Average Error: 25.7 → 1.4
Time: 1.1m
Precision: 64
Internal Precision: 384
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{b}{\sqrt{c^2 + d^2}^*}}{\frac{\sqrt{c^2 + d^2}^*}{c}} + \frac{\frac{d}{\sqrt{c^2 + d^2}^*}}{\frac{\sqrt{c^2 + d^2}^*}{-a}} \le 6.149619111825869 \cdot 10^{+241}:\\ \;\;\;\;\frac{\frac{b}{\sqrt{c^2 + d^2}^*}}{\frac{\sqrt{c^2 + d^2}^*}{c}} + \frac{\frac{d}{\sqrt{c^2 + d^2}^*}}{\frac{\sqrt{c^2 + d^2}^*}{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original25.7
Target0.5
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ (/ (/ b (hypot c d)) (/ (hypot c d) c)) (/ (/ d (hypot c d)) (/ (hypot c d) (- a)))) < 6.149619111825869e+241

    1. Initial program 26.1

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt26.1

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

    4. Applied *-un-lft-identity26.1

      \[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]

    5. Applied times-frac26.1

      \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]

    6. Applied simplify26.1

      \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]

    7. Applied simplify17.0

      \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}}\]
    8. Using strategy rm

    9. Applied div-sub17.0

      \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\left(\frac{c \cdot b}{\sqrt{c^2 + d^2}^*} - \frac{a \cdot d}{\sqrt{c^2 + d^2}^*}\right)}\]
    10. Using strategy rm

    11. Applied sub-neg17.0

      \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\left(\frac{c \cdot b}{\sqrt{c^2 + d^2}^*} + \left(-\frac{a \cdot d}{\sqrt{c^2 + d^2}^*}\right)\right)}\]

    12. Applied distribute-rgt-in17.0

      \[\leadsto \color{blue}{\frac{c \cdot b}{\sqrt{c^2 + d^2}^*} \cdot \frac{1}{\sqrt{c^2 + d^2}^*} + \left(-\frac{a \cdot d}{\sqrt{c^2 + d^2}^*}\right) \cdot \frac{1}{\sqrt{c^2 + d^2}^*}}\]

    13. Applied simplify9.9

      \[\leadsto \color{blue}{\frac{\frac{b}{\sqrt{c^2 + d^2}^*}}{\frac{\sqrt{c^2 + d^2}^*}{c}}} + \left(-\frac{a \cdot d}{\sqrt{c^2 + d^2}^*}\right) \cdot \frac{1}{\sqrt{c^2 + d^2}^*}\]

    14. Applied simplify1.4

      \[\leadsto \frac{\frac{b}{\sqrt{c^2 + d^2}^*}}{\frac{\sqrt{c^2 + d^2}^*}{c}} + \color{blue}{\frac{\frac{d}{\sqrt{c^2 + d^2}^*}}{\frac{\sqrt{c^2 + d^2}^*}{-a}}}\]

    if 6.149619111825869e+241 < (+ (/ (/ b (hypot c d)) (/ (hypot c d) c)) (/ (/ d (hypot c d)) (/ (hypot c d) (- a))))

    1. Initial program 13.5

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt13.5

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

    4. Applied *-un-lft-identity13.5

      \[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]

    5. Applied times-frac13.5

      \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]

    6. Applied simplify13.4

      \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]

    7. Applied simplify3.3

      \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}}\]
    8. Using strategy rm

    9. Applied associate-*r/3.2

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{c^2 + d^2}^*} \cdot \left(c \cdot b - a \cdot d\right)}{\sqrt{c^2 + d^2}^*}}\]

    10. Applied simplify3.1

      \[\leadsto \frac{\color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1070609872 3456127585 2380521889 2328837196 1765472538 734540918)' +o rules:numerics
(FPCore (a b c d)
  :name "Complex division, imag part"

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (- b (* a (/ d c))) (+ c (* d (/ d c)))) (/ (+ (- a) (* b (/ c d))) (+ d (* c (/ c d)))))

  (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))