Average Error: 25.6 → 13.2
Time: 1.1m
Precision: 64
Internal Precision: 576
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{d^2 + c^2}^*} \cdot \left((b \cdot d + \left(c \cdot a\right))_* \cdot \frac{1}{\sqrt{d^2 + c^2}^*}\right) \le -1.7829501442338802 \cdot 10^{+308}:\\ \;\;\;\;\frac{-b}{\sqrt{d^2 + c^2}^*}\\ \mathbf{if}\;\frac{1}{\sqrt{d^2 + c^2}^*} \cdot \left((b \cdot d + \left(c \cdot a\right))_* \cdot \frac{1}{\sqrt{d^2 + c^2}^*}\right) \le 1.7748952502892744 \cdot 10^{+308}:\\ \;\;\;\;\frac{1}{\sqrt{d^2 + c^2}^*} \cdot \frac{(b \cdot d + \left(c \cdot a\right))_*}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original25.6
Target0.6
Herbie13.2
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* (/ 1 (hypot d c)) (* (fma b d (* c a)) (/ 1 (hypot d c)))) < -1.7829501442338802e+308

    1. Initial program 63.0

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

    2. Applied simplify63.0

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

    4. Applied add-sqr-sqrt63.0

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

    5. Applied *-un-lft-identity63.0

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

    6. Applied times-frac63.0

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

    7. Applied simplify63.0

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

    8. Applied simplify62.0

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

    9. Taylor expanded around -inf 49.0

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

    10. Applied simplify49.0

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

    if -1.7829501442338802e+308 < (* (/ 1 (hypot d c)) (* (fma b d (* c a)) (/ 1 (hypot d c)))) < 1.7748952502892744e+308

    1. Initial program 13.3

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

    2. Applied simplify13.3

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

    4. Applied add-sqr-sqrt13.3

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

    5. Applied *-un-lft-identity13.3

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

    6. Applied times-frac13.3

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

    7. Applied simplify13.3

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

    8. Applied simplify1.5

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

    if 1.7748952502892744e+308 < (* (/ 1 (hypot d c)) (* (fma b d (* c a)) (/ 1 (hypot d c))))

    1. Initial program 61.9

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

    2. Applied simplify61.9

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

    4. Applied add-sqr-sqrt61.9

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

    5. Applied *-un-lft-identity61.9

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

    6. Applied times-frac61.9

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

    7. Applied simplify61.9

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

    8. Applied simplify61.9

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

    9. Taylor expanded around 0 47.4

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

    10. Applied simplify47.3

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018198 +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))))