Expression, p6

Average Error: 3.7 → 2.8

Time: 51.5s

Precision: 64

\[-14 \le a \le -13 \land -3 \le b \le -2 \land 3 \le c \le 3.5 \land 12.5 \le d \le 13.5\]

Math

\[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]

↓

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

C

double f(double a, double b, double c, double d) {
        double r11153118 = a;
        double r11153119 = b;
        double r11153120 = c;
        double r11153121 = d;
        double r11153122 = r11153120 + r11153121;
        double r11153123 = r11153119 + r11153122;
        double r11153124 = r11153118 + r11153123;
        double r11153125 = 2.0;
        double r11153126 = r11153124 * r11153125;
        return r11153126;
}

↓

double f(double a, double b, double c, double d) {
        double r11153127 = 2.0;
        double r11153128 = b;
        double r11153129 = d;
        double r11153130 = r11153128 + r11153129;
        double r11153131 = a;
        double r11153132 = c;
        double r11153133 = r11153131 + r11153132;
        double r11153134 = r11153130 + r11153133;
        double r11153135 = exp(r11153134);
        double r11153136 = log(r11153135);
        double r11153137 = r11153128 + r11153132;
        double r11153138 = r11153137 + r11153129;
        double r11153139 = r11153138 + r11153131;
        double r11153140 = r11153139 * r11153139;
        double r11153141 = r11153136 * r11153140;
        double r11153142 = cbrt(r11153141);
        double r11153143 = r11153127 * r11153142;
        return r11153143;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	3.7
Target	3.9
Herbie	2.8

\[\left(a + b\right) \cdot 2 + \left(c + d\right) \cdot 2\]

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-cube 2.9

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

7. Applied associate-+r+ 2.8

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

9. Applied add-log-exp 2.8

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

10. Applied add-log-exp 2.8

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

11. Applied add-log-exp 2.8

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

12. Applied sum-log 2.8

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

13. Applied add-log-exp 2.8

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

14. Applied sum-log 2.8

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

15. Applied sum-log 2.7

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

16. Simplified 2.8

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

17. Final simplification 2.8

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

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(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))