Expression, p14

Average Error: 0.0 → 0.1

Time: 13.7s

Precision: 64

\[56789 \le a \le 98765 \land 0 \le b \le 1 \land 0 \le c \le 0.0016773 \land 0 \le d \le 0.0016773\]

Math

\[a \cdot \left(\left(b + c\right) + d\right)\]

↓

\[\left(\left(b - d\right) - c\right) \cdot \frac{a}{\frac{c + \left(b - d\right)}{c + \left(d + b\right)} \cdot \frac{\left(b - d\right) - c}{c + \left(b - d\right)}}\]

C

double f(double a, double b, double c, double d) {
        double r2688220 = a;
        double r2688221 = b;
        double r2688222 = c;
        double r2688223 = r2688221 + r2688222;
        double r2688224 = d;
        double r2688225 = r2688223 + r2688224;
        double r2688226 = r2688220 * r2688225;
        return r2688226;
}

↓

double f(double a, double b, double c, double d) {
        double r2688227 = b;
        double r2688228 = d;
        double r2688229 = r2688227 - r2688228;
        double r2688230 = c;
        double r2688231 = r2688229 - r2688230;
        double r2688232 = a;
        double r2688233 = r2688230 + r2688229;
        double r2688234 = r2688228 + r2688227;
        double r2688235 = r2688230 + r2688234;
        double r2688236 = r2688233 / r2688235;
        double r2688237 = r2688231 / r2688233;
        double r2688238 = r2688236 * r2688237;
        double r2688239 = r2688232 / r2688238;
        double r2688240 = r2688231 * r2688239;
        return r2688240;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	0.0
Target	0.0
Herbie	0.1

\[a \cdot b + a \cdot \left(c + d\right)\]

Derivation

1. Initial program 0.0

   \[a \cdot \left(\left(b + c\right) + d\right)\]
2. Using strategy rm

3. Applied add-sqr-sqrt 0.5

   \[\leadsto a \cdot \color{blue}{\left(\sqrt{\left(b + c\right) + d} \cdot \sqrt{\left(b + c\right) + d}\right)}\]

4. Applied add-sqr-sqrt 0.7

   \[\leadsto \color{blue}{\left(\sqrt{a} \cdot \sqrt{a}\right)} \cdot \left(\sqrt{\left(b + c\right) + d} \cdot \sqrt{\left(b + c\right) + d}\right)\]

5. Applied unswap-sqr 0.8

   \[\leadsto \color{blue}{\left(\sqrt{a} \cdot \sqrt{\left(b + c\right) + d}\right) \cdot \left(\sqrt{a} \cdot \sqrt{\left(b + c\right) + d}\right)}\]
6. Using strategy rm

7. Applied flip-+ 7.6

   \[\leadsto \left(\sqrt{a} \cdot \sqrt{\left(b + c\right) + d}\right) \cdot \left(\sqrt{a} \cdot \sqrt{\color{blue}{\frac{\left(b + c\right) \cdot \left(b + c\right) - d \cdot d}{\left(b + c\right) - d}}}\right)\]

8. Applied sqrt-div 26.4

   \[\leadsto \left(\sqrt{a} \cdot \sqrt{\left(b + c\right) + d}\right) \cdot \left(\sqrt{a} \cdot \color{blue}{\frac{\sqrt{\left(b + c\right) \cdot \left(b + c\right) - d \cdot d}}{\sqrt{\left(b + c\right) - d}}}\right)\]

9. Applied associate-*r/ 26.4

   \[\leadsto \left(\sqrt{a} \cdot \sqrt{\left(b + c\right) + d}\right) \cdot \color{blue}{\frac{\sqrt{a} \cdot \sqrt{\left(b + c\right) \cdot \left(b + c\right) - d \cdot d}}{\sqrt{\left(b + c\right) - d}}}\]

10. Applied flip-+ 26.4

    \[\leadsto \left(\sqrt{a} \cdot \sqrt{\color{blue}{\frac{\left(b + c\right) \cdot \left(b + c\right) - d \cdot d}{\left(b + c\right) - d}}}\right) \cdot \frac{\sqrt{a} \cdot \sqrt{\left(b + c\right) \cdot \left(b + c\right) - d \cdot d}}{\sqrt{\left(b + c\right) - d}}\]

11. Applied sqrt-div 26.5

    \[\leadsto \left(\sqrt{a} \cdot \color{blue}{\frac{\sqrt{\left(b + c\right) \cdot \left(b + c\right) - d \cdot d}}{\sqrt{\left(b + c\right) - d}}}\right) \cdot \frac{\sqrt{a} \cdot \sqrt{\left(b + c\right) \cdot \left(b + c\right) - d \cdot d}}{\sqrt{\left(b + c\right) - d}}\]

12. Applied associate-*r/ 26.5

    \[\leadsto \color{blue}{\frac{\sqrt{a} \cdot \sqrt{\left(b + c\right) \cdot \left(b + c\right) - d \cdot d}}{\sqrt{\left(b + c\right) - d}}} \cdot \frac{\sqrt{a} \cdot \sqrt{\left(b + c\right) \cdot \left(b + c\right) - d \cdot d}}{\sqrt{\left(b + c\right) - d}}\]

13. Applied frac-times 26.5

    \[\leadsto \color{blue}{\frac{\left(\sqrt{a} \cdot \sqrt{\left(b + c\right) \cdot \left(b + c\right) - d \cdot d}\right) \cdot \left(\sqrt{a} \cdot \sqrt{\left(b + c\right) \cdot \left(b + c\right) - d \cdot d}\right)}{\sqrt{\left(b + c\right) - d} \cdot \sqrt{\left(b + c\right) - d}}}\]

14. Simplified 26.4

    \[\leadsto \frac{\color{blue}{a \cdot \left(\left(c + b\right) \cdot \left(c + b\right) - d \cdot d\right)}}{\sqrt{\left(b + c\right) - d} \cdot \sqrt{\left(b + c\right) - d}}\]

15. Simplified 7.2

    \[\leadsto \frac{a \cdot \left(\left(c + b\right) \cdot \left(c + b\right) - d \cdot d\right)}{\color{blue}{\left(b - d\right) + c}}\]
16. Using strategy rm

17. Applied flip-+ 7.2

    \[\leadsto \frac{a \cdot \left(\left(c + b\right) \cdot \left(c + b\right) - d \cdot d\right)}{\color{blue}{\frac{\left(b - d\right) \cdot \left(b - d\right) - c \cdot c}{\left(b - d\right) - c}}}\]

18. Applied associate-/r/ 7.0

    \[\leadsto \color{blue}{\frac{a \cdot \left(\left(c + b\right) \cdot \left(c + b\right) - d \cdot d\right)}{\left(b - d\right) \cdot \left(b - d\right) - c \cdot c} \cdot \left(\left(b - d\right) - c\right)}\]

19. Simplified 0.1

    \[\leadsto \color{blue}{\frac{a}{\frac{c + \left(b - d\right)}{\left(b + d\right) + c} \cdot \frac{\left(b - d\right) - c}{c + \left(b - d\right)}}} \cdot \left(\left(b - d\right) - c\right)\]

20. Final simplification 0.1

    \[\leadsto \left(\left(b - d\right) - c\right) \cdot \frac{a}{\frac{c + \left(b - d\right)}{c + \left(d + b\right)} \cdot \frac{\left(b - d\right) - c}{c + \left(b - d\right)}}\]

Reproduce

herbie shell --seed 2019143 
(FPCore (a b c d)
  :name "Expression, p14"
  :pre (and (<= 56789 a 98765) (<= 0 b 1) (<= 0 c 0.0016773) (<= 0 d 0.0016773))

  :herbie-target
  (+ (* a b) (* a (+ c d)))

  (* a (+ (+ b c) d)))