VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 96.8%

Time: 40.1s

Alternatives: 4

Speedup: 4.9×

• could not determine a ground truth

  1. f = 2.00000000000000007e154

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.7% 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: 96.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{-\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{4}{f \cdot \pi}\right)\right)}{\pi \cdot 0.25} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/ (- (log (fma f (* PI 0.08333333333333333) (/ 4.0 (* f PI))))) (* PI 0.25)))

C

double code(double f) {
	return -log(fma(f, (((double) M_PI) * 0.08333333333333333), (4.0 / (f * ((double) M_PI))))) / (((double) M_PI) * 0.25);
}

Julia

function code(f)
	return Float64(Float64(-log(fma(f, Float64(pi * 0.08333333333333333), Float64(4.0 / Float64(f * pi))))) / Float64(pi * 0.25))
end

Wolfram

code[f_] := N[((-N[Log[N[(f * N[(Pi * 0.08333333333333333), $MachinePrecision] + N[(4.0 / N[(f * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.2%

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

3. Taylor expanded in f around 0 96.0%

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

5. Simplified 96.0%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{0.5 \cdot \frac{0.5}{\pi}}, -2, 0.0625 \cdot \left(2 \cdot \pi\right)\right), \frac{\frac{4}{\pi}}{f}\right)\right)} \]
6. Step-by-step derivation

   1. associate-*l/ 96.1%

      \[\leadsto -\color{blue}{\frac{1 \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{0.5 \cdot \frac{0.5}{\pi}}, -2, 0.0625 \cdot \left(2 \cdot \pi\right)\right), \frac{\frac{4}{\pi}}{f}\right)\right)}{\frac{\pi}{4}}} \]

   2. *-un-lft-identity 96.1%

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

   3. associate-*r/ 96.1%

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

   4. metadata-eval 96.1%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{\frac{\color{blue}{0.25}}{\pi}}, -2, 0.0625 \cdot \left(2 \cdot \pi\right)\right), \frac{\frac{4}{\pi}}{f}\right)\right)}{\frac{\pi}{4}} \]

   5. associate-*r* 96.1%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{\frac{0.25}{\pi}}, -2, \color{blue}{\left(0.0625 \cdot 2\right) \cdot \pi}\right), \frac{\frac{4}{\pi}}{f}\right)\right)}{\frac{\pi}{4}} \]

   6. metadata-eval 96.1%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{\frac{0.25}{\pi}}, -2, \color{blue}{0.125} \cdot \pi\right), \frac{\frac{4}{\pi}}{f}\right)\right)}{\frac{\pi}{4}} \]

   7. div-inv 96.1%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{\frac{0.25}{\pi}}, -2, 0.125 \cdot \pi\right), \frac{\frac{4}{\pi}}{f}\right)\right)}{\color{blue}{\pi \cdot \frac{1}{4}}} \]

   8. metadata-eval 96.1%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{\frac{0.25}{\pi}}, -2, 0.125 \cdot \pi\right), \frac{\frac{4}{\pi}}{f}\right)\right)}{\pi \cdot \color{blue}{0.25}} \]

7. Applied egg-rr 96.1%

   \[\leadsto -\color{blue}{\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{\frac{0.25}{\pi}}, -2, 0.125 \cdot \pi\right), \frac{\frac{4}{\pi}}{f}\right)\right)}{\pi \cdot 0.25}} \]
8. Step-by-step derivation

   1. fma-def 96.1%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \color{blue}{\frac{0.005208333333333333}{\frac{0.25}{\pi}} \cdot -2 + 0.125 \cdot \pi}, \frac{\frac{4}{\pi}}{f}\right)\right)}{\pi \cdot 0.25} \]

   2. *-commutative 96.1%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \color{blue}{-2 \cdot \frac{0.005208333333333333}{\frac{0.25}{\pi}}} + 0.125 \cdot \pi, \frac{\frac{4}{\pi}}{f}\right)\right)}{\pi \cdot 0.25} \]

   3. associate-/r/ 96.1%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, -2 \cdot \color{blue}{\left(\frac{0.005208333333333333}{0.25} \cdot \pi\right)} + 0.125 \cdot \pi, \frac{\frac{4}{\pi}}{f}\right)\right)}{\pi \cdot 0.25} \]

   4. metadata-eval 96.1%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, -2 \cdot \left(\color{blue}{0.020833333333333332} \cdot \pi\right) + 0.125 \cdot \pi, \frac{\frac{4}{\pi}}{f}\right)\right)}{\pi \cdot 0.25} \]

   5. associate-*r* 96.1%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \color{blue}{\left(-2 \cdot 0.020833333333333332\right) \cdot \pi} + 0.125 \cdot \pi, \frac{\frac{4}{\pi}}{f}\right)\right)}{\pi \cdot 0.25} \]

   6. metadata-eval 96.1%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \color{blue}{-0.041666666666666664} \cdot \pi + 0.125 \cdot \pi, \frac{\frac{4}{\pi}}{f}\right)\right)}{\pi \cdot 0.25} \]

   7. distribute-rgt-out 96.1%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(-0.041666666666666664 + 0.125\right)}, \frac{\frac{4}{\pi}}{f}\right)\right)}{\pi \cdot 0.25} \]

   8. metadata-eval 96.1%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.08333333333333333}, \frac{\frac{4}{\pi}}{f}\right)\right)}{\pi \cdot 0.25} \]

   9. associate-/l/ 96.1%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \color{blue}{\frac{4}{f \cdot \pi}}\right)\right)}{\pi \cdot 0.25} \]

   10. *-commutative 96.1%

       \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{4}{\color{blue}{\pi \cdot f}}\right)\right)}{\pi \cdot 0.25} \]

9. Simplified 96.1%

   \[\leadsto -\color{blue}{\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{4}{\pi \cdot f}\right)\right)}{\pi \cdot 0.25}} \]

10. Final simplification 96.1%

    \[\leadsto \frac{-\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{4}{f \cdot \pi}\right)\right)}{\pi \cdot 0.25} \]
11. Add Preprocessing

Alternative 2: 96.3% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.2%

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

   1. distribute-lft-neg-in 5.2%

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

   2. *-commutative 5.2%

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

3. Simplified 5.2%

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

5. Taylor expanded in f around 0 95.9%

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

   1. *-commutative 95.9%

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

   2. associate-*l/ 95.9%

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

   3. mul-1-neg 95.9%

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

   4. unsub-neg 95.9%

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

   5. distribute-rgt-out-- 95.9%

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

   6. metadata-eval 95.9%

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

7. Simplified 95.9%

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

8. Final simplification 95.9%

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

Alternative 3: 96.3% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.2%

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

   1. distribute-lft-neg-in 5.2%

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

   2. *-commutative 5.2%

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

3. Simplified 5.2%

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

5. Taylor expanded in f around 0 95.9%

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

   1. *-commutative 95.9%

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

   2. associate-*l/ 95.9%

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

   3. mul-1-neg 95.9%

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

   4. unsub-neg 95.9%

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

   5. distribute-rgt-out-- 95.9%

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

   6. metadata-eval 95.9%

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

7. Simplified 95.9%

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

8. Taylor expanded in f around 0 95.9%

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

   1. div-sub 95.8%

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

   2. remove-double-neg 95.8%

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

   3. mul-1-neg 95.8%

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

   4. log-rec 95.8%

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

   5. div-sub 95.9%

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

   6. div-sub 95.8%

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

   7. log-rec 95.8%

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

   8. mul-1-neg 95.8%

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

   9. remove-double-neg 95.8%

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

   10. metadata-eval 95.8%

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

   11. associate-/r* 95.8%

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

   12. div-sub 95.9%

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

10. Simplified 95.8%

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

11. Final simplification 95.8%

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

Alternative 4: 0.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.2%

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

3. Taylor expanded in f around 0 95.7%

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

4. Taylor expanded in f around inf 0.7%

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

   1. distribute-rgt-out 0.7%

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

   2. distribute-rgt-out-- 0.7%

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

   3. metadata-eval 0.7%

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

   4. *-commutative 0.7%

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

   5. metadata-eval 0.7%

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

   6. mul0-rgt 0.7%

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

6. Simplified 0.7%

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

7. Final simplification 0.7%

   \[\leadsto 4 \cdot \frac{-\log 0}{\pi} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(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)))))))))