Average Error: 9.4 → 0.1
Time: 52.0s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\left(\frac{2}{t} + \left(\frac{\frac{\sqrt{2}}{\frac{1}{\sqrt{\sqrt{2}}}}}{\frac{z}{\frac{\sqrt{\sqrt{2}}}{t}}} - 2\right)\right) + \frac{x}{y}\]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\left(\frac{2}{t} + \left(\frac{\frac{\sqrt{2}}{\frac{1}{\sqrt{\sqrt{2}}}}}{\frac{z}{\frac{\sqrt{\sqrt{2}}}{t}}} - 2\right)\right) + \frac{x}{y}
double f(double x, double y, double z, double t) {
        double r991123 = x;
        double r991124 = y;
        double r991125 = r991123 / r991124;
        double r991126 = 2.0;
        double r991127 = z;
        double r991128 = r991127 * r991126;
        double r991129 = 1.0;
        double r991130 = t;
        double r991131 = r991129 - r991130;
        double r991132 = r991128 * r991131;
        double r991133 = r991126 + r991132;
        double r991134 = r991130 * r991127;
        double r991135 = r991133 / r991134;
        double r991136 = r991125 + r991135;
        return r991136;
}
↓
double f(double x, double y, double z, double t) {
        double r991137 = 2.0;
        double r991138 = t;
        double r991139 = r991137 / r991138;
        double r991140 = sqrt(r991137);
        double r991141 = 1.0;
        double r991142 = sqrt(r991140);
        double r991143 = r991141 / r991142;
        double r991144 = r991140 / r991143;
        double r991145 = z;
        double r991146 = r991142 / r991138;
        double r991147 = r991145 / r991146;
        double r991148 = r991144 / r991147;
        double r991149 = r991148 - r991137;
        double r991150 = r991139 + r991149;
        double r991151 = x;
        double r991152 = y;
        double r991153 = r991151 / r991152;
        double r991154 = r991150 + r991153;
        return r991154;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.4
Target0.1
Herbie0.1
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 9.4

    \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]

  2. Taylor expanded around 0 0.1

    \[\leadsto \frac{x}{y} + \color{blue}{\left(\left(2 \cdot \frac{1}{t \cdot z} + 2 \cdot \frac{1}{t}\right) - 2\right)}\]

  3. Simplified0.1

    \[\leadsto \frac{x}{y} + \color{blue}{\left(\frac{2}{t} + \left(\frac{\frac{2}{t}}{z} - 2\right)\right)}\]
  4. Using strategy rm

  5. Applied *-un-lft-identity0.1

    \[\leadsto \frac{x}{y} + \left(\frac{2}{t} + \left(\frac{\frac{2}{\color{blue}{1 \cdot t}}}{z} - 2\right)\right)\]

  6. Applied add-sqr-sqrt0.3

    \[\leadsto \frac{x}{y} + \left(\frac{2}{t} + \left(\frac{\frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{1 \cdot t}}{z} - 2\right)\right)\]

  7. Applied times-frac0.2

    \[\leadsto \frac{x}{y} + \left(\frac{2}{t} + \left(\frac{\color{blue}{\frac{\sqrt{2}}{1} \cdot \frac{\sqrt{2}}{t}}}{z} - 2\right)\right)\]

  8. Applied associate-/l*0.2

    \[\leadsto \frac{x}{y} + \left(\frac{2}{t} + \left(\color{blue}{\frac{\frac{\sqrt{2}}{1}}{\frac{z}{\frac{\sqrt{2}}{t}}}} - 2\right)\right)\]
  9. Using strategy rm

  10. Applied *-un-lft-identity0.2

    \[\leadsto \frac{x}{y} + \left(\frac{2}{t} + \left(\frac{\frac{\sqrt{2}}{1}}{\frac{z}{\frac{\sqrt{2}}{\color{blue}{1 \cdot t}}}} - 2\right)\right)\]

  11. Applied add-sqr-sqrt0.2

    \[\leadsto \frac{x}{y} + \left(\frac{2}{t} + \left(\frac{\frac{\sqrt{2}}{1}}{\frac{z}{\frac{\sqrt{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}}{1 \cdot t}}} - 2\right)\right)\]

  12. Applied sqrt-prod0.1

    \[\leadsto \frac{x}{y} + \left(\frac{2}{t} + \left(\frac{\frac{\sqrt{2}}{1}}{\frac{z}{\frac{\color{blue}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}}{1 \cdot t}}} - 2\right)\right)\]

  13. Applied times-frac0.1

    \[\leadsto \frac{x}{y} + \left(\frac{2}{t} + \left(\frac{\frac{\sqrt{2}}{1}}{\frac{z}{\color{blue}{\frac{\sqrt{\sqrt{2}}}{1} \cdot \frac{\sqrt{\sqrt{2}}}{t}}}} - 2\right)\right)\]

  14. Applied *-un-lft-identity0.1

    \[\leadsto \frac{x}{y} + \left(\frac{2}{t} + \left(\frac{\frac{\sqrt{2}}{1}}{\frac{\color{blue}{1 \cdot z}}{\frac{\sqrt{\sqrt{2}}}{1} \cdot \frac{\sqrt{\sqrt{2}}}{t}}} - 2\right)\right)\]

  15. Applied times-frac0.1

    \[\leadsto \frac{x}{y} + \left(\frac{2}{t} + \left(\frac{\frac{\sqrt{2}}{1}}{\color{blue}{\frac{1}{\frac{\sqrt{\sqrt{2}}}{1}} \cdot \frac{z}{\frac{\sqrt{\sqrt{2}}}{t}}}} - 2\right)\right)\]

  16. Applied associate-/r*0.1

    \[\leadsto \frac{x}{y} + \left(\frac{2}{t} + \left(\color{blue}{\frac{\frac{\frac{\sqrt{2}}{1}}{\frac{1}{\frac{\sqrt{\sqrt{2}}}{1}}}}{\frac{z}{\frac{\sqrt{\sqrt{2}}}{t}}}} - 2\right)\right)\]

  17. Final simplification0.1

    \[\leadsto \left(\frac{2}{t} + \left(\frac{\frac{\sqrt{2}}{\frac{1}{\sqrt{\sqrt{2}}}}}{\frac{z}{\frac{\sqrt{\sqrt{2}}}{t}}} - 2\right)\right) + \frac{x}{y}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (x y z t)
  :name "Data.HashTable.ST.Basic:computeOverhead from hashtables-1.2.0.2"
  :precision binary64

  :herbie-target
  (- (/ (+ (/ 2 z) 2) t) (- 2 (/ x y)))

  (+ (/ x y) (/ (+ 2 (* (* z 2) (- 1 t))) (* t z))))