Average Error: 25.9 → 22.2
Time: 3.2s
Precision: binary64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{b \cdot \frac{c}{\sqrt{c \cdot c + d \cdot d}} - a \cdot \frac{d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\frac{b \cdot \frac{c}{\sqrt{c \cdot c + d \cdot d}} - a \cdot \frac{d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}
double code(double a, double b, double c, double d) {
	return ((double) (((double) (((double) (b * c)) - ((double) (a * d)))) / ((double) (((double) (c * c)) + ((double) (d * d))))));
}
↓
double code(double a, double b, double c, double d) {
	return ((double) (((double) (((double) (b * ((double) (c / ((double) sqrt(((double) (((double) (c * c)) + ((double) (d * d)))))))))) - ((double) (a * ((double) (d / ((double) sqrt(((double) (((double) (c * c)) + ((double) (d * d)))))))))))) / ((double) sqrt(((double) (((double) (c * c)) + ((double) (d * d))))))));
}

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

Original25.9
Target0.5
Herbie22.2
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 25.9

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

  3. Applied add-sqr-sqrt25.9

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

  4. Applied associate-/r*25.9

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

  6. Applied div-sub25.9

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

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

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

  9. Applied sqrt-prod25.9

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

  10. Applied times-frac24.2

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

  11. Simplified24.2

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

  13. Applied *-un-lft-identity24.2

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

  14. Applied sqrt-prod24.2

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

  15. Applied times-frac22.2

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

  16. Simplified22.2

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

  17. Final simplification22.2

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

Reproduce

herbie shell --seed 2020162 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

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

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