Average Error: 25.6 → 3.3
Time: 11.0s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{b}{\frac{d}{\frac{c}{d}} + c} - \frac{a}{\frac{c}{\frac{d}{c}} + d}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\frac{b}{\frac{d}{\frac{c}{d}} + c} - \frac{a}{\frac{c}{\frac{d}{c}} + d}
double f(double a, double b, double c, double d) {
        double r94207 = b;
        double r94208 = c;
        double r94209 = r94207 * r94208;
        double r94210 = a;
        double r94211 = d;
        double r94212 = r94210 * r94211;
        double r94213 = r94209 - r94212;
        double r94214 = r94208 * r94208;
        double r94215 = r94211 * r94211;
        double r94216 = r94214 + r94215;
        double r94217 = r94213 / r94216;
        return r94217;
}
↓
double f(double a, double b, double c, double d) {
        double r94218 = b;
        double r94219 = d;
        double r94220 = c;
        double r94221 = r94220 / r94219;
        double r94222 = r94219 / r94221;
        double r94223 = r94222 + r94220;
        double r94224 = r94218 / r94223;
        double r94225 = a;
        double r94226 = r94219 / r94220;
        double r94227 = r94220 / r94226;
        double r94228 = r94227 + r94219;
        double r94229 = r94225 / r94228;
        double r94230 = r94224 - r94229;
        return r94230;
}

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

Original25.6
Target0.4
Herbie3.3
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 25.6

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

  3. Applied div-sub25.6

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

  4. Simplified24.1

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

  5. Taylor expanded around 0 17.2

    \[\leadsto \frac{b \cdot c}{c \cdot c + d \cdot d} - \frac{a}{\color{blue}{\frac{{c}^{2}}{d} + d}}\]

  6. Simplified17.2

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

  8. Applied *-un-lft-identity17.2

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

  9. Applied *-un-lft-identity17.2

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

  10. Applied distribute-lft-out--17.2

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

  11. Simplified13.8

    \[\leadsto 1 \cdot \color{blue}{\left(\frac{b}{\frac{{c}^{2} + d \cdot d}{c}} - \frac{a}{d + \frac{c}{\frac{d}{c}}}\right)}\]

  12. Taylor expanded around 0 5.3

    \[\leadsto 1 \cdot \left(\frac{b}{\color{blue}{\frac{{d}^{2}}{c} + c}} - \frac{a}{d + \frac{c}{\frac{d}{c}}}\right)\]

  13. Simplified3.3

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

  14. Final simplification3.3

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

Reproduce

herbie shell --seed 2019194 
(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))))