Average Error: 26.8 → 25.8
Time: 9.0s
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 r121058 = a;
        double r121059 = c;
        double r121060 = r121058 * r121059;
        double r121061 = b;
        double r121062 = d;
        double r121063 = r121061 * r121062;
        double r121064 = r121060 + r121063;
        double r121065 = r121059 * r121059;
        double r121066 = r121062 * r121062;
        double r121067 = r121065 + r121066;
        double r121068 = r121064 / r121067;
        return r121068;
}
↓
double f(double a, double b, double c, double d) {
        double r121069 = d;
        double r121070 = c;
        double r121071 = r121070 * r121070;
        double r121072 = r121069 * r121069;
        double r121073 = r121071 + r121072;
        double r121074 = cbrt(r121073);
        double r121075 = fabs(r121074);
        double r121076 = b;
        double r121077 = r121075 / r121076;
        double r121078 = r121069 / r121077;
        double r121079 = a;
        double r121080 = r121079 * r121070;
        double r121081 = r121080 / r121075;
        double r121082 = r121078 + r121081;
        double r121083 = sqrt(r121074);
        double r121084 = r121082 / r121083;
        double r121085 = sqrt(r121073);
        double r121086 = r121084 / r121085;
        return r121086;
}

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))))