Complex division, imag part

Average Error: 26.1 → 22.8

Time: 20.1s

Precision: 64

Math

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r3839539 = b;
        double r3839540 = c;
        double r3839541 = r3839539 * r3839540;
        double r3839542 = a;
        double r3839543 = d;
        double r3839544 = r3839542 * r3839543;
        double r3839545 = r3839541 - r3839544;
        double r3839546 = r3839540 * r3839540;
        double r3839547 = r3839543 * r3839543;
        double r3839548 = r3839546 + r3839547;
        double r3839549 = r3839545 / r3839548;
        return r3839549;
}

↓

double f(double a, double b, double c, double d) {
        double r3839550 = b;
        double r3839551 = c;
        double r3839552 = d;
        double r3839553 = r3839551 * r3839551;
        double r3839554 = fma(r3839552, r3839552, r3839553);
        double r3839555 = sqrt(r3839554);
        double r3839556 = r3839551 / r3839555;
        double r3839557 = r3839550 * r3839556;
        double r3839558 = r3839552 / r3839555;
        double r3839559 = a;
        double r3839560 = r3839558 * r3839559;
        double r3839561 = r3839557 - r3839560;
        double r3839562 = r3839561 / r3839555;
        return r3839562;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.1
Target	0.5
Herbie	22.8

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

Derivation

1. Initial program 26.1

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

2. Simplified 26.1

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

4. Applied add-sqr-sqrt 26.1

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

5. Applied associate-/r* 26.1

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

7. Applied div-sub 26.1

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

9. Applied *-un-lft-identity 26.1

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

10. Applied sqrt-prod 26.1

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

11. Applied times-frac 24.5

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

12. Simplified 24.5

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

14. Applied *-un-lft-identity 24.5

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

15. Applied sqrt-prod 24.5

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

16. Applied times-frac 22.8

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

17. Simplified 22.8

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

18. Final simplification 22.8

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

Reproduce

herbie shell --seed 2019133 +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))))