VandenBroeck and Keller, Equation (20)

Average Accuracy: 4.0% → 98.4%

Time: 34.0s

Precision: binary64

Cost: 39428

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 20:\\ \;\;\;\;\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{4}{\pi \cdot f}\right)\right)}{\pi} \cdot \left(-4\right)\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \frac{-1}{\frac{\pi}{4}}\\ \end{array} \]

FPCore

(FPCore (f)
 :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)))))))))

↓

(FPCore (f)
 :precision binary64
 (if (<= (* (/ PI 4.0) f) 20.0)
   (* (/ (log (fma f (* PI 0.08333333333333333) (/ 4.0 (* PI f)))) PI) (- 4.0))
   (* 0.0 (/ -1.0 (/ PI 4.0)))))

C

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

↓

double code(double f) {
	double tmp;
	if (((((double) M_PI) / 4.0) * f) <= 20.0) {
		tmp = (log(fma(f, (((double) M_PI) * 0.08333333333333333), (4.0 / (((double) M_PI) * f)))) / ((double) M_PI)) * -4.0;
	} else {
		tmp = 0.0 * (-1.0 / (((double) M_PI) / 4.0));
	}
	return tmp;
}

Julia

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

↓

function code(f)
	tmp = 0.0
	if (Float64(Float64(pi / 4.0) * f) <= 20.0)
		tmp = Float64(Float64(log(fma(f, Float64(pi * 0.08333333333333333), Float64(4.0 / Float64(pi * f)))) / pi) * Float64(-4.0));
	else
		tmp = Float64(0.0 * Float64(-1.0 / Float64(pi / 4.0)));
	end
	return tmp
end

Wolfram

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

↓

code[f_] := If[LessEqual[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision], 20.0], N[(N[(N[Log[N[(f * N[(Pi * 0.08333333333333333), $MachinePrecision] + N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * (-4.0)), $MachinePrecision], N[(0.0 * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (PI.f64) 4) f) < 20

   1. Initial program 2.3%

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

   2. Taylor expanded in f around 0 98.6%

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

   3. Simplified 98.6%

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

      [Start] 98.6	\[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + \left(2 \cdot \frac{1}{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f} + \left(0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + f \cdot \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right)\right)\right)\right) \]
      associate-+r+ [=>] 98.6	\[ -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 2 \cdot \frac{1}{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f}\right) + \left(0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + f \cdot \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right)\right)\right)} \]
      +-commutative [=>] 98.6	\[ -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\left(0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + f \cdot \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right)\right) + \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 2 \cdot \frac{1}{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f}\right)\right)} \]

   4. Applied egg-rr 97.4%

      \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{4}{\pi} \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \pi \cdot 2, \left(\pi \cdot 0.010416666666666666\right) \cdot -4\right), \frac{4}{\pi \cdot f}\right)\right)\right)} - 1\right)} \]
      Proof

      [Start] 98.6	\[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \frac{\pi}{0.5}, \frac{0.005208333333333333 \cdot \frac{\pi}{0.5}}{0.5} \cdot -2\right), \frac{\frac{4}{\pi}}{f}\right)\right) \]
      expm1-log1p-u [=>] 97.4	\[ -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \frac{\pi}{0.5}, \frac{0.005208333333333333 \cdot \frac{\pi}{0.5}}{0.5} \cdot -2\right), \frac{\frac{4}{\pi}}{f}\right)\right)\right)\right)} \]
      expm1-udef [=>] 97.4	\[ -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \frac{\pi}{0.5}, \frac{0.005208333333333333 \cdot \frac{\pi}{0.5}}{0.5} \cdot -2\right), \frac{\frac{4}{\pi}}{f}\right)\right)\right)} - 1\right)} \]

   5. Simplified 98.7%

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

      [Start] 97.4	\[ -\left(e^{\mathsf{log1p}\left(\frac{4}{\pi} \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \pi \cdot 2, \left(\pi \cdot 0.010416666666666666\right) \cdot -4\right), \frac{4}{\pi \cdot f}\right)\right)\right)} - 1\right) \]
      expm1-def [=>] 97.4	\[ -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{4}{\pi} \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \pi \cdot 2, \left(\pi \cdot 0.010416666666666666\right) \cdot -4\right), \frac{4}{\pi \cdot f}\right)\right)\right)\right)} \]
      expm1-log1p [=>] 98.6	\[ -\color{blue}{\frac{4}{\pi} \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \pi \cdot 2, \left(\pi \cdot 0.010416666666666666\right) \cdot -4\right), \frac{4}{\pi \cdot f}\right)\right)} \]
      associate-*l/ [=>] 98.7	\[ -\color{blue}{\frac{4 \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \pi \cdot 2, \left(\pi \cdot 0.010416666666666666\right) \cdot -4\right), \frac{4}{\pi \cdot f}\right)\right)}{\pi}} \]
      *-commutative [=>] 98.7	\[ -\frac{\color{blue}{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \pi \cdot 2, \left(\pi \cdot 0.010416666666666666\right) \cdot -4\right), \frac{4}{\pi \cdot f}\right)\right) \cdot 4}}{\pi} \]
      associate-*l/ [<=] 98.7	\[ -\color{blue}{\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \pi \cdot 2, \left(\pi \cdot 0.010416666666666666\right) \cdot -4\right), \frac{4}{\pi \cdot f}\right)\right)}{\pi} \cdot 4} \]
      *-commutative [=>] 98.7	\[ -\color{blue}{4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \pi \cdot 2, \left(\pi \cdot 0.010416666666666666\right) \cdot -4\right), \frac{4}{\pi \cdot f}\right)\right)}{\pi}} \]

   if 20 < (*.f64 (/.f64 (PI.f64) 4) f)

   1. Initial program 16.7%

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

   2. Applied egg-rr 0.5%

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

      No proof available- proof too large to flatten.

   3. Applied egg-rr 100.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{0} \]
      Proof

      No proof available- proof too large to flatten.
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 20:\\ \;\;\;\;\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{4}{\pi \cdot f}\right)\right)}{\pi} \cdot \left(-4\right)\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \frac{-1}{\frac{\pi}{4}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.7%
Cost	19780

\[\begin{array}{l} \mathbf{if}\;f \leq 1.3:\\ \;\;\;\;\frac{4}{\pi} \cdot \log \left(\pi \cdot \frac{f}{4}\right)\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \frac{-1}{\frac{\pi}{4}}\\ \end{array} \]

Alternative 2
Accuracy	97.8%
Cost	19780

\[\begin{array}{l} \mathbf{if}\;f \leq 1.3:\\ \;\;\;\;\frac{-4 \cdot \log \left(\frac{\frac{4}{f}}{\pi}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \frac{-1}{\frac{\pi}{4}}\\ \end{array} \]

Alternative 3
Accuracy	97.8%
Cost	19780

\[\begin{array}{l} \mathbf{if}\;f \leq 1.3:\\ \;\;\;\;\frac{\log \left(\frac{\frac{4}{\pi}}{f}\right) \cdot -4}{\pi}\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \frac{-1}{\frac{\pi}{4}}\\ \end{array} \]

Alternative 4
Accuracy	11.8%
Cost	6916

\[\begin{array}{l} t_0 := \frac{-1}{\frac{\pi}{4}}\\ \mathbf{if}\;f \leq 235:\\ \;\;\;\;1.4551915228366852 \cdot 10^{-11} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0 \cdot t_0\\ \end{array} \]

Alternative 5
Accuracy	15.8%
Cost	6916

\[\begin{array}{l} t_0 := \frac{-1}{\frac{\pi}{4}}\\ \mathbf{if}\;f \leq 225:\\ \;\;\;\;1.5 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0 \cdot t_0\\ \end{array} \]

Alternative 6
Accuracy	16.1%
Cost	6916

\[\begin{array}{l} t_0 := \frac{-1}{\frac{\pi}{4}}\\ \mathbf{if}\;f \leq 225:\\ \;\;\;\;3 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0 \cdot t_0\\ \end{array} \]

Alternative 7
Accuracy	16.7%
Cost	6916

\[\begin{array}{l} t_0 := \frac{-1}{\frac{\pi}{4}}\\ \mathbf{if}\;f \leq 225:\\ \;\;\;\;8 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0 \cdot t_0\\ \end{array} \]

Alternative 8
Accuracy	16.8%
Cost	6916

\[\begin{array}{l} t_0 := \frac{-1}{\frac{\pi}{4}}\\ \mathbf{if}\;f \leq 225:\\ \;\;\;\;9 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0 \cdot t_0\\ \end{array} \]

Alternative 9
Accuracy	17.2%
Cost	6916

\[\begin{array}{l} t_0 := \frac{-1}{\frac{\pi}{4}}\\ \mathbf{if}\;f \leq 225:\\ \;\;\;\;16 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0 \cdot t_0\\ \end{array} \]

Alternative 10
Accuracy	17.7%
Cost	6916

\[\begin{array}{l} t_0 := \frac{-1}{\frac{\pi}{4}}\\ \mathbf{if}\;f \leq 225:\\ \;\;\;\;27 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0 \cdot t_0\\ \end{array} \]

Alternative 11
Accuracy	18.7%
Cost	6916

\[\begin{array}{l} t_0 := \frac{-1}{\frac{\pi}{4}}\\ \mathbf{if}\;f \leq 225:\\ \;\;\;\;64 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0 \cdot t_0\\ \end{array} \]

Alternative 12
Accuracy	20.8%
Cost	6916

\[\begin{array}{l} t_0 := \frac{-1}{\frac{\pi}{4}}\\ \mathbf{if}\;f \leq 225:\\ \;\;\;\;256 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0 \cdot t_0\\ \end{array} \]

Alternative 13
Accuracy	21.5%
Cost	6916

\[\begin{array}{l} t_0 := \frac{-1}{\frac{\pi}{4}}\\ \mathbf{if}\;f \leq 225:\\ \;\;\;\;512 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0 \cdot t_0\\ \end{array} \]

Alternative 14
Accuracy	4.8%
Cost	6784

\[0 \cdot \frac{-1}{\frac{\pi}{4}} \]

Reproduce

herbie shell --seed 2023157 
(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)))))))))