Average Error: 9.3 → 0.1
Time: 15.8s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\left(\frac{2}{t} + \frac{\sqrt{\sqrt{2}}}{t} \cdot \frac{\sqrt{2}}{\frac{z}{\sqrt{\sqrt{2}}}}\right) + \left(\frac{x}{y} - 2\right)\]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\left(\frac{2}{t} + \frac{\sqrt{\sqrt{2}}}{t} \cdot \frac{\sqrt{2}}{\frac{z}{\sqrt{\sqrt{2}}}}\right) + \left(\frac{x}{y} - 2\right)
double f(double x, double y, double z, double t) {
        double r911220 = x;
        double r911221 = y;
        double r911222 = r911220 / r911221;
        double r911223 = 2.0;
        double r911224 = z;
        double r911225 = r911224 * r911223;
        double r911226 = 1.0;
        double r911227 = t;
        double r911228 = r911226 - r911227;
        double r911229 = r911225 * r911228;
        double r911230 = r911223 + r911229;
        double r911231 = r911227 * r911224;
        double r911232 = r911230 / r911231;
        double r911233 = r911222 + r911232;
        return r911233;
}
↓
double f(double x, double y, double z, double t) {
        double r911234 = 2.0;
        double r911235 = t;
        double r911236 = r911234 / r911235;
        double r911237 = sqrt(r911234);
        double r911238 = sqrt(r911237);
        double r911239 = r911238 / r911235;
        double r911240 = z;
        double r911241 = r911240 / r911238;
        double r911242 = r911237 / r911241;
        double r911243 = r911239 * r911242;
        double r911244 = r911236 + r911243;
        double r911245 = x;
        double r911246 = y;
        double r911247 = r911245 / r911246;
        double r911248 = r911247 - r911234;
        double r911249 = r911244 + r911248;
        return r911249;
}

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

Original9.3
Target0.1
Herbie0.1
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 9.3

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

  2. Simplified0.1

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

  3. Taylor expanded around 0 0.1

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

  4. Simplified0.1

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

  6. Applied add-sqr-sqrt0.3

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

  7. Applied associate-/l*0.2

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

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

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

  10. Applied sqrt-prod0.2

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

  11. Applied times-frac0.2

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

  12. Applied add-sqr-sqrt0.2

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

  13. Applied sqrt-prod0.1

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

  14. Applied times-frac0.1

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

  15. Simplified0.1

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

  16. Simplified0.1

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

  17. Final simplification0.1

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

Reproduce

herbie shell --seed 2019350 +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))))