Average Error: 25.6 → 12.6
Time: 24.8s
Precision: 64
Internal Precision: 576
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le -4.681990609948178 \cdot 10^{+154}:\\ \;\;\;\;\frac{a}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;d \le 4.494951878835627 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{(\left(-d\right) \cdot a + \left(c \cdot b\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original25.6
Target0.4
Herbie12.6
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if d < -4.681990609948178e+154

    1. Initial program 43.3

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]

    2. Initial simplification43.3

      \[\leadsto \frac{(d \cdot \left(-a\right) + \left(b \cdot c\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt43.3

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

    5. Applied *-un-lft-identity43.3

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

    6. Applied times-frac43.3

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

    7. Simplified43.3

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

    8. Simplified27.9

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

    10. Applied associate-*l/27.8

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

    11. Simplified27.8

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

    12. Taylor expanded around -inf 14.2

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

    if -4.681990609948178e+154 < d < 4.494951878835627e+102

    1. Initial program 18.3

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]

    2. Initial simplification18.3

      \[\leadsto \frac{(d \cdot \left(-a\right) + \left(b \cdot c\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt18.3

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

    5. Applied *-un-lft-identity18.3

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

    6. Applied times-frac18.4

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

    7. Simplified18.3

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

    8. Simplified11.3

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

    10. Applied associate-*l/11.1

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

    11. Simplified11.1

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

    if 4.494951878835627e+102 < d

    1. Initial program 40.3

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]

    2. Initial simplification40.3

      \[\leadsto \frac{(d \cdot \left(-a\right) + \left(b \cdot c\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt40.3

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

    5. Applied *-un-lft-identity40.3

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

    6. Applied times-frac40.3

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

    7. Simplified40.3

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

    8. Simplified27.4

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

    10. Applied associate-*l/27.4

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

    11. Simplified27.4

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

    12. Taylor expanded around inf 17.0

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

    13. Simplified17.0

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

  4. Final simplification12.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -4.681990609948178 \cdot 10^{+154}:\\ \;\;\;\;\frac{a}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;d \le 4.494951878835627 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{(\left(-d\right) \cdot a + \left(c \cdot b\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Runtime

Time bar (total: 24.8s)Debug logProfile

herbie shell --seed 2018221 +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))))