Average Error: 26.6 → 13.6
Time: 4.4s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le -1033969720.49365747:\\ \;\;\;\;\frac{b}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{c}} - \frac{a}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{d}}\\ \mathbf{elif}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le 7.8735811135706561 \cdot 10^{304}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1}} \cdot \left(\sqrt{\frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1}} \cdot \frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\begin{array}{l}
\mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le -1033969720.49365747:\\
\;\;\;\;\frac{b}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{c}} - \frac{a}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{d}}\\

\mathbf{elif}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le 7.8735811135706561 \cdot 10^{304}:\\
\;\;\;\;\sqrt{\frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1}} \cdot \left(\sqrt{\frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1}} \cdot \frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{-1 \cdot a}{\mathsf{hypot}\left(c, d\right)}\\

\end{array}
double f(double a, double b, double c, double d) {
        double r118640 = b;
        double r118641 = c;
        double r118642 = r118640 * r118641;
        double r118643 = a;
        double r118644 = d;
        double r118645 = r118643 * r118644;
        double r118646 = r118642 - r118645;
        double r118647 = r118641 * r118641;
        double r118648 = r118644 * r118644;
        double r118649 = r118647 + r118648;
        double r118650 = r118646 / r118649;
        return r118650;
}
↓
double f(double a, double b, double c, double d) {
        double r118651 = b;
        double r118652 = c;
        double r118653 = r118651 * r118652;
        double r118654 = a;
        double r118655 = d;
        double r118656 = r118654 * r118655;
        double r118657 = r118653 - r118656;
        double r118658 = r118652 * r118652;
        double r118659 = r118655 * r118655;
        double r118660 = r118658 + r118659;
        double r118661 = r118657 / r118660;
        double r118662 = -1033969720.4936575;
        bool r118663 = r118661 <= r118662;
        double r118664 = fma(r118652, r118652, r118659);
        double r118665 = r118664 / r118652;
        double r118666 = r118651 / r118665;
        double r118667 = r118664 / r118655;
        double r118668 = r118654 / r118667;
        double r118669 = r118666 - r118668;
        double r118670 = 7.873581113570656e+304;
        bool r118671 = r118661 <= r118670;
        double r118672 = 1.0;
        double r118673 = hypot(r118652, r118655);
        double r118674 = r118673 * r118672;
        double r118675 = r118672 / r118674;
        double r118676 = sqrt(r118675);
        double r118677 = r118657 / r118673;
        double r118678 = r118676 * r118677;
        double r118679 = r118676 * r118678;
        double r118680 = -1.0;
        double r118681 = r118680 * r118654;
        double r118682 = r118681 / r118673;
        double r118683 = r118671 ? r118679 : r118682;
        double r118684 = r118663 ? r118669 : r118683;
        return r118684;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original26.6
Target0.5
Herbie13.6
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (- (* b c) (* a d)) (+ (* c c) (* d d))) < -1033969720.4936575

    1. Initial program 17.1

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

    3. Applied div-sub17.1

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

    4. Simplified11.1

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

    5. Simplified5.8

      \[\leadsto \frac{b}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{c}} - \color{blue}{\frac{a}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{d}}}\]

    if -1033969720.4936575 < (/ (- (* b c) (* a d)) (+ (* c c) (* d d))) < 7.873581113570656e+304

    1. Initial program 13.4

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

    3. Applied add-sqr-sqrt13.4

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

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

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

    5. Applied times-frac13.4

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

    6. Simplified13.4

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

    7. Simplified0.9

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}\]
    8. Using strategy rm

    9. Applied add-sqr-sqrt1.1

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

    10. Applied associate-*l*1.1

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

    if 7.873581113570656e+304 < (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))

    1. Initial program 63.7

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

    3. Applied add-sqr-sqrt63.7

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

    4. Applied *-un-lft-identity63.7

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

    5. Applied times-frac63.7

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

    6. Simplified63.7

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

    7. Simplified61.2

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}\]
    8. Using strategy rm

    9. Applied associate-*r/61.2

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

    10. Simplified61.2

      \[\leadsto \frac{\color{blue}{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}}{\mathsf{hypot}\left(c, d\right)}\]

    11. Taylor expanded around 0 48.5

      \[\leadsto \frac{\color{blue}{-1 \cdot a}}{\mathsf{hypot}\left(c, d\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification13.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le -1033969720.49365747:\\ \;\;\;\;\frac{b}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{c}} - \frac{a}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{d}}\\ \mathbf{elif}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le 7.8735811135706561 \cdot 10^{304}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1}} \cdot \left(\sqrt{\frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1}} \cdot \frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :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))))