Cubic critical, wide range

Average Error: 52.9 → 52.1

Time: 39.0s

Precision: 64

• unused variable d

\[4.930380657631324 \cdot 10^{-32} \lt a \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt b \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt c \lt 2.028240960365167 \cdot 10^{+31}\]

Math

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

↓

\[\sqrt[3]{\left(\frac{(\left(\sqrt{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{3 \cdot a} \cdot \frac{(\left(\sqrt{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{3 \cdot a}\right) \cdot \frac{(\left(\sqrt{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{3 \cdot a}}\]

C

double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r14617642 = b;
        double r14617643 = -r14617642;
        double r14617644 = r14617642 * r14617642;
        double r14617645 = 3.0;
        double r14617646 = a;
        double r14617647 = r14617645 * r14617646;
        double r14617648 = c;
        double r14617649 = r14617647 * r14617648;
        double r14617650 = r14617644 - r14617649;
        double r14617651 = sqrt(r14617650);
        double r14617652 = r14617643 + r14617651;
        double r14617653 = r14617652 / r14617647;
        return r14617653;
}

↓

double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r14617654 = -3.0;
        double r14617655 = a;
        double r14617656 = c;
        double r14617657 = r14617655 * r14617656;
        double r14617658 = b;
        double r14617659 = r14617658 * r14617658;
        double r14617660 = fma(r14617654, r14617657, r14617659);
        double r14617661 = sqrt(r14617660);
        double r14617662 = sqrt(r14617661);
        double r14617663 = -r14617658;
        double r14617664 = fma(r14617662, r14617662, r14617663);
        double r14617665 = 3.0;
        double r14617666 = r14617665 * r14617655;
        double r14617667 = r14617664 / r14617666;
        double r14617668 = r14617667 * r14617667;
        double r14617669 = r14617668 * r14617667;
        double r14617670 = cbrt(r14617669);
        return r14617670;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

1. Initial program 52.9

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

2. Simplified 52.9

   \[\leadsto \color{blue}{\frac{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}{3 \cdot a}}\]
3. Using strategy rm

4. Applied add-sqr-sqrt 52.7

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

5. Applied fma-neg 52.1

   \[\leadsto \frac{\color{blue}{(\left(\sqrt{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}}{3 \cdot a}\]
6. Using strategy rm

7. Applied add-cbrt-cube 52.1

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

8. Final simplification 52.1

   \[\leadsto \sqrt[3]{\left(\frac{(\left(\sqrt{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{3 \cdot a} \cdot \frac{(\left(\sqrt{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{3 \cdot a}\right) \cdot \frac{(\left(\sqrt{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{3 \cdot a}}\]

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))