VandenBroeck and Keller, Equation (20)

?

Percentage Accurate: 7.0% → 98.3%
Time: 32.7s
Precision: binary64
Cost: 52420

?

\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 200:\\ \;\;\;\;\frac{\frac{-\log \left(\mathsf{fma}\left(\pi \cdot 0.08333333333333333, f, {\left(\sqrt[3]{\frac{-2}{\left(\pi \cdot f\right) \cdot -0.5}}\right)}^{3}\right)\right)}{\pi}}{0.25}\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \frac{-1}{\frac{\pi}{4}}\\ \end{array} \]
(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) 200.0)
   (/
    (/
     (-
      (log
       (fma
        (* PI 0.08333333333333333)
        f
        (pow (cbrt (/ -2.0 (* (* PI f) -0.5))) 3.0))))
     PI)
    0.25)
   (* 0.0 (/ -1.0 (/ PI 4.0)))))
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) <= 200.0) {
		tmp = (-log(fma((((double) M_PI) * 0.08333333333333333), f, pow(cbrt((-2.0 / ((((double) M_PI) * f) * -0.5))), 3.0))) / ((double) M_PI)) / 0.25;
	} else {
		tmp = 0.0 * (-1.0 / (((double) M_PI) / 4.0));
	}
	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) <= 200.0)
		tmp = Float64(Float64(Float64(-log(fma(Float64(pi * 0.08333333333333333), f, (cbrt(Float64(-2.0 / Float64(Float64(pi * f) * -0.5))) ^ 3.0)))) / pi) / 0.25);
	else
		tmp = Float64(0.0 * Float64(-1.0 / Float64(pi / 4.0)));
	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], 200.0], N[(N[((-N[Log[N[(N[(Pi * 0.08333333333333333), $MachinePrecision] * f + N[Power[N[Power[N[(-2.0 / N[(N[(Pi * f), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / Pi), $MachinePrecision] / 0.25), $MachinePrecision], N[(0.0 * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $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 200:\\
\;\;\;\;\frac{\frac{-\log \left(\mathsf{fma}\left(\pi \cdot 0.08333333333333333, f, {\left(\sqrt[3]{\frac{-2}{\left(\pi \cdot f\right) \cdot -0.5}}\right)}^{3}\right)\right)}{\pi}}{0.25}\\

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


\end{array}

Local Percentage Accuracy?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 19 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each of Herbie's proposed alternatives. Up and to the right is better. Each dot represents an alternative program; the red square represents the initial program.

Derivation?

  1. Split input into 2 regimes

  2. if (*.f64 (/.f64 (PI.f64) 4) f) < 200

    1. Initial program 6.6%

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

    2. Taylor expanded in f around 0 97.8%

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

    3. Simplified97.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, 1 \cdot \frac{\pi}{0.5}, \frac{-2}{\frac{{\pi}^{2}}{{\pi}^{3}} \cdot 48}\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)} \]
      Step-by-step derivation

      [Start]97.8

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

      associate-+r+ [=>]97.8

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

      +-commutative [=>]97.8

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

    4. Applied egg-rr97.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\log \left(\sqrt[3]{e^{\mathsf{fma}\left(-2 \cdot \pi, 0.020833333333333332, \frac{0.0625}{\frac{0.5}{\pi}}\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(-2 \cdot \pi, 0.020833333333333332, \frac{0.0625}{\frac{0.5}{\pi}}\right)}}\right) + \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-2 \cdot \pi, 0.020833333333333332, \frac{0.0625}{\frac{0.5}{\pi}}\right)}}\right)}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]
      Step-by-step derivation

      [Start]97.8

      \[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, 1 \cdot \frac{\pi}{0.5}, \frac{-2}{\frac{{\pi}^{2}}{{\pi}^{3}} \cdot 48}\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

      add-log-exp [=>]97.8

      \[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\log \left(e^{\mathsf{fma}\left(0.0625, 1 \cdot \frac{\pi}{0.5}, \frac{-2}{\frac{{\pi}^{2}}{{\pi}^{3}} \cdot 48}\right)}\right)}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

      add-cube-cbrt [=>]97.8

      \[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \log \color{blue}{\left(\left(\sqrt[3]{e^{\mathsf{fma}\left(0.0625, 1 \cdot \frac{\pi}{0.5}, \frac{-2}{\frac{{\pi}^{2}}{{\pi}^{3}} \cdot 48}\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(0.0625, 1 \cdot \frac{\pi}{0.5}, \frac{-2}{\frac{{\pi}^{2}}{{\pi}^{3}} \cdot 48}\right)}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(0.0625, 1 \cdot \frac{\pi}{0.5}, \frac{-2}{\frac{{\pi}^{2}}{{\pi}^{3}} \cdot 48}\right)}}\right)}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

      log-prod [=>]97.8

      \[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\log \left(\sqrt[3]{e^{\mathsf{fma}\left(0.0625, 1 \cdot \frac{\pi}{0.5}, \frac{-2}{\frac{{\pi}^{2}}{{\pi}^{3}} \cdot 48}\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(0.0625, 1 \cdot \frac{\pi}{0.5}, \frac{-2}{\frac{{\pi}^{2}}{{\pi}^{3}} \cdot 48}\right)}}\right) + \log \left(\sqrt[3]{e^{\mathsf{fma}\left(0.0625, 1 \cdot \frac{\pi}{0.5}, \frac{-2}{\frac{{\pi}^{2}}{{\pi}^{3}} \cdot 48}\right)}}\right)}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

    5. Simplified97.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{3 \cdot \log \left(\sqrt[3]{e^{\pi \cdot 0.08333333333333333}}\right)}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]
      Step-by-step derivation

      [Start]97.8

      \[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-2 \cdot \pi, 0.020833333333333332, \frac{0.0625}{\frac{0.5}{\pi}}\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(-2 \cdot \pi, 0.020833333333333332, \frac{0.0625}{\frac{0.5}{\pi}}\right)}}\right) + \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-2 \cdot \pi, 0.020833333333333332, \frac{0.0625}{\frac{0.5}{\pi}}\right)}}\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

      log-prod [=>]97.8

      \[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\left(\log \left(\sqrt[3]{e^{\mathsf{fma}\left(-2 \cdot \pi, 0.020833333333333332, \frac{0.0625}{\frac{0.5}{\pi}}\right)}}\right) + \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-2 \cdot \pi, 0.020833333333333332, \frac{0.0625}{\frac{0.5}{\pi}}\right)}}\right)\right)} + \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-2 \cdot \pi, 0.020833333333333332, \frac{0.0625}{\frac{0.5}{\pi}}\right)}}\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

      count-2 [=>]97.8

      \[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{2 \cdot \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-2 \cdot \pi, 0.020833333333333332, \frac{0.0625}{\frac{0.5}{\pi}}\right)}}\right)} + \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-2 \cdot \pi, 0.020833333333333332, \frac{0.0625}{\frac{0.5}{\pi}}\right)}}\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

      distribute-lft1-in [=>]97.8

      \[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-2 \cdot \pi, 0.020833333333333332, \frac{0.0625}{\frac{0.5}{\pi}}\right)}}\right)}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

      metadata-eval [=>]97.8

      \[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{3} \cdot \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-2 \cdot \pi, 0.020833333333333332, \frac{0.0625}{\frac{0.5}{\pi}}\right)}}\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

      fma-udef [=>]97.8

      \[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, 3 \cdot \log \left(\sqrt[3]{e^{\color{blue}{\left(-2 \cdot \pi\right) \cdot 0.020833333333333332 + \frac{0.0625}{\frac{0.5}{\pi}}}}}\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

      exp-sum [=>]97.8

      \[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, 3 \cdot \log \left(\sqrt[3]{\color{blue}{e^{\left(-2 \cdot \pi\right) \cdot 0.020833333333333332} \cdot e^{\frac{0.0625}{\frac{0.5}{\pi}}}}}\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

    6. Applied egg-rr97.9%

      \[\leadsto -\color{blue}{\frac{\frac{\log \left(\mathsf{fma}\left(\pi \cdot 0.08333333333333333, f, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\pi}}{0.25}} \]
      Step-by-step derivation

      [Start]97.8

      \[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, 3 \cdot \log \left(\sqrt[3]{e^{\pi \cdot 0.08333333333333333}}\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

      associate-*l/ [=>]97.9

      \[ -\color{blue}{\frac{1 \cdot \log \left(\mathsf{fma}\left(f, 3 \cdot \log \left(\sqrt[3]{e^{\pi \cdot 0.08333333333333333}}\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\frac{\pi}{4}}} \]

      *-un-lft-identity [<=]97.9

      \[ -\frac{\color{blue}{\log \left(\mathsf{fma}\left(f, 3 \cdot \log \left(\sqrt[3]{e^{\pi \cdot 0.08333333333333333}}\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}}{\frac{\pi}{4}} \]

      div-inv [=>]97.9

      \[ -\frac{\log \left(\mathsf{fma}\left(f, 3 \cdot \log \left(\sqrt[3]{e^{\pi \cdot 0.08333333333333333}}\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\color{blue}{\pi \cdot \frac{1}{4}}} \]

      metadata-eval [=>]97.9

      \[ -\frac{\log \left(\mathsf{fma}\left(f, 3 \cdot \log \left(\sqrt[3]{e^{\pi \cdot 0.08333333333333333}}\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\pi \cdot \color{blue}{0.25}} \]

      metadata-eval [<=]97.9

      \[ -\frac{\log \left(\mathsf{fma}\left(f, 3 \cdot \log \left(\sqrt[3]{e^{\pi \cdot 0.08333333333333333}}\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\pi \cdot \color{blue}{\left(0.5 \cdot 0.5\right)}} \]

      associate-/r* [=>]97.9

      \[ -\color{blue}{\frac{\frac{\log \left(\mathsf{fma}\left(f, 3 \cdot \log \left(\sqrt[3]{e^{\pi \cdot 0.08333333333333333}}\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\pi}}{0.5 \cdot 0.5}} \]

    7. Applied egg-rr97.9%

      \[\leadsto -\frac{\frac{\log \left(\mathsf{fma}\left(\pi \cdot 0.08333333333333333, f, \color{blue}{{\left(\sqrt[3]{\frac{-2}{\left(\pi \cdot f\right) \cdot -0.5}}\right)}^{3}}\right)\right)}{\pi}}{0.25} \]
      Step-by-step derivation

      [Start]97.9

      \[ -\frac{\frac{\log \left(\mathsf{fma}\left(\pi \cdot 0.08333333333333333, f, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\pi}}{0.25} \]

      add-cube-cbrt [=>]97.9

      \[ -\frac{\frac{\log \left(\mathsf{fma}\left(\pi \cdot 0.08333333333333333, f, \color{blue}{\left(\sqrt[3]{\frac{2}{\pi \cdot \left(f \cdot 0.5\right)}} \cdot \sqrt[3]{\frac{2}{\pi \cdot \left(f \cdot 0.5\right)}}\right) \cdot \sqrt[3]{\frac{2}{\pi \cdot \left(f \cdot 0.5\right)}}}\right)\right)}{\pi}}{0.25} \]

      pow3 [=>]97.9

      \[ -\frac{\frac{\log \left(\mathsf{fma}\left(\pi \cdot 0.08333333333333333, f, \color{blue}{{\left(\sqrt[3]{\frac{2}{\pi \cdot \left(f \cdot 0.5\right)}}\right)}^{3}}\right)\right)}{\pi}}{0.25} \]

      frac-2neg [=>]97.9

      \[ -\frac{\frac{\log \left(\mathsf{fma}\left(\pi \cdot 0.08333333333333333, f, {\left(\sqrt[3]{\color{blue}{\frac{-2}{-\pi \cdot \left(f \cdot 0.5\right)}}}\right)}^{3}\right)\right)}{\pi}}{0.25} \]

      metadata-eval [=>]97.9

      \[ -\frac{\frac{\log \left(\mathsf{fma}\left(\pi \cdot 0.08333333333333333, f, {\left(\sqrt[3]{\frac{\color{blue}{-2}}{-\pi \cdot \left(f \cdot 0.5\right)}}\right)}^{3}\right)\right)}{\pi}}{0.25} \]

      associate-*r* [=>]97.9

      \[ -\frac{\frac{\log \left(\mathsf{fma}\left(\pi \cdot 0.08333333333333333, f, {\left(\sqrt[3]{\frac{-2}{-\color{blue}{\left(\pi \cdot f\right) \cdot 0.5}}}\right)}^{3}\right)\right)}{\pi}}{0.25} \]

      distribute-rgt-neg-in [=>]97.9

      \[ -\frac{\frac{\log \left(\mathsf{fma}\left(\pi \cdot 0.08333333333333333, f, {\left(\sqrt[3]{\frac{-2}{\color{blue}{\left(\pi \cdot f\right) \cdot \left(-0.5\right)}}}\right)}^{3}\right)\right)}{\pi}}{0.25} \]

      metadata-eval [=>]97.9

      \[ -\frac{\frac{\log \left(\mathsf{fma}\left(\pi \cdot 0.08333333333333333, f, {\left(\sqrt[3]{\frac{-2}{\left(\pi \cdot f\right) \cdot \color{blue}{-0.5}}}\right)}^{3}\right)\right)}{\pi}}{0.25} \]

    if 200 < (*.f64 (/.f64 (PI.f64) 4) f)

    1. Initial program 0.0%

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

    2. Applied egg-rr1.6%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{5}}\right) \]

    3. Applied egg-rr100.0%

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

  4. Final simplification98.0%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 200:\\ \;\;\;\;\frac{\frac{-\log \left(\mathsf{fma}\left(\pi \cdot 0.08333333333333333, f, {\left(\sqrt[3]{\frac{-2}{\left(\pi \cdot f\right) \cdot -0.5}}\right)}^{3}\right)\right)}{\pi}}{0.25}\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \frac{-1}{\frac{\pi}{4}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy98.2%
Cost39428
\[\begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 200:\\ \;\;\;\;\frac{-\frac{\log \left(\mathsf{fma}\left(\pi \cdot 0.08333333333333333, f, \frac{\frac{4}{f}}{\pi}\right)\right)}{\pi}}{0.25}\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \frac{-1}{\frac{\pi}{4}}\\ \end{array} \]
Alternative 2
Accuracy97.8%
Cost26180
\[\begin{array}{l} \mathbf{if}\;f \leq 1.26:\\ \;\;\;\;4 \cdot \frac{\log f - \log \left(\frac{4}{\pi}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \frac{-1}{\frac{\pi}{4}}\\ \end{array} \]
Alternative 3
Accuracy97.6%
Cost19844
\[\begin{array}{l} \mathbf{if}\;f \leq 1.26:\\ \;\;\;\;\frac{4}{\pi} \cdot \left(-\log \left(\frac{4}{\pi \cdot f}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \frac{-1}{\frac{\pi}{4}}\\ \end{array} \]
Alternative 4
Accuracy97.6%
Cost19844
\[\begin{array}{l} \mathbf{if}\;f \leq 1.26:\\ \;\;\;\;\frac{-4}{\frac{\pi}{\log \left(\frac{4}{\pi \cdot f}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \frac{-1}{\frac{\pi}{4}}\\ \end{array} \]
Alternative 5
Accuracy97.7%
Cost19844
\[\begin{array}{l} \mathbf{if}\;f \leq 1.26:\\ \;\;\;\;-\frac{\frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}}{0.25}\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \frac{-1}{\frac{\pi}{4}}\\ \end{array} \]
Alternative 6
Accuracy14.3%
Cost6916
\[\begin{array}{l} \mathbf{if}\;f \leq 230:\\ \;\;\;\;\frac{-0.032}{\pi}\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \frac{-1}{\frac{\pi}{4}}\\ \end{array} \]
Alternative 7
Accuracy14.7%
Cost6916
\[\begin{array}{l} t_0 := \frac{-1}{\frac{\pi}{4}}\\ \mathbf{if}\;f \leq 230:\\ \;\;\;\;0.04 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0 \cdot t_0\\ \end{array} \]
Alternative 8
Accuracy15.5%
Cost6916
\[\begin{array}{l} t_0 := \frac{-1}{\frac{\pi}{4}}\\ \mathbf{if}\;f \leq 230:\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0 \cdot t_0\\ \end{array} \]
Alternative 9
Accuracy16.1%
Cost6916
\[\begin{array}{l} t_0 := \frac{-1}{\frac{\pi}{4}}\\ \mathbf{if}\;f \leq 230:\\ \;\;\;\;2 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0 \cdot t_0\\ \end{array} \]
Alternative 10
Accuracy17.4%
Cost6916
\[\begin{array}{l} t_0 := \frac{-1}{\frac{\pi}{4}}\\ \mathbf{if}\;f \leq 230:\\ \;\;\;\;15.625 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0 \cdot t_0\\ \end{array} \]
Alternative 11
Accuracy17.4%
Cost6916
\[\begin{array}{l} t_0 := \frac{-1}{\frac{\pi}{4}}\\ \mathbf{if}\;f \leq 230:\\ \;\;\;\;16 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0 \cdot t_0\\ \end{array} \]
Alternative 12
Accuracy17.8%
Cost6916
\[\begin{array}{l} t_0 := \frac{-1}{\frac{\pi}{4}}\\ \mathbf{if}\;f \leq 230:\\ \;\;\;\;25 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0 \cdot t_0\\ \end{array} \]
Alternative 13
Accuracy17.9%
Cost6916
\[\begin{array}{l} t_0 := \frac{-1}{\frac{\pi}{4}}\\ \mathbf{if}\;f \leq 230:\\ \;\;\;\;27 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0 \cdot t_0\\ \end{array} \]
Alternative 14
Accuracy18.8%
Cost6916
\[\begin{array}{l} t_0 := \frac{-1}{\frac{\pi}{4}}\\ \mathbf{if}\;f \leq 230:\\ \;\;\;\;64 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0 \cdot t_0\\ \end{array} \]
Alternative 15
Accuracy19.8%
Cost6916
\[\begin{array}{l} t_0 := \frac{-1}{\frac{\pi}{4}}\\ \mathbf{if}\;f \leq 230:\\ \;\;\;\;125 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0 \cdot t_0\\ \end{array} \]
Alternative 16
Accuracy21.0%
Cost6916
\[\begin{array}{l} t_0 := \frac{-1}{\frac{\pi}{4}}\\ \mathbf{if}\;f \leq 230:\\ \;\;\;\;256 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0 \cdot t_0\\ \end{array} \]
Alternative 17
Accuracy21.4%
Cost6916
\[\begin{array}{l} t_0 := \frac{-1}{\frac{\pi}{4}}\\ \mathbf{if}\;f \leq 230:\\ \;\;\;\;625 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0 \cdot t_0\\ \end{array} \]
Alternative 18
Accuracy12.4%
Cost6592
\[\frac{-0.032}{\pi} \]

Reproduce?

herbie shell --seed 2023160 
(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)))))))))