Average Error: 0.4 → 0.3
Time: 3.5s
Precision: 64
\[1 \le a \le 2 \le b \le 4 \le c \le 8 \le d \le 16 \le e \le 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
↓
\[\left(e + d\right) + \mathsf{fma}\left(\sqrt[3]{a} \cdot \sqrt[3]{a}, \sqrt[3]{a}, b + c\right)\]
\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
\left(e + d\right) + \mathsf{fma}\left(\sqrt[3]{a} \cdot \sqrt[3]{a}, \sqrt[3]{a}, b + c\right)
double f(double a, double b, double c, double d, double e) {
        double r121791 = e;
        double r121792 = d;
        double r121793 = r121791 + r121792;
        double r121794 = c;
        double r121795 = r121793 + r121794;
        double r121796 = b;
        double r121797 = r121795 + r121796;
        double r121798 = a;
        double r121799 = r121797 + r121798;
        return r121799;
}
↓
double f(double a, double b, double c, double d, double e) {
        double r121800 = e;
        double r121801 = d;
        double r121802 = r121800 + r121801;
        double r121803 = a;
        double r121804 = cbrt(r121803);
        double r121805 = r121804 * r121804;
        double r121806 = b;
        double r121807 = c;
        double r121808 = r121806 + r121807;
        double r121809 = fma(r121805, r121804, r121808);
        double r121810 = r121802 + r121809;
        return r121810;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Bits error versus e

Target

Original0.4
Target0.2
Herbie0.3
\[\left(d + \left(c + \left(a + b\right)\right)\right) + e\]

Derivation

  1. Initial program 0.4

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

  3. Applied associate-+l+0.3

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

  5. Applied associate-+l+0.3

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

  6. Simplified0.3

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

  8. Applied add-cube-cbrt0.3

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

  9. Applied fma-def0.3

    \[\leadsto \left(e + d\right) + \color{blue}{\mathsf{fma}\left(\sqrt[3]{a} \cdot \sqrt[3]{a}, \sqrt[3]{a}, b + c\right)}\]

  10. Final simplification0.3

    \[\leadsto \left(e + d\right) + \mathsf{fma}\left(\sqrt[3]{a} \cdot \sqrt[3]{a}, \sqrt[3]{a}, b + c\right)\]

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (a b c d e)
  :name "Expression 1, p15"
  :precision binary64
  :pre (<= 1 a 2 b 4 c 8 d 16 e 32)

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

  (+ (+ (+ (+ e d) c) b) a))