Average Error: 25.6 → 25.7
Time: 19.7s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{b \cdot c - d \cdot a}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\frac{b \cdot c - d \cdot a}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}
double f(double a, double b, double c, double d) {
        double r31630557 = b;
        double r31630558 = c;
        double r31630559 = r31630557 * r31630558;
        double r31630560 = a;
        double r31630561 = d;
        double r31630562 = r31630560 * r31630561;
        double r31630563 = r31630559 - r31630562;
        double r31630564 = r31630558 * r31630558;
        double r31630565 = r31630561 * r31630561;
        double r31630566 = r31630564 + r31630565;
        double r31630567 = r31630563 / r31630566;
        return r31630567;
}
↓
double f(double a, double b, double c, double d) {
        double r31630568 = b;
        double r31630569 = c;
        double r31630570 = r31630568 * r31630569;
        double r31630571 = d;
        double r31630572 = a;
        double r31630573 = r31630571 * r31630572;
        double r31630574 = r31630570 - r31630573;
        double r31630575 = r31630569 * r31630569;
        double r31630576 = fma(r31630571, r31630571, r31630575);
        double r31630577 = sqrt(r31630576);
        double r31630578 = r31630574 / r31630577;
        double r31630579 = 1.0;
        double r31630580 = r31630579 / r31630577;
        double r31630581 = r31630578 * r31630580;
        return r31630581;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original25.6
Target0.3
Herbie25.7
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 25.6

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

  2. Simplified25.6

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

  4. Applied add-sqr-sqrt25.7

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

  5. Applied *-un-lft-identity25.7

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

  6. Applied times-frac25.7

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

  7. Final simplification25.7

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

Reproduce

herbie shell --seed 2019112 +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))))