VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.5% → 96.3%

Time: 29.0s

Alternatives: 5

Speedup: 3.3×

• could not determine a ground truth

  1. f = -1.9511338864540421e+65

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: 6.5% 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: 96.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{4}{\pi \cdot f}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log (fma f (* PI 0.08333333333333333) (/ 4.0 (* PI f))))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(fma(f, (((double) M_PI) * 0.08333333333333333), (4.0 / (((double) M_PI) * f)))) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(log(fma(f, Float64(pi * 0.08333333333333333), Float64(4.0 / Float64(pi * f)))) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

code[f_] := N[(N[Log[N[(f * N[(Pi * 0.08333333333333333), $MachinePrecision] + N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.1%

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

2. Taylor expanded in f around 0 96.0%

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

4. Simplified 96.0%

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

5. Taylor expanded in f around 0 96.0%

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

   1. *-commutative 96.0%

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

7. Simplified 96.0%

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

   1. fma-udef 96.0%

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

   2. +-commutative 96.0%

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

   3. associate-*l/ 96.0%

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

   4. associate-*r/ 96.0%

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

   5. associate-/r/ 96.0%

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

   6. metadata-eval 96.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \frac{\color{blue}{-0.010416666666666666}}{0.5 \cdot 0.5} \cdot \pi + 0.0625 \cdot \left(2 \cdot \pi\right), \frac{4}{\pi \cdot f}\right)\right) \]

   7. metadata-eval 96.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \frac{-0.010416666666666666}{\color{blue}{0.25}} \cdot \pi + 0.0625 \cdot \left(2 \cdot \pi\right), \frac{4}{\pi \cdot f}\right)\right) \]

   8. metadata-eval 96.0%

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

   9. associate-*r* 96.0%

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

   10. metadata-eval 96.0%

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

9. Applied egg-rr 96.0%

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

    1. distribute-rgt-out 96.0%

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

    2. metadata-eval 96.0%

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

11. Simplified 96.0%

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

12. Final simplification 96.0%

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

Alternative 2: 96.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.1%

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

2. Taylor expanded in f around 0 95.7%

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

   1. mul-1-neg 95.7%

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

   2. unsub-neg 95.7%

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

   3. distribute-rgt-out-- 95.7%

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

   4. metadata-eval 95.7%

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

4. Simplified 95.7%

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

5. Final simplification 95.7%

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

Alternative 3: 1.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double f) {
	return (Math.log((f * (Math.PI * 0.08333333333333333))) / Math.PI) * -4.0;
}

Python

def code(f):
	return (math.log((f * (math.pi * 0.08333333333333333))) / math.pi) * -4.0

Julia

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

MATLAB

function tmp = code(f)
	tmp = (log((f * (pi * 0.08333333333333333))) / pi) * -4.0;
end

Wolfram

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

Derivation

1. Initial program 6.1%

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

2. Taylor expanded in f around 0 96.0%

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

4. Simplified 96.0%

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

5. Taylor expanded in f around 0 96.0%

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

   1. *-commutative 96.0%

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

7. Simplified 96.0%

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

8. Taylor expanded in f around inf 1.7%

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

   1. mul-1-neg 1.7%

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

   2. log-rec 1.7%

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

   3. remove-double-neg 1.7%

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

   4. log-prod 1.7%

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

   5. distribute-rgt-out 1.7%

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

   6. metadata-eval 1.7%

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

   7. *-commutative 1.7%

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

   8. *-lft-identity 1.7%

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

   9. *-lft-identity 1.7%

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

10. Simplified 1.7%

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

11. Final simplification 1.7%

    \[\leadsto \frac{\log \left(f \cdot \left(\pi \cdot 0.08333333333333333\right)\right)}{\pi} \cdot \left(-4\right) \]

Alternative 4: 95.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.1%

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

2. Taylor expanded in f around 0 95.6%

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

   1. distribute-rgt-out-- 95.6%

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

   2. metadata-eval 95.6%

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

4. Simplified 95.6%

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

5. Taylor expanded in f around 0 95.7%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(0.125 \cdot \left(f \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)} \]
6. Step-by-step derivation

   1. add-exp-log 95.6%

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

   2. rec-exp 95.6%

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

   3. *-commutative 95.6%

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

7. Applied egg-rr 95.6%

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

8. Taylor expanded in f around 0 95.6%

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

   1. associate-*r/ 95.6%

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

   2. associate-/l* 95.5%

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

   3. exp-neg 95.5%

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

   4. +-commutative 95.5%

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

   5. log-prod 95.6%

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

   6. rem-exp-log 95.6%

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

   7. associate-*r/ 95.6%

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

   8. metadata-eval 95.6%

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

10. Simplified 95.6%

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

11. Final simplification 95.6%

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

Alternative 5: 96.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.1%

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

2. Taylor expanded in f around inf 6.1%

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

3. Taylor expanded in f around 0 95.7%

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

   1. associate-+r+ 95.7%

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

   2. +-commutative 95.7%

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

   3. associate-*r/ 95.7%

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

   4. metadata-eval 95.7%

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

   5. distribute-rgt-out-- 95.7%

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

   6. metadata-eval 95.7%

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

   7. associate-*r* 95.7%

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

   8. *-commutative 95.7%

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

   9. associate-/r* 95.7%

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

   10. metadata-eval 95.7%

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

   11. associate-/l/ 95.7%

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

5. Simplified 95.7%

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

6. Final simplification 95.7%

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

Reproduce

herbie shell --seed 2023301 
(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)))))))))