + (0.020833333333333332 * ((f ^ 2.0) * (pi ^ 2.0)))) - log(f)) * (-4.0 / pi);
end

Wolfram

code[f_] := N[(N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] + N[(0.020833333333333332 * N[(N[Power[f, 2.0], $MachinePrecision] * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.8%

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

   1. distribute-lft-neg-in 5.8%

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

   2. *-commutative 5.8%

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

3. Simplified 5.8%

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

4. Taylor expanded in f around 0 96.7%

   \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\color{blue}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}}\right) \cdot \frac{-4}{\pi} \]
5. Step-by-step derivation

   1. associate-+r+ 96.7%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\color{blue}{\left(f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)}}\right) \cdot \frac{-4}{\pi} \]

6. Simplified 96.7%

   \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left(f, \pi \cdot 0.5, {\left(f \cdot \pi\right)}^{3} \cdot 0.005208333333333333\right) + \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, {f}^{7} \cdot \left({\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}\right)\right)}}\right) \cdot \frac{-4}{\pi} \]

7. Taylor expanded in f around 0 96.8%

   \[\leadsto \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(-1 \cdot \log f + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right)\right)\right)} \cdot \frac{-4}{\pi} \]
8. Step-by-step derivation

   1. +-commutative 96.8%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \color{blue}{\left(\left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right) + -1 \cdot \log f\right)}\right) \cdot \frac{-4}{\pi} \]

   2. mul-1-neg 96.8%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \left(\left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right) + \color{blue}{\left(-\log f\right)}\right)\right) \cdot \frac{-4}{\pi} \]

   3. unsub-neg 96.8%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \color{blue}{\left(\left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right) - \log f\right)}\right) \cdot \frac{-4}{\pi} \]

9. Simplified 96.8%

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

10. Taylor expanded in f around 0 96.8%

    \[\leadsto \color{blue}{\left(\left(\log \left(\frac{4}{\pi}\right) + 0.020833333333333332 \cdot \left({f}^{2} \cdot {\pi}^{2}\right)\right) - \log f\right)} \cdot \frac{-4}{\pi} \]

11. Final simplification 96.8%

    \[\leadsto \left(\left(\log \left(\frac{4}{\pi}\right) + 0.020833333333333332 \cdot \left({f}^{2} \cdot {\pi}^{2}\right)\right) - \log f\right) \cdot \frac{-4}{\pi} \]

Alternative 2: 96.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}, {f}^{2} \cdot \left(\pi \cdot -0.08333333333333333\right)\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (fma
  -4.0
  (/ (log (/ (/ 4.0 PI) f)) PI)
  (* (pow f 2.0) (* PI -0.08333333333333333))))

C

double code(double f) {
	return fma(-4.0, (log(((4.0 / ((double) M_PI)) / f)) / ((double) M_PI)), (pow(f, 2.0) * (((double) M_PI) * -0.08333333333333333)));
}

Julia

function code(f)
	return fma(-4.0, Float64(log(Float64(Float64(4.0 / pi) / f)) / pi), Float64((f ^ 2.0) * Float64(pi * -0.08333333333333333)))
end

Wolfram

code[f_] := N[(-4.0 * N[(N[Log[N[(N[(4.0 / Pi), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] + N[(N[Power[f, 2.0], $MachinePrecision] * N[(Pi * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.8%

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

   1. distribute-lft-neg-in 5.8%

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

   2. *-commutative 5.8%

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

3. Simplified 5.8%

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

4. Taylor expanded in f around 0 96.7%

   \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\color{blue}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}}\right) \cdot \frac{-4}{\pi} \]
5. Step-by-step derivation

   1. associate-+r+ 96.7%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\color{blue}{\left(f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)}}\right) \cdot \frac{-4}{\pi} \]

6. Simplified 96.7%

   \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left(f, \pi \cdot 0.5, {\left(f \cdot \pi\right)}^{3} \cdot 0.005208333333333333\right) + \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, {f}^{7} \cdot \left({\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}\right)\right)}}\right) \cdot \frac{-4}{\pi} \]

7. Taylor expanded in f around 0 96.8%

   \[\leadsto \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(-1 \cdot \log f + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right)\right)\right)} \cdot \frac{-4}{\pi} \]
8. Step-by-step derivation

   1. +-commutative 96.8%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \color{blue}{\left(\left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right) + -1 \cdot \log f\right)}\right) \cdot \frac{-4}{\pi} \]

   2. mul-1-neg 96.8%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \left(\left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right) + \color{blue}{\left(-\log f\right)}\right)\right) \cdot \frac{-4}{\pi} \]

   3. unsub-neg 96.8%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \color{blue}{\left(\left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right) - \log f\right)}\right) \cdot \frac{-4}{\pi} \]

9. Simplified 96.8%

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

10. Taylor expanded in f around 0 96.8%

    \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} + -0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)} \]
11. Step-by-step derivation

    1. fma-def 96.8%

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

12. Simplified 96.7%

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

13. Final simplification 96.7%

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

Alternative 3: 96.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{-4}{\pi} \cdot \left(\log \left(\frac{\frac{4}{\pi}}{f}\right) + 0.020833333333333332 \cdot {\left(\pi \cdot f\right)}^{2}\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (/ -4.0 PI)
  (+ (log (/ (/ 4.0 PI) f)) (* 0.020833333333333332 (pow (* PI f) 2.0)))))

C

double code(double f) {
	return (-4.0 / ((double) M_PI)) * (log(((4.0 / ((double) M_PI)) / f)) + (0.020833333333333332 * pow((((double) M_PI) * f), 2.0)));
}

Java

public static double code(double f) {
	return (-4.0 / Math.PI) * (Math.log(((4.0 / Math.PI) / f)) + (0.020833333333333332 * Math.pow((Math.PI * f), 2.0)));
}

Python

def code(f):
	return (-4.0 / math.pi) * (math.log(((4.0 / math.pi) / f)) + (0.020833333333333332 * math.pow((math.pi * f), 2.0)))

Julia

function code(f)
	return Float64(Float64(-4.0 / pi) * Float64(log(Float64(Float64(4.0 / pi) / f)) + Float64(0.020833333333333332 * (Float64(pi * f) ^ 2.0))))
end

MATLAB

function tmp = code(f)
	tmp = (-4.0 / pi) * (log(((4.0 / pi) / f)) + (0.020833333333333332 * ((pi * f) ^ 2.0)));
end

Wolfram

code[f_] := N[(N[(-4.0 / Pi), $MachinePrecision] * N[(N[Log[N[(N[(4.0 / Pi), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision] + N[(0.020833333333333332 * N[Power[N[(Pi * f), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.8%

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

   1. distribute-lft-neg-in 5.8%

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

   2. *-commutative 5.8%

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

3. Simplified 5.8%

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

4. Taylor expanded in f around 0 96.7%

   \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\color{blue}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}}\right) \cdot \frac{-4}{\pi} \]
5. Step-by-step derivation

   1. associate-+r+ 96.7%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\color{blue}{\left(f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)}}\right) \cdot \frac{-4}{\pi} \]

6. Simplified 96.7%

   \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left(f, \pi \cdot 0.5, {\left(f \cdot \pi\right)}^{3} \cdot 0.005208333333333333\right) + \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, {f}^{7} \cdot \left({\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}\right)\right)}}\right) \cdot \frac{-4}{\pi} \]

7. Taylor expanded in f around 0 96.8%

   \[\leadsto \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(-1 \cdot \log f + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right)\right)\right)} \cdot \frac{-4}{\pi} \]
8. Step-by-step derivation

   1. +-commutative 96.8%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \color{blue}{\left(\left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right) + -1 \cdot \log f\right)}\right) \cdot \frac{-4}{\pi} \]

   2. mul-1-neg 96.8%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \left(\left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right) + \color{blue}{\left(-\log f\right)}\right)\right) \cdot \frac{-4}{\pi} \]

   3. unsub-neg 96.8%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \color{blue}{\left(\left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right) - \log f\right)}\right) \cdot \frac{-4}{\pi} \]

9. Simplified 96.8%

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

10. Taylor expanded in f around 0 96.8%

    \[\leadsto \color{blue}{\left(\left(\log \left(\frac{4}{\pi}\right) + 0.020833333333333332 \cdot \left({f}^{2} \cdot {\pi}^{2}\right)\right) - \log f\right)} \cdot \frac{-4}{\pi} \]
11. Step-by-step derivation

    1. +-commutative 96.8%

       \[\leadsto \left(\color{blue}{\left(0.020833333333333332 \cdot \left({f}^{2} \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right)} - \log f\right) \cdot \frac{-4}{\pi} \]

    2. associate--l+ 96.8%

       \[\leadsto \color{blue}{\left(0.020833333333333332 \cdot \left({f}^{2} \cdot {\pi}^{2}\right) + \left(\log \left(\frac{4}{\pi}\right) - \log f\right)\right)} \cdot \frac{-4}{\pi} \]

    3. *-commutative 96.8%

       \[\leadsto \left(0.020833333333333332 \cdot \color{blue}{\left({\pi}^{2} \cdot {f}^{2}\right)} + \left(\log \left(\frac{4}{\pi}\right) - \log f\right)\right) \cdot \frac{-4}{\pi} \]

    4. unpow2 96.8%

       \[\leadsto \left(0.020833333333333332 \cdot \left(\color{blue}{\left(\pi \cdot \pi\right)} \cdot {f}^{2}\right) + \left(\log \left(\frac{4}{\pi}\right) - \log f\right)\right) \cdot \frac{-4}{\pi} \]

    5. unpow2 96.8%

       \[\leadsto \left(0.020833333333333332 \cdot \left(\left(\pi \cdot \pi\right) \cdot \color{blue}{\left(f \cdot f\right)}\right) + \left(\log \left(\frac{4}{\pi}\right) - \log f\right)\right) \cdot \frac{-4}{\pi} \]

    6. swap-sqr 96.8%

       \[\leadsto \left(0.020833333333333332 \cdot \color{blue}{\left(\left(\pi \cdot f\right) \cdot \left(\pi \cdot f\right)\right)} + \left(\log \left(\frac{4}{\pi}\right) - \log f\right)\right) \cdot \frac{-4}{\pi} \]

    7. unpow2 96.8%

       \[\leadsto \left(0.020833333333333332 \cdot \color{blue}{{\left(\pi \cdot f\right)}^{2}} + \left(\log \left(\frac{4}{\pi}\right) - \log f\right)\right) \cdot \frac{-4}{\pi} \]

    8. metadata-eval 96.8%

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

    9. associate-/r* 96.8%

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

    10. *-commutative 96.8%

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

    11. log-div 96.8%

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

    12. associate--l- 96.8%

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

    13. log-prod 96.6%

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

    14. associate-*r* 96.6%

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

    15. log-div 96.6%

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

12. Simplified 96.6%

    \[\leadsto \color{blue}{\left(0.020833333333333332 \cdot {\left(\pi \cdot f\right)}^{2} + \log \left(\frac{\frac{4}{\pi}}{f}\right)\right)} \cdot \frac{-4}{\pi} \]

13. Final simplification 96.6%

    \[\leadsto \frac{-4}{\pi} \cdot \left(\log \left(\frac{\frac{4}{\pi}}{f}\right) + 0.020833333333333332 \cdot {\left(\pi \cdot f\right)}^{2}\right) \]

Alternative 4: 95.6% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.8%

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

   1. distribute-lft-neg-in 5.8%

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

   2. *-commutative 5.8%

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

3. Simplified 5.8%

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

4. Taylor expanded in f around 0 96.2%

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

   1. associate-/r* 96.2%

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

   2. distribute-rgt-out-- 96.2%

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

   3. metadata-eval 96.2%

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

6. Simplified 96.2%

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

7. Taylor expanded in f around 0 96.4%

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

   1. mul-1-neg 96.4%

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

   2. unsub-neg 96.4%

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

9. Simplified 96.4%

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

10. Final simplification 96.4%

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

Alternative 5: 95.7% accurate, 2.5× speedup

Math

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

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

Derivation

1. Initial program 5.8%

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

   1. distribute-lft-neg-in 5.8%

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

   2. *-commutative 5.8%

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

3. Simplified 5.8%

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

4. Taylor expanded in f around 0 96.2%

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

   1. associate-/r* 96.2%

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

   2. distribute-rgt-out-- 96.2%

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

   3. metadata-eval 96.2%

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

6. Simplified 96.2%

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

7. Taylor expanded in f around 0 96.5%

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

   1. associate-*r/ 96.5%

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

   2. +-commutative 96.5%

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

   3. mul-1-neg 96.5%

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

   4. log-rec 96.5%

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

   5. log-rec 96.5%

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

   6. mul-1-neg 96.5%

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

   7. +-commutative 96.5%

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

   8. mul-1-neg 96.5%

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

   9. unsub-neg 96.5%

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

9. Simplified 96.5%

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

10. Final simplification 96.5%

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

Alternative 6: 95.7% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.8%

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

   1. distribute-lft-neg-in 5.8%

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

   2. *-commutative 5.8%

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

3. Simplified 5.8%

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

4. Taylor expanded in f around 0 96.2%

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

5. Taylor expanded in f around 0 96.5%

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

7. Simplified 96.4%

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

8. Final simplification 96.4%

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

Alternative 7: 95.7% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.8%

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

   1. distribute-lft-neg-in 5.8%

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

   2. *-commutative 5.8%

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

3. Simplified 5.8%

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

4. Taylor expanded in f around 0 96.7%

   \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\color{blue}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}}\right) \cdot \frac{-4}{\pi} \]
5. Step-by-step derivation

   1. associate-+r+ 96.7%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\color{blue}{\left(f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)}}\right) \cdot \frac{-4}{\pi} \]

6. Simplified 96.7%

   \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left(f, \pi \cdot 0.5, {\left(f \cdot \pi\right)}^{3} \cdot 0.005208333333333333\right) + \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, {f}^{7} \cdot \left({\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}\right)\right)}}\right) \cdot \frac{-4}{\pi} \]

7. Taylor expanded in f around 0 96.8%

   \[\leadsto \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(-1 \cdot \log f + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right)\right)\right)} \cdot \frac{-4}{\pi} \]
8. Step-by-step derivation

   1. +-commutative 96.8%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \color{blue}{\left(\left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right) + -1 \cdot \log f\right)}\right) \cdot \frac{-4}{\pi} \]

   2. mul-1-neg 96.8%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \left(\left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right) + \color{blue}{\left(-\log f\right)}\right)\right) \cdot \frac{-4}{\pi} \]

   3. unsub-neg 96.8%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \color{blue}{\left(\left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right) - \log f\right)}\right) \cdot \frac{-4}{\pi} \]

9. Simplified 96.8%

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

10. Taylor expanded in f around 0 96.5%

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

    1. sub-neg 96.5%

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

    2. log-rec 96.5%

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

    3. +-commutative 96.5%

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

    4. metadata-eval 96.5%

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

    5. associate-/r* 96.5%

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

    6. *-commutative 96.5%

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

    7. log-div 96.5%

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

    8. associate--l+ 96.5%

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

    9. +-commutative 96.5%

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

    10. log-rec 96.5%

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

    11. mul-1-neg 96.5%

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

    12. *-commutative 96.5%

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

12. Simplified 96.4%

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

13. Final simplification 96.4%

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

Alternative 8: 4.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ -0.08333333333333333 \cdot \left(\pi \cdot {f}^{2}\right) \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* -0.08333333333333333 (* PI (pow f 2.0))))

C

double code(double f) {
	return -0.08333333333333333 * (((double) M_PI) * pow(f, 2.0));
}

Java

public static double code(double f) {
	return -0.08333333333333333 * (Math.PI * Math.pow(f, 2.0));
}

Python

def code(f):
	return -0.08333333333333333 * (math.pi * math.pow(f, 2.0))

Julia

function code(f)
	return Float64(-0.08333333333333333 * Float64(pi * (f ^ 2.0)))
end

MATLAB

function tmp = code(f)
	tmp = -0.08333333333333333 * (pi * (f ^ 2.0));
end

Wolfram

code[f_] := N[(-0.08333333333333333 * N[(Pi * N[Power[f, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.8%

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

   1. distribute-lft-neg-in 5.8%

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

   2. *-commutative 5.8%

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

3. Simplified 5.8%

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

4. Taylor expanded in f around 0 96.7%

   \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\color{blue}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}}\right) \cdot \frac{-4}{\pi} \]
5. Step-by-step derivation

   1. associate-+r+ 96.7%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\color{blue}{\left(f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)}}\right) \cdot \frac{-4}{\pi} \]

6. Simplified 96.7%

   \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left(f, \pi \cdot 0.5, {\left(f \cdot \pi\right)}^{3} \cdot 0.005208333333333333\right) + \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, {f}^{7} \cdot \left({\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}\right)\right)}}\right) \cdot \frac{-4}{\pi} \]

7. Taylor expanded in f around 0 96.8%

   \[\leadsto \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(-1 \cdot \log f + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right)\right)\right)} \cdot \frac{-4}{\pi} \]
8. Step-by-step derivation

   1. +-commutative 96.8%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \color{blue}{\left(\left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right) + -1 \cdot \log f\right)}\right) \cdot \frac{-4}{\pi} \]

   2. mul-1-neg 96.8%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \left(\left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right) + \color{blue}{\left(-\log f\right)}\right)\right) \cdot \frac{-4}{\pi} \]

   3. unsub-neg 96.8%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \color{blue}{\left(\left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right) - \log f\right)}\right) \cdot \frac{-4}{\pi} \]

9. Simplified 96.8%

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

10. Taylor expanded in f around inf 4.2%

    \[\leadsto \color{blue}{-0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)} \]

11. Final simplification 4.2%

    \[\leadsto -0.08333333333333333 \cdot \left(\pi \cdot {f}^{2}\right) \]

Alternative 9: 0.7% accurate, 5.0× speedup

Math

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

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(Float64(-4.0 * log(0.0)) / pi)
end

MATLAB

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

Wolfram

code[f_] := N[(N[(-4.0 * N[Log[0.0], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 5.8%

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

   1. distribute-lft-neg-in 5.8%

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

   2. *-commutative 5.8%

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

3. Simplified 5.8%

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

4. Taylor expanded in f around 0 96.2%

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

5. Taylor expanded in f around inf 0.7%

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

   1. associate-*r/ 0.7%

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

   2. distribute-rgt-out 0.7%

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

   3. distribute-rgt-out-- 0.7%

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

   4. metadata-eval 0.7%

      \[\leadsto \frac{-4 \cdot \log \left(\frac{\pi}{\pi \cdot \color{blue}{0.5}} \cdot \left(-0.25 + 0.25\right)\right)}{\pi} \]

   5. metadata-eval 0.7%

      \[\leadsto \frac{-4 \cdot \log \left(\frac{\pi}{\pi \cdot 0.5} \cdot \color{blue}{0}\right)}{\pi} \]

   6. mul0-rgt 0.7%

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

7. Simplified 0.7%

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

8. Final simplification 0.7%

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

Reproduce

herbie shell --seed 2023318 
(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)))))))))