Average Error: 25.4 → 25.4
Time: 13.5s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\frac{1}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}} \cdot \mathsf{fma}\left(b, c, d \cdot \left(-a\right)\right)}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\frac{\frac{1}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}} \cdot \mathsf{fma}\left(b, c, d \cdot \left(-a\right)\right)}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}
double f(double a, double b, double c, double d) {
        double r1722718 = b;
        double r1722719 = c;
        double r1722720 = r1722718 * r1722719;
        double r1722721 = a;
        double r1722722 = d;
        double r1722723 = r1722721 * r1722722;
        double r1722724 = r1722720 - r1722723;
        double r1722725 = r1722719 * r1722719;
        double r1722726 = r1722722 * r1722722;
        double r1722727 = r1722725 + r1722726;
        double r1722728 = r1722724 / r1722727;
        return r1722728;
}
↓
double f(double a, double b, double c, double d) {
        double r1722729 = 1.0;
        double r1722730 = d;
        double r1722731 = c;
        double r1722732 = r1722731 * r1722731;
        double r1722733 = fma(r1722730, r1722730, r1722732);
        double r1722734 = sqrt(r1722733);
        double r1722735 = r1722729 / r1722734;
        double r1722736 = b;
        double r1722737 = a;
        double r1722738 = -r1722737;
        double r1722739 = r1722730 * r1722738;
        double r1722740 = fma(r1722736, r1722731, r1722739);
        double r1722741 = r1722735 * r1722740;
        double r1722742 = r1722741 / r1722734;
        return r1722742;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original25.4
Target0.4
Herbie25.4
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 25.4

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

  2. Simplified25.4

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

  4. Applied add-sqr-sqrt25.4

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

  5. Applied associate-/r*25.3

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

  7. Applied fma-neg25.3

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

  9. Applied div-inv25.4

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

  10. Final simplification25.4

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

Reproduce

herbie shell --seed 2019152 +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))))