Average Error: 25.2 → 12.6
Time: 1.2m
Precision: 64
Internal Precision: 576
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le -8.198260356844967 \cdot 10^{+154}:\\ \;\;\;\;\frac{-b}{\sqrt{d^2 + c^2}^*}\\ \mathbf{if}\;d \le 3.85836339497122 \cdot 10^{+95}:\\ \;\;\;\;\frac{1}{\sqrt{d^2 + c^2}^*} \cdot \left((b \cdot d + \left(c \cdot a\right))_* \cdot \frac{1}{\sqrt{d^2 + c^2}^*}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original25.2
Target0.5
Herbie12.6
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if d < -8.198260356844967e+154

    1. Initial program 44.8

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

    2. Applied simplify44.8

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

    4. Applied add-sqr-sqrt44.8

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

    5. Applied *-un-lft-identity44.8

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

    6. Applied times-frac44.8

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

    7. Applied simplify44.8

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

    8. Applied simplify29.3

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

    9. Taylor expanded around -inf 12.8

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

    10. Applied simplify12.6

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

    if -8.198260356844967e+154 < d < 3.85836339497122e+95

    1. Initial program 18.3

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

    2. Applied simplify18.3

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

    4. Applied add-sqr-sqrt18.3

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

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

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

    6. Applied times-frac18.3

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

    7. Applied simplify18.3

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

    8. Applied simplify11.3

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

    10. Applied div-inv11.4

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

    if 3.85836339497122e+95 < d

    1. Initial program 36.8

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

    2. Applied simplify36.8

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

    4. Applied add-sqr-sqrt36.8

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

    5. Applied *-un-lft-identity36.8

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

    6. Applied times-frac36.8

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

    7. Applied simplify36.8

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

    8. Applied simplify23.9

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

    9. Taylor expanded around inf 16.8

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

    10. Applied simplify16.7

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018166 +o rules:numerics
(FPCore (a b c d)
  :name "Complex division, real part"

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

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