Average Error: 26.8 → 25.8
Time: 9.5s
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\frac{\frac{d}{\frac{\left|\sqrt[3]{c \cdot c + d \cdot d}\right|}{b}} + \frac{a \cdot c}{\left|\sqrt[3]{c \cdot c + d \cdot d}\right|}}{\sqrt{\sqrt[3]{c \cdot c + d \cdot d}}}}{\sqrt{c \cdot c + d \cdot d}}\]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\frac{\frac{\frac{d}{\frac{\left|\sqrt[3]{c \cdot c + d \cdot d}\right|}{b}} + \frac{a \cdot c}{\left|\sqrt[3]{c \cdot c + d \cdot d}\right|}}{\sqrt{\sqrt[3]{c \cdot c + d \cdot d}}}}{\sqrt{c \cdot c + d \cdot d}}
double f(double a, double b, double c, double d) {
        double r146294 = a;
        double r146295 = c;
        double r146296 = r146294 * r146295;
        double r146297 = b;
        double r146298 = d;
        double r146299 = r146297 * r146298;
        double r146300 = r146296 + r146299;
        double r146301 = r146295 * r146295;
        double r146302 = r146298 * r146298;
        double r146303 = r146301 + r146302;
        double r146304 = r146300 / r146303;
        return r146304;
}
↓
double f(double a, double b, double c, double d) {
        double r146305 = d;
        double r146306 = c;
        double r146307 = r146306 * r146306;
        double r146308 = r146305 * r146305;
        double r146309 = r146307 + r146308;
        double r146310 = cbrt(r146309);
        double r146311 = fabs(r146310);
        double r146312 = b;
        double r146313 = r146311 / r146312;
        double r146314 = r146305 / r146313;
        double r146315 = a;
        double r146316 = r146315 * r146306;
        double r146317 = r146316 / r146311;
        double r146318 = r146314 + r146317;
        double r146319 = sqrt(r146310);
        double r146320 = r146318 / r146319;
        double r146321 = sqrt(r146309);
        double r146322 = r146320 / r146321;
        return r146322;
}

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

Original26.8
Target0.5
Herbie25.8
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 26.8

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

  3. Applied add-sqr-sqrt26.8

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

  4. Applied associate-/r*26.7

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

  6. Applied add-cube-cbrt27.0

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

  7. Applied sqrt-prod27.0

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

  8. Applied associate-/r*27.0

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

  9. Simplified27.0

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

  10. Taylor expanded around 0 28.7

    \[\leadsto \frac{\frac{\color{blue}{\frac{a \cdot c}{\left|{\left({d}^{2} + {c}^{2}\right)}^{\frac{1}{3}}\right|} + \frac{d \cdot b}{\left|{\left({d}^{2} + {c}^{2}\right)}^{\frac{1}{3}}\right|}}}{\sqrt{\sqrt[3]{c \cdot c + d \cdot d}}}}{\sqrt{c \cdot c + d \cdot d}}\]

  11. Simplified25.8

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

  12. Final simplification25.8

    \[\leadsto \frac{\frac{\frac{d}{\frac{\left|\sqrt[3]{c \cdot c + d \cdot d}\right|}{b}} + \frac{a \cdot c}{\left|\sqrt[3]{c \cdot c + d \cdot d}\right|}}{\sqrt{\sqrt[3]{c \cdot c + d \cdot d}}}}{\sqrt{c \cdot c + d \cdot d}}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :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))))