Average Error: 25.9 → 13.1
Time: 1.2m
Precision: 64
Internal Precision: 576
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{c^2 + d^2}^*} \cdot \left(\frac{1}{\sqrt{\sqrt{c^2 + d^2}^*}} \cdot \frac{(b \cdot d + \left(c \cdot a\right))_*}{\sqrt{\sqrt{c^2 + d^2}^*}}\right) \le -1.7776580676770771 \cdot 10^{+308}:\\ \;\;\;\;\frac{-b}{\sqrt{c^2 + d^2}^*}\\ \mathbf{if}\;\frac{1}{\sqrt{c^2 + d^2}^*} \cdot \left(\frac{1}{\sqrt{\sqrt{c^2 + d^2}^*}} \cdot \frac{(b \cdot d + \left(c \cdot a\right))_*}{\sqrt{\sqrt{c^2 + d^2}^*}}\right) \le 1.7822238799756147 \cdot 10^{+308}:\\ \;\;\;\;\frac{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\sqrt{c^2 + d^2}^*}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original25.9
Target0.5
Herbie13.1
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* (/ 1 (hypot c d)) (* (/ 1 (sqrt (hypot c d))) (/ (fma b d (* c a)) (sqrt (hypot c d))))) < -1.7776580676770771e+308

    1. Initial program 63.0

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

    3. Applied add-sqr-sqrt63.0

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

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

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

    5. Applied times-frac63.0

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

    6. Applied simplify63.0

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

    7. Applied simplify62.0

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

    8. Taylor expanded around -inf 48.0

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

    9. Applied simplify47.9

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

    if -1.7776580676770771e+308 < (* (/ 1 (hypot c d)) (* (/ 1 (sqrt (hypot c d))) (/ (fma b d (* c a)) (sqrt (hypot c d))))) < 1.7822238799756147e+308

    1. Initial program 13.7

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

    3. Applied add-sqr-sqrt13.7

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

    4. Applied *-un-lft-identity13.7

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

    5. Applied times-frac13.7

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

    6. Applied simplify13.7

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

    7. Applied simplify1.5

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

    9. Applied associate-*r/1.4

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

    10. Applied simplify1.3

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

    if 1.7822238799756147e+308 < (* (/ 1 (hypot c d)) (* (/ 1 (sqrt (hypot c d))) (/ (fma b d (* c a)) (sqrt (hypot c d)))))

    1. Initial program 62.0

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

    3. Applied add-sqr-sqrt62.0

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

    4. Applied *-un-lft-identity62.0

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

    5. Applied times-frac62.0

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

    6. Applied simplify62.0

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

    7. Applied simplify62.0

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

    8. Taylor expanded around 0 49.0

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

    9. Applied simplify48.9

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018199 +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))))