?

Average Accuracy: 4.0% → 98.4%
Time: 33.8s
Precision: binary64
Cost: 39428

?

\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 20:\\ \;\;\;\;-\frac{\frac{\log \left(\mathsf{fma}\left(\pi \cdot 0.08333333333333333, f, \frac{4}{\pi \cdot f}\right)\right)}{\pi}}{0.25}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot 0\\ \end{array} \]
(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 (* PI 0.08333333333333333) f (/ 4.0 (* PI f)))) PI) 0.25))
   (* (/ 1.0 (/ PI 4.0)) 0.0)))
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((((double) M_PI) * 0.08333333333333333), f, (4.0 / (((double) M_PI) * f)))) / ((double) M_PI)) / 0.25);
	} else {
		tmp = (1.0 / (((double) M_PI) / 4.0)) * 0.0;
	}
	return tmp;
}

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(Float64(log(fma(Float64(pi * 0.08333333333333333), f, Float64(4.0 / Float64(pi * f)))) / pi) / 0.25));
	else
		tmp = Float64(Float64(1.0 / Float64(pi / 4.0)) * 0.0);
	end
	return tmp
end

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[(N[(Pi * 0.08333333333333333), $MachinePrecision] * f + N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] / 0.25), $MachinePrecision]), N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision]]

-\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{\frac{\log \left(\mathsf{fma}\left(\pi \cdot 0.08333333333333333, f, \frac{4}{\pi \cdot f}\right)\right)}{\pi}}{0.25}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot 0\\


\end{array}

Error?

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. Simplified98.6%

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

      [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

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

    4. Applied egg-rr98.7%

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

      [Start]98.6

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

      associate-*l/ [=>]98.7

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

      *-un-lft-identity [<=]98.7

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

      div-inv [=>]98.7

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

      metadata-eval [=>]98.7

      \[ -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, 2 \cdot \pi, \frac{-2}{\frac{0.5 \cdot \frac{0.5}{\pi}}{0.005208333333333333}}\right), \frac{\frac{4}{\pi}}{f}\right)\right)}{\pi \cdot \color{blue}{0.25}} \]

      metadata-eval [<=]98.7

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

      associate-/r* [=>]98.7

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

    5. Applied egg-rr97.5%

      \[\leadsto -\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\log \left(\mathsf{fma}\left(\pi \cdot 0.08333333333333333, f, \frac{4}{\pi \cdot f}\right)\right)}{\pi}\right)} - 1}}{0.25} \]
      Step-by-step derivation

      [Start]98.7

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

      expm1-log1p-u [=>]97.5

      \[ -\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\mathsf{fma}\left(f, -2 \cdot \left(0.020833333333333332 \cdot \pi\right) + 0.125 \cdot \pi, \frac{4}{\pi \cdot f}\right)\right)}{\pi}\right)\right)}}{0.25} \]

      expm1-udef [=>]97.5

      \[ -\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\log \left(\mathsf{fma}\left(f, -2 \cdot \left(0.020833333333333332 \cdot \pi\right) + 0.125 \cdot \pi, \frac{4}{\pi \cdot f}\right)\right)}{\pi}\right)} - 1}}{0.25} \]

    6. Simplified98.7%

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

      [Start]97.5

      \[ -\frac{e^{\mathsf{log1p}\left(\frac{\log \left(\mathsf{fma}\left(\pi \cdot 0.08333333333333333, f, \frac{4}{\pi \cdot f}\right)\right)}{\pi}\right)} - 1}{0.25} \]

      expm1-def [=>]97.5

      \[ -\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\mathsf{fma}\left(\pi \cdot 0.08333333333333333, f, \frac{4}{\pi \cdot f}\right)\right)}{\pi}\right)\right)}}{0.25} \]

      expm1-log1p [=>]98.7

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

    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-rr0.5%

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

    3. Applied egg-rr100.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{0} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification98.8%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 20:\\ \;\;\;\;-\frac{\frac{\log \left(\mathsf{fma}\left(\pi \cdot 0.08333333333333333, f, \frac{4}{\pi \cdot f}\right)\right)}{\pi}}{0.25}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot 0\\ \end{array} \]

Alternatives

Alternative 1
Accuracy98.3%
Cost32836
\[\begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;\frac{4}{\pi} \cdot \left(-\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{4}{\pi \cdot f}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot 0\\ \end{array} \]
Alternative 2
Accuracy97.9%
Cost26308
\[\begin{array}{l} \mathbf{if}\;f \leq 1.3:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot 0\\ \end{array} \]
Alternative 3
Accuracy97.8%
Cost19780
\[\begin{array}{l} \mathbf{if}\;f \leq 1.3:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{\frac{4}{f}}{\pi}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot 0\\ \end{array} \]
Alternative 4
Accuracy11.8%
Cost6916
\[\begin{array}{l} \mathbf{if}\;f \leq 235:\\ \;\;\;\;1.4551915228366852 \cdot 10^{-11} \cdot \frac{-1}{\frac{\pi}{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot 0\\ \end{array} \]
Alternative 5
Accuracy15.8%
Cost6916
\[\begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;1.5 \cdot \frac{-1}{\frac{\pi}{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot 0\\ \end{array} \]
Alternative 6
Accuracy16.1%
Cost6916
\[\begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;3 \cdot \frac{-1}{\frac{\pi}{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot 0\\ \end{array} \]
Alternative 7
Accuracy16.7%
Cost6916
\[\begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;8 \cdot \frac{-1}{\frac{\pi}{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot 0\\ \end{array} \]
Alternative 8
Accuracy16.8%
Cost6916
\[\begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;9 \cdot \frac{-1}{\frac{\pi}{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot 0\\ \end{array} \]
Alternative 9
Accuracy17.2%
Cost6916
\[\begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;16 \cdot \frac{-1}{\frac{\pi}{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot 0\\ \end{array} \]
Alternative 10
Accuracy17.7%
Cost6916
\[\begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;27 \cdot \frac{-1}{\frac{\pi}{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot 0\\ \end{array} \]
Alternative 11
Accuracy18.7%
Cost6916
\[\begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;64 \cdot \frac{-1}{\frac{\pi}{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot 0\\ \end{array} \]
Alternative 12
Accuracy20.8%
Cost6916
\[\begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;256 \cdot \frac{-1}{\frac{\pi}{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot 0\\ \end{array} \]
Alternative 13
Accuracy21.5%
Cost6916
\[\begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;512 \cdot \frac{-1}{\frac{\pi}{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot 0\\ \end{array} \]
Alternative 14
Accuracy4.8%
Cost6784
\[\frac{1}{\frac{\pi}{4}} \cdot 0 \]

Error

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)))))))))