Average Error: 25.8 → 25.8
Time: 11.9s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{b \cdot c - d \cdot a}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}} \cdot \frac{1}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\frac{b \cdot c - d \cdot a}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}} \cdot \frac{1}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}
double f(double a, double b, double c, double d) {
        double r1752172 = b;
        double r1752173 = c;
        double r1752174 = r1752172 * r1752173;
        double r1752175 = a;
        double r1752176 = d;
        double r1752177 = r1752175 * r1752176;
        double r1752178 = r1752174 - r1752177;
        double r1752179 = r1752173 * r1752173;
        double r1752180 = r1752176 * r1752176;
        double r1752181 = r1752179 + r1752180;
        double r1752182 = r1752178 / r1752181;
        return r1752182;
}
↓
double f(double a, double b, double c, double d) {
        double r1752183 = b;
        double r1752184 = c;
        double r1752185 = r1752183 * r1752184;
        double r1752186 = d;
        double r1752187 = a;
        double r1752188 = r1752186 * r1752187;
        double r1752189 = r1752185 - r1752188;
        double r1752190 = r1752184 * r1752184;
        double r1752191 = fma(r1752186, r1752186, r1752190);
        double r1752192 = sqrt(r1752191);
        double r1752193 = r1752189 / r1752192;
        double r1752194 = 1.0;
        double r1752195 = r1752194 / r1752192;
        double r1752196 = r1752193 * r1752195;
        return r1752196;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original25.8
Target0.5
Herbie25.8
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 25.8

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

  2. Simplified25.8

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

  4. Applied add-sqr-sqrt25.8

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

  5. Applied *-un-lft-identity25.8

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

  6. Applied times-frac25.8

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

  7. Final simplification25.8

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

Reproduce

herbie shell --seed 2019155 +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))))