Kahan's Unum-Targeted Monster

Average Error: 60.8 → 32.6

Time: 15.2s

Precision: 64

\[1 \le y \le 9999\]

Math

\[\begin{array}{l} \mathbf{if}\;\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)} = 0.0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)}} - 1}{\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)}}\\ \end{array}\]

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \log \left(e^{\sqrt[3]{{\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right)}^{3}}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)} = 0.0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)}} - 1}{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)}}\\ \end{array}\]

C

double f(double y) {
        double r101151 = y;
        double r101152 = r101151 * r101151;
        double r101153 = 1.0;
        double r101154 = r101152 + r101153;
        double r101155 = sqrt(r101154);
        double r101156 = r101151 - r101155;
        double r101157 = fabs(r101156);
        double r101158 = r101151 + r101155;
        double r101159 = r101153 / r101158;
        double r101160 = r101157 - r101159;
        double r101161 = r101160 * r101160;
        double r101162 = 10.0;
        double r101163 = -300.0;
        double r101164 = pow(r101162, r101163);
        double r101165 = 10000.0;
        double r101166 = r101151 + r101153;
        double r101167 = r101165 * r101166;
        double r101168 = pow(r101164, r101167);
        double r101169 = r101161 + r101168;
        double r101170 = 0.0;
        double r101171 = r101169 == r101170;
        double r101172 = exp(r101169);
        double r101173 = r101172 - r101153;
        double r101174 = r101173 / r101169;
        double r101175 = r101171 ? r101153 : r101174;
        return r101175;
}

↓

double f(double y) {
        double r101176 = y;
        double r101177 = r101176 * r101176;
        double r101178 = 1.0;
        double r101179 = r101177 + r101178;
        double r101180 = sqrt(r101179);
        double r101181 = r101176 - r101180;
        double r101182 = fabs(r101181);
        double r101183 = r101176 + r101180;
        double r101184 = r101178 / r101183;
        double r101185 = r101182 - r101184;
        double r101186 = 3.0;
        double r101187 = pow(r101185, r101186);
        double r101188 = cbrt(r101187);
        double r101189 = exp(r101188);
        double r101190 = log(r101189);
        double r101191 = r101185 * r101190;
        double r101192 = 10.0;
        double r101193 = -300.0;
        double r101194 = pow(r101192, r101193);
        double r101195 = 10000.0;
        double r101196 = r101176 + r101178;
        double r101197 = r101195 * r101196;
        double r101198 = pow(r101194, r101197);
        double r101199 = r101191 + r101198;
        double r101200 = 0.0;
        double r101201 = r101199 == r101200;
        double r101202 = 0.5;
        double r101203 = 1.0;
        double r101204 = r101203 / r101176;
        double r101205 = r101202 * r101204;
        double r101206 = 2.0;
        double r101207 = r101206 * r101176;
        double r101208 = 2.0;
        double r101209 = pow(r101176, r101208);
        double r101210 = r101209 + r101178;
        double r101211 = sqrt(r101210);
        double r101212 = r101176 - r101211;
        double r101213 = fabs(r101212);
        double r101214 = r101207 + r101213;
        double r101215 = r101205 + r101214;
        double r101216 = r101215 * r101185;
        double r101217 = r101216 + r101198;
        double r101218 = exp(r101217);
        double r101219 = r101218 - r101178;
        double r101220 = r101219 / r101217;
        double r101221 = r101201 ? r101178 : r101220;
        return r101221;
}

Error

Bits error versus y

Derivation

1. Initial program 60.8

   \[\begin{array}{l} \mathbf{if}\;\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)} = 0.0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)}} - 1}{\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)}}\\ \end{array}\]

2. Taylor expanded around -inf 59.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)} = 0.0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)}} - 1}{\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)}}\\ \end{array}\]

3. Taylor expanded around -inf 37.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)} = 0.0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{e^{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)}}} - 1}{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)}}\\ \end{array}\]
4. Using strategy rm

5. Applied add-log-exp 36.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \color{blue}{\log \left(e^{\frac{1}{y + \sqrt{y \cdot y + 1}}}\right)}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)} = 0.0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)}} - 1}{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)}}\\ \end{array}\]

6. Applied add-log-exp 33.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(\color{blue}{\log \left(e^{\left|y - \sqrt{y \cdot y + 1}\right|}\right)} - \log \left(e^{\frac{1}{y + \sqrt{y \cdot y + 1}}}\right)\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)} = 0.0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)}} - 1}{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)}}\\ \end{array}\]

7. Applied diff-log 33.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \color{blue}{\log \left(\frac{e^{\left|y - \sqrt{y \cdot y + 1}\right|}}{e^{\frac{1}{y + \sqrt{y \cdot y + 1}}}}\right)} + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)} = 0.0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)}} - 1}{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)}}\\ \end{array}\]

8. Simplified 33.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \log \color{blue}{\left(e^{\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}}\right)} + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)} = 0.0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)}} - 1}{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)}}\\ \end{array}\]
9. Using strategy rm

10. Applied add-cbrt-cube 32.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \log \left(e^{\color{blue}{\sqrt[3]{\left(\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right)}}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)} = 0.0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)}} - 1}{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)}}\\ \end{array}\]

11. Simplified 32.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \log \left(e^{\sqrt[3]{\color{blue}{{\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right)}^{3}}}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)} = 0.0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)}} - 1}{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)}}\\ \end{array}\]

12. Final simplification 32.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) \cdot \log \left(e^{\sqrt[3]{{\left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right)}^{3}}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)} = 0.0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)}} - 1}{\left(0.5 \cdot \frac{1}{y} + \left(2 \cdot y + \left|y - \sqrt{{y}^{2} + 1}\right|\right)\right) \cdot \left(\left|y - \sqrt{y \cdot y + 1}\right| - \frac{1}{y + \sqrt{y \cdot y + 1}}\right) + {\left({10}^{-300}\right)}^{\left(10^{4} \cdot \left(y + 1\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 
(FPCore (y)
  :name "Kahan's Unum-Targeted 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)))))) (pow (pow 10 -300) (* 10000.0 (+ 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)))))) (pow (pow 10 -300) (* 10000.0 (+ 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)))))) (pow (pow 10 -300) (* 10000.0 (+ y 1)))))))