Average Error: 26.1 → 25.3
Time: 3.6s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\frac{b}{\left|e^{\log \left(\sqrt[3]{c \cdot c + d \cdot d}\right)}\right|} \cdot \frac{c}{\sqrt{\sqrt[3]{c \cdot c + d \cdot d}}} - \frac{a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\frac{\frac{b}{\left|e^{\log \left(\sqrt[3]{c \cdot c + d \cdot d}\right)}\right|} \cdot \frac{c}{\sqrt{\sqrt[3]{c \cdot c + d \cdot d}}} - \frac{a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}
double f(double a, double b, double c, double d) {
        double r112657 = b;
        double r112658 = c;
        double r112659 = r112657 * r112658;
        double r112660 = a;
        double r112661 = d;
        double r112662 = r112660 * r112661;
        double r112663 = r112659 - r112662;
        double r112664 = r112658 * r112658;
        double r112665 = r112661 * r112661;
        double r112666 = r112664 + r112665;
        double r112667 = r112663 / r112666;
        return r112667;
}
↓
double f(double a, double b, double c, double d) {
        double r112668 = b;
        double r112669 = c;
        double r112670 = r112669 * r112669;
        double r112671 = d;
        double r112672 = r112671 * r112671;
        double r112673 = r112670 + r112672;
        double r112674 = cbrt(r112673);
        double r112675 = log(r112674);
        double r112676 = exp(r112675);
        double r112677 = fabs(r112676);
        double r112678 = r112668 / r112677;
        double r112679 = sqrt(r112674);
        double r112680 = r112669 / r112679;
        double r112681 = r112678 * r112680;
        double r112682 = a;
        double r112683 = r112682 * r112671;
        double r112684 = sqrt(r112673);
        double r112685 = r112683 / r112684;
        double r112686 = r112681 - r112685;
        double r112687 = r112686 / r112684;
        return r112687;
}

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

Original26.1
Target0.5
Herbie25.3
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 26.1

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

  3. Applied add-sqr-sqrt26.1

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

  4. Applied associate-/r*26.0

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

  6. Applied div-sub26.0

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

  8. Applied add-cube-cbrt26.2

    \[\leadsto \frac{\frac{b \cdot c}{\sqrt{\color{blue}{\left(\sqrt[3]{c \cdot c + d \cdot d} \cdot \sqrt[3]{c \cdot c + d \cdot d}\right) \cdot \sqrt[3]{c \cdot c + d \cdot d}}}} - \frac{a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\]

  9. Applied sqrt-prod26.2

    \[\leadsto \frac{\frac{b \cdot c}{\color{blue}{\sqrt{\sqrt[3]{c \cdot c + d \cdot d} \cdot \sqrt[3]{c \cdot c + d \cdot d}} \cdot \sqrt{\sqrt[3]{c \cdot c + d \cdot d}}}} - \frac{a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\]

  10. Applied times-frac24.5

    \[\leadsto \frac{\color{blue}{\frac{b}{\sqrt{\sqrt[3]{c \cdot c + d \cdot d} \cdot \sqrt[3]{c \cdot c + d \cdot d}}} \cdot \frac{c}{\sqrt{\sqrt[3]{c \cdot c + d \cdot d}}}} - \frac{a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\]

  11. Simplified24.5

    \[\leadsto \frac{\color{blue}{\frac{b}{\left|\sqrt[3]{c \cdot c + d \cdot d}\right|}} \cdot \frac{c}{\sqrt{\sqrt[3]{c \cdot c + d \cdot d}}} - \frac{a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\]
  12. Using strategy rm

  13. Applied add-exp-log25.3

    \[\leadsto \frac{\frac{b}{\left|\color{blue}{e^{\log \left(\sqrt[3]{c \cdot c + d \cdot d}\right)}}\right|} \cdot \frac{c}{\sqrt{\sqrt[3]{c \cdot c + d \cdot d}}} - \frac{a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\]

  14. Final simplification25.3

    \[\leadsto \frac{\frac{b}{\left|e^{\log \left(\sqrt[3]{c \cdot c + d \cdot d}\right)}\right|} \cdot \frac{c}{\sqrt{\sqrt[3]{c \cdot c + d \cdot d}}} - \frac{a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\]

Reproduce

herbie shell --seed 2019354 
(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)))) (/ (+ (- a) (* b (/ c d))) (+ d (* c (/ c d)))))

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