Average Error: 60.8 → 32.6
Time: 9.6s
Precision: 64
\[1 \le y \le 9999\]
\[\begin{array}{l} \mathbf{if}\;\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) = 0.0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right)} - 1}{\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right)}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \log \left(e^{\sqrt[3]{{\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right)}^{3}}}\right) = 0.0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right)} - 1}{\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right)}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) = 0.0:\\
\;\;\;\;1\\

\mathbf{else}:\\
\;\;\;\;\frac{e^{\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right)} - 1}{\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right)}\\

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \log \left(e^{\sqrt[3]{{\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right)}^{3}}}\right) = 0.0:\\
\;\;\;\;1\\

\mathbf{else}:\\
\;\;\;\;\frac{e^{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right)} - 1}{\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right)}\\

\end{array}
double f(double y) {
        double r87091 = y;
        double r87092 = r87091 * r87091;
        double r87093 = 1.0;
        double r87094 = r87092 + r87093;
        double r87095 = sqrt(r87094);
        double r87096 = r87091 - r87095;
        double r87097 = fabs(r87096);
        double r87098 = r87091 + r87095;
        double r87099 = r87093 / r87098;
        double r87100 = r87097 - r87099;
        double r87101 = r87100 * r87100;
        double r87102 = 0.0;
        double r87103 = r87101 == r87102;
        double r87104 = exp(r87101);
        double r87105 = r87104 - r87093;
        double r87106 = r87105 / r87101;
        double r87107 = r87103 ? r87093 : r87106;
        return r87107;
}
↓
double f(double y) {
        double r87108 = y;
        double r87109 = r87108 * r87108;
        double r87110 = 1.0;
        double r87111 = r87109 + r87110;
        double r87112 = sqrt(r87111);
        double r87113 = r87108 - r87112;
        double r87114 = fabs(r87113);
        double r87115 = r87108 + r87112;
        double r87116 = r87110 / r87115;
        double r87117 = r87114 - r87116;
        double r87118 = 3.0;
        double r87119 = pow(r87117, r87118);
        double r87120 = cbrt(r87119);
        double r87121 = exp(r87120);
        double r87122 = log(r87121);
        double r87123 = r87117 * r87122;
        double r87124 = 0.0;
        double r87125 = r87123 == r87124;
        double r87126 = 0.5;
        double r87127 = 1.0;
        double r87128 = r87127 / r87108;
        double r87129 = r87126 * r87128;
        double r87130 = 2.0;
        double r87131 = r87130 * r87108;
        double r87132 = 2.0;
        double r87133 = pow(r87108, r87132);
        double r87134 = r87133 + r87110;
        double r87135 = sqrt(r87134);
        double r87136 = r87108 - r87135;
        double r87137 = fabs(r87136);
        double r87138 = r87131 + r87137;
        double r87139 = r87129 + r87138;
        double r87140 = r87139 * r87117;
        double r87141 = exp(r87140);
        double r87142 = r87141 - r87110;
        double r87143 = r87117 * r87139;
        double r87144 = r87142 / r87143;
        double r87145 = r87125 ? r87110 : r87144;
        return r87145;
}

Error

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 60.8

    \[\begin{array}{l} \mathbf{if}\;\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) = 0.0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right)} - 1}{\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right)}\\ \end{array}\]

  2. Taylor expanded around -inf 59.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) = 0.0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right)} - 1}{\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right)}\\ \end{array}\]

  3. Taylor expanded around -inf 37.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) = 0.0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right)} - 1}{\color{blue}{\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right)}}\\ \end{array}\]
  4. Using strategy rm

  5. Applied add-log-exp36.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \color{blue}{\log \left(e^{\frac{1}{y + \sqrt{y \cdot y + 1}}}\right)}\right) = 0.0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right)} - 1}{\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right)}\\ \end{array}\]

  6. Applied add-log-exp33.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(\color{blue}{\log \left(e^{\left|y - \sqrt{y \cdot y + 1}\right|}\right)} - \log \left(e^{\frac{1}{y + \sqrt{y \cdot y + 1}}}\right)\right) = 0.0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right)} - 1}{\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right)}\\ \end{array}\]

  7. Applied diff-log33.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \color{blue}{\log \left(\frac{e^{\left|y - \sqrt{y \cdot y + 1}\right|}}{e^{\frac{1}{y + \sqrt{y \cdot y + 1}}}}\right)} = 0.0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right)} - 1}{\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right)}\\ \end{array}\]

  8. Simplified33.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \log \color{blue}{\left(e^{\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}}\right)} = 0.0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right)} - 1}{\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right)}\\ \end{array}\]
  9. Using strategy rm

  10. Applied add-cbrt-cube32.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \log \left(e^{\color{blue}{\sqrt[3]{\left(\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right)}}}\right) = 0.0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right)} - 1}{\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right)}\\ \end{array}\]

  11. Simplified32.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \log \left(e^{\sqrt[3]{\color{blue}{{\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right)}^{3}}}}\right) = 0.0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right)} - 1}{\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right)}\\ \end{array}\]

  12. Final simplification32.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \log \left(e^{\sqrt[3]{{\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right)}^{3}}}\right) = 0.0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right)} - 1}{\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 
(FPCore (y)
  :name "Kahan's Monster"
  :precision binary64
  :pre (<= 1 y 9999)
  (if (== (* (- (fabs (- y (sqrt (+ (* y y) 1)))) (/ 1 (+ y (sqrt (+ (* y y) 1))))) (- (fabs (- y (sqrt (+ (* y y) 1)))) (/ 1 (+ y (sqrt (+ (* y y) 1)))))) 0.0) 1 (/ (- (exp (* (- (fabs (- y (sqrt (+ (* y y) 1)))) (/ 1 (+ y (sqrt (+ (* y y) 1))))) (- (fabs (- y (sqrt (+ (* y y) 1)))) (/ 1 (+ y (sqrt (+ (* y y) 1))))))) 1) (* (- (fabs (- y (sqrt (+ (* y y) 1)))) (/ 1 (+ y (sqrt (+ (* y y) 1))))) (- (fabs (- y (sqrt (+ (* y y) 1)))) (/ 1 (+ y (sqrt (+ (* y y) 1)))))))))