Average Error: 3.7 → 2.8
Time: 32.4s
Precision: 64
\[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]
↓
\[\sqrt[3]{\left(\log \left(e^{b + \left(a + \left(c + d\right)\right)}\right) \cdot \left(\left(\left(b + c\right) + d\right) + a\right)\right) \cdot \left(d + \left(\left(b + c\right) + a\right)\right)} \cdot 2\]
double f(double a, double b, double c, double d) {
        double r15197048 = a;
        double r15197049 = b;
        double r15197050 = c;
        double r15197051 = d;
        double r15197052 = r15197050 + r15197051;
        double r15197053 = r15197049 + r15197052;
        double r15197054 = r15197048 + r15197053;
        double r15197055 = 2.0;
        double r15197056 = r15197054 * r15197055;
        return r15197056;
}
↓
double f(double a, double b, double c, double d) {
        double r15197057 = b;
        double r15197058 = a;
        double r15197059 = c;
        double r15197060 = d;
        double r15197061 = r15197059 + r15197060;
        double r15197062 = r15197058 + r15197061;
        double r15197063 = r15197057 + r15197062;
        double r15197064 = exp(r15197063);
        double r15197065 = log(r15197064);
        double r15197066 = r15197057 + r15197059;
        double r15197067 = r15197066 + r15197060;
        double r15197068 = r15197067 + r15197058;
        double r15197069 = r15197065 * r15197068;
        double r15197070 = r15197066 + r15197058;
        double r15197071 = r15197060 + r15197070;
        double r15197072 = r15197069 * r15197071;
        double r15197073 = cbrt(r15197072);
        double r15197074 = 2.0;
        double r15197075 = r15197073 * r15197074;
        return r15197075;
}


\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2
↓
\sqrt[3]{\left(\log \left(e^{b + \left(a + \left(c + d\right)\right)}\right) \cdot \left(\left(\left(b + c\right) + d\right) + a\right)\right) \cdot \left(d + \left(\left(b + c\right) + a\right)\right)} \cdot 2

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original3.7
Target3.9
Herbie2.8
\[\left(a + b\right) \cdot 2 + \left(c + d\right) \cdot 2\]

Derivation

  1. Initial program 3.7

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

  3. Applied associate-+r+2.8

    \[\leadsto \left(a + \color{blue}{\left(\left(b + c\right) + d\right)}\right) \cdot 2\]
  4. Using strategy rm

  5. Applied add-cbrt-cube2.9

    \[\leadsto \color{blue}{\sqrt[3]{\left(\left(a + \left(\left(b + c\right) + d\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)}} \cdot 2\]
  6. Using strategy rm

  7. Applied add-log-exp2.9

    \[\leadsto \sqrt[3]{\left(\left(a + \left(\left(b + c\right) + d\right)\right) \cdot \left(a + \left(\left(b + c\right) + \color{blue}{\log \left(e^{d}\right)}\right)\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)} \cdot 2\]

  8. Applied add-log-exp2.9

    \[\leadsto \sqrt[3]{\left(\left(a + \left(\left(b + c\right) + d\right)\right) \cdot \left(a + \left(\color{blue}{\log \left(e^{b + c}\right)} + \log \left(e^{d}\right)\right)\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)} \cdot 2\]

  9. Applied sum-log2.8

    \[\leadsto \sqrt[3]{\left(\left(a + \left(\left(b + c\right) + d\right)\right) \cdot \left(a + \color{blue}{\log \left(e^{b + c} \cdot e^{d}\right)}\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)} \cdot 2\]

  10. Applied add-log-exp2.8

    \[\leadsto \sqrt[3]{\left(\left(a + \left(\left(b + c\right) + d\right)\right) \cdot \left(\color{blue}{\log \left(e^{a}\right)} + \log \left(e^{b + c} \cdot e^{d}\right)\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)} \cdot 2\]

  11. Applied sum-log2.6

    \[\leadsto \sqrt[3]{\left(\left(a + \left(\left(b + c\right) + d\right)\right) \cdot \color{blue}{\log \left(e^{a} \cdot \left(e^{b + c} \cdot e^{d}\right)\right)}\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)} \cdot 2\]

  12. Simplified2.9

    \[\leadsto \sqrt[3]{\left(\left(a + \left(\left(b + c\right) + d\right)\right) \cdot \log \color{blue}{\left(e^{\left(\left(c + d\right) + a\right) + b}\right)}\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)} \cdot 2\]
  13. Using strategy rm

  14. Applied associate-+r+2.8

    \[\leadsto \sqrt[3]{\left(\left(a + \left(\left(b + c\right) + d\right)\right) \cdot \log \left(e^{\left(\left(c + d\right) + a\right) + b}\right)\right) \cdot \color{blue}{\left(\left(a + \left(b + c\right)\right) + d\right)}} \cdot 2\]

  15. Final simplification2.8

    \[\leadsto \sqrt[3]{\left(\log \left(e^{b + \left(a + \left(c + d\right)\right)}\right) \cdot \left(\left(\left(b + c\right) + d\right) + a\right)\right) \cdot \left(d + \left(\left(b + c\right) + a\right)\right)} \cdot 2\]

Reproduce

herbie shell --seed 2019102 
(FPCore (a b c d)
  :name "Expression, p6"
  :pre (and (<= -14 a -13) (<= -3 b -2) (<= 3 c 3.5) (<= 12.5 d 13.5))

  :herbie-target
  (+ (* (+ a b) 2) (* (+ c d) 2))

  (* (+ a (+ b (+ c d))) 2))