Average Error: 26.1 → 26.0
Time: 16.5s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\frac{b \cdot c - a \cdot d}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\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)}}
double f(double a, double b, double c, double d) {
        double r3768276 = b;
        double r3768277 = c;
        double r3768278 = r3768276 * r3768277;
        double r3768279 = a;
        double r3768280 = d;
        double r3768281 = r3768279 * r3768280;
        double r3768282 = r3768278 - r3768281;
        double r3768283 = r3768277 * r3768277;
        double r3768284 = r3768280 * r3768280;
        double r3768285 = r3768283 + r3768284;
        double r3768286 = r3768282 / r3768285;
        return r3768286;
}
↓
double f(double a, double b, double c, double d) {
        double r3768287 = b;
        double r3768288 = c;
        double r3768289 = r3768287 * r3768288;
        double r3768290 = a;
        double r3768291 = d;
        double r3768292 = r3768290 * r3768291;
        double r3768293 = r3768289 - r3768292;
        double r3768294 = r3768288 * r3768288;
        double r3768295 = fma(r3768291, r3768291, r3768294);
        double r3768296 = sqrt(r3768295);
        double r3768297 = r3768293 / r3768296;
        double r3768298 = r3768297 / r3768296;
        return r3768298;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original26.1
Target0.4
Herbie26.0
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 26.1

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

  2. Simplified26.1

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

  4. Applied add-sqr-sqrt26.1

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

  5. Applied associate-/r*26.0

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

  6. Final simplification26.0

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

Reproduce

herbie shell --seed 2019163 +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))))