Average Error: 25.7 → 25.7
Time: 17.3s
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot d + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}\]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot d + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}
double f(double a, double b, double c, double d) {
        double r2427423 = a;
        double r2427424 = c;
        double r2427425 = r2427423 * r2427424;
        double r2427426 = b;
        double r2427427 = d;
        double r2427428 = r2427426 * r2427427;
        double r2427429 = r2427425 + r2427428;
        double r2427430 = r2427424 * r2427424;
        double r2427431 = r2427427 * r2427427;
        double r2427432 = r2427430 + r2427431;
        double r2427433 = r2427429 / r2427432;
        return r2427433;
}
↓
double f(double a, double b, double c, double d) {
        double r2427434 = 1.0;
        double r2427435 = c;
        double r2427436 = r2427435 * r2427435;
        double r2427437 = d;
        double r2427438 = r2427437 * r2427437;
        double r2427439 = r2427436 + r2427438;
        double r2427440 = sqrt(r2427439);
        double r2427441 = r2427434 / r2427440;
        double r2427442 = b;
        double r2427443 = r2427442 * r2427437;
        double r2427444 = a;
        double r2427445 = r2427444 * r2427435;
        double r2427446 = r2427443 + r2427445;
        double r2427447 = r2427446 / r2427440;
        double r2427448 = r2427441 * r2427447;
        return r2427448;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original25.7
Target0.4
Herbie25.7
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 25.7

    \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt25.7

    \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

  4. Applied associate-/r*25.7

    \[\leadsto \color{blue}{\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
  5. Using strategy rm

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

    \[\leadsto \frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\color{blue}{1 \cdot \sqrt{c \cdot c + d \cdot d}}}\]

  7. Applied associate-/r*25.7

    \[\leadsto \color{blue}{\frac{\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{1}}{\sqrt{c \cdot c + d \cdot d}}}\]

  8. Simplified25.7

    \[\leadsto \frac{\color{blue}{\frac{b \cdot d + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}}}{\sqrt{c \cdot c + d \cdot d}}\]
  9. Using strategy rm

  10. Applied add-sqr-sqrt25.7

    \[\leadsto \frac{\frac{b \cdot d + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}}\]

  11. Applied sqrt-prod25.8

    \[\leadsto \frac{\frac{b \cdot d + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}}{\color{blue}{\sqrt{\sqrt{c \cdot c + d \cdot d}} \cdot \sqrt{\sqrt{c \cdot c + d \cdot d}}}}\]

  12. Applied add-sqr-sqrt25.8

    \[\leadsto \frac{\frac{b \cdot d + a \cdot c}{\sqrt{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}}}{\sqrt{\sqrt{c \cdot c + d \cdot d}} \cdot \sqrt{\sqrt{c \cdot c + d \cdot d}}}\]

  13. Applied sqrt-prod25.9

    \[\leadsto \frac{\frac{b \cdot d + a \cdot c}{\color{blue}{\sqrt{\sqrt{c \cdot c + d \cdot d}} \cdot \sqrt{\sqrt{c \cdot c + d \cdot d}}}}}{\sqrt{\sqrt{c \cdot c + d \cdot d}} \cdot \sqrt{\sqrt{c \cdot c + d \cdot d}}}\]

  14. Applied *-un-lft-identity25.9

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

  15. Applied times-frac26.0

    \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt{\sqrt{c \cdot c + d \cdot d}}} \cdot \frac{b \cdot d + a \cdot c}{\sqrt{\sqrt{c \cdot c + d \cdot d}}}}}{\sqrt{\sqrt{c \cdot c + d \cdot d}} \cdot \sqrt{\sqrt{c \cdot c + d \cdot d}}}\]

  16. Applied times-frac26.0

    \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\sqrt{c \cdot c + d \cdot d}}}}{\sqrt{\sqrt{c \cdot c + d \cdot d}}} \cdot \frac{\frac{b \cdot d + a \cdot c}{\sqrt{\sqrt{c \cdot c + d \cdot d}}}}{\sqrt{\sqrt{c \cdot c + d \cdot d}}}}\]

  17. Simplified25.9

    \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}}} \cdot \frac{\frac{b \cdot d + a \cdot c}{\sqrt{\sqrt{c \cdot c + d \cdot d}}}}{\sqrt{\sqrt{c \cdot c + d \cdot d}}}\]

  18. Simplified25.7

    \[\leadsto \frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \color{blue}{\frac{c \cdot a + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]

  19. Final simplification25.7

    \[\leadsto \frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot d + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (a b c d)
  :name "Complex division, real part"

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d)))))

  (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))