Average Error: 0.4 → 0.5
Time: 12.2s
Precision: 64
\[1 \le a \le 2 \le b \le 4 \le c \le 8 \le d \le 16 \le e \le 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
↓
\[\mathsf{fma}\left(\sqrt{\left(e + d\right) + c}, \sqrt{\left(e + d\right) + c}, b + a\right)\]
\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
\mathsf{fma}\left(\sqrt{\left(e + d\right) + c}, \sqrt{\left(e + d\right) + c}, b + a\right)
double f(double a, double b, double c, double d, double e) {
        double r115072 = e;
        double r115073 = d;
        double r115074 = r115072 + r115073;
        double r115075 = c;
        double r115076 = r115074 + r115075;
        double r115077 = b;
        double r115078 = r115076 + r115077;
        double r115079 = a;
        double r115080 = r115078 + r115079;
        return r115080;
}
↓
double f(double a, double b, double c, double d, double e) {
        double r115081 = e;
        double r115082 = d;
        double r115083 = r115081 + r115082;
        double r115084 = c;
        double r115085 = r115083 + r115084;
        double r115086 = sqrt(r115085);
        double r115087 = b;
        double r115088 = a;
        double r115089 = r115087 + r115088;
        double r115090 = fma(r115086, r115086, r115089);
        return r115090;
}

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
Herbie0.5
\[\left(d + \left(c + \left(a + b\right)\right)\right) + e\]

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 add-sqr-sqrt0.5

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

  6. Applied fma-def0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2019212 +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))