Average Error: 1.6 → 0.2
Time: 21.5s
Precision: 64
Internal Precision: 384
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{\frac{y}{z}} \le -1.3314788140349553 \cdot 10^{-47}:\\ \;\;\;\;\left|(\left(x + 4\right) \cdot \left(\frac{1}{y}\right) + \left(-\frac{x}{y} \cdot z\right))_*\right|\\ \mathbf{if}\;\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{\frac{y}{z}} \le 2.0698612563574678 \cdot 10^{+270}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|(\left(x + 4\right) \cdot \left(\frac{1}{y}\right) + \left(-\frac{x}{y} \cdot z\right))_*\right|\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Split input into 2 regimes

  2. if (- (+ (/ 4 y) (/ x y)) (/ x (/ y z))) < -1.3314788140349553e-47 or 2.0698612563574678e+270 < (- (+ (/ 4 y) (/ x y)) (/ x (/ y z)))

    1. Initial program 0.2

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    2. Using strategy rm

    3. Applied div-inv0.2

      \[\leadsto \left|\color{blue}{\left(x + 4\right) \cdot \frac{1}{y}} - \frac{x}{y} \cdot z\right|\]

    4. Applied fma-neg0.2

      \[\leadsto \left|\color{blue}{(\left(x + 4\right) \cdot \left(\frac{1}{y}\right) + \left(-\frac{x}{y} \cdot z\right))_*}\right|\]

    if -1.3314788140349553e-47 < (- (+ (/ 4 y) (/ x y)) (/ x (/ y z))) < 2.0698612563574678e+270

    1. Initial program 2.4

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    2. Using strategy rm

    3. Applied div-inv2.5

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\left(x \cdot \frac{1}{y}\right)} \cdot z\right|\]

    4. Applied associate-*l*0.3

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{x \cdot \left(\frac{1}{y} \cdot z\right)}\right|\]

    5. Applied simplify0.2

      \[\leadsto \left|\frac{x + 4}{y} - x \cdot \color{blue}{\frac{z}{y}}\right|\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 21.5s)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))