Average Error: 51.4 → 51.4
Time: 18.2s
Precision: binary64
\[\frac{\left(\left({b}^{2} \cdot \left(\log \left(\left|b + 2 \cdot \left(\sqrt{a} \cdot \sqrt{c}\right)\right|\right) - \log \left(\left|\left(2 \cdot a + 2 \cdot \left(\sqrt{\left(a + b\right) + c} \cdot \sqrt{a}\right)\right) + b\right|\right)\right) - 4 \cdot \left(a \cdot \left(c \cdot \left(\log \left(\left|b + 2 \cdot \left(\sqrt{a} \cdot \sqrt{c}\right)\right|\right) - \log \left(\left|\left(2 \cdot a + 2 \cdot \left(\sqrt{\left(a + b\right) + c} \cdot \sqrt{a}\right)\right) + b\right|\right)\right)\right)\right)\right) - 2 \cdot \left(\sqrt{a} \cdot \left(b \cdot \sqrt{c}\right)\right)\right) + 2 \cdot \left(\sqrt{a} \cdot \left(\left(2 \cdot a + b\right) \cdot \sqrt{\left(a + b\right) + c}\right)\right)}{8 \cdot {a}^{\left(\frac{3}{2}\right)}}\]
↓
\[\frac{\frac{\left(2 \cdot \left(\sqrt{a} \cdot \left(\left(2 \cdot a + b\right) \cdot \sqrt{\left(a + b\right) + c}\right)\right) + \left(-2 \cdot \left(\sqrt{a} \cdot \left(b \cdot \sqrt{c}\right)\right)\right)\right) + \left(\log \left(\left|b + 2 \cdot \left(\sqrt{a} \cdot \sqrt{c}\right)\right|\right) - \log \left(\left|\left(2 \cdot a + 2 \cdot \left(\sqrt{\left(a + b\right) + c} \cdot \sqrt{a}\right)\right) + b\right|\right)\right) \cdot \left({b}^{2} - \left(4 \cdot a\right) \cdot c\right)}{8 \cdot {a}^{\left(\frac{\frac{3}{2}}{2}\right)}}}{{a}^{\left(\frac{\frac{3}{2}}{2}\right)}}\]

Error

Bits error versus b

Bits error versus a

Bits error versus c

Derivation

  1. Initial program 51.4

    \[\frac{\left(\left({b}^{2} \cdot \left(\log \left(\left|b + 2 \cdot \left(\sqrt{a} \cdot \sqrt{c}\right)\right|\right) - \log \left(\left|\left(2 \cdot a + 2 \cdot \left(\sqrt{\left(a + b\right) + c} \cdot \sqrt{a}\right)\right) + b\right|\right)\right) - 4 \cdot \left(a \cdot \left(c \cdot \left(\log \left(\left|b + 2 \cdot \left(\sqrt{a} \cdot \sqrt{c}\right)\right|\right) - \log \left(\left|\left(2 \cdot a + 2 \cdot \left(\sqrt{\left(a + b\right) + c} \cdot \sqrt{a}\right)\right) + b\right|\right)\right)\right)\right)\right) - 2 \cdot \left(\sqrt{a} \cdot \left(b \cdot \sqrt{c}\right)\right)\right) + 2 \cdot \left(\sqrt{a} \cdot \left(\left(2 \cdot a + b\right) \cdot \sqrt{\left(a + b\right) + c}\right)\right)}{8 \cdot {a}^{\left(\frac{3}{2}\right)}}\]

  2. Simplified51.4

    \[\leadsto \color{blue}{\frac{\frac{\left(2 \cdot \left(\sqrt{a} \cdot \left(\left(2 \cdot a + b\right) \cdot \sqrt{\left(a + b\right) + c}\right)\right) + \left(-2 \cdot \left(\sqrt{a} \cdot \left(b \cdot \sqrt{c}\right)\right)\right)\right) + \left(\log \left(\left|b + 2 \cdot \left(\sqrt{a} \cdot \sqrt{c}\right)\right|\right) - \log \left(\left|\left(2 \cdot a + 2 \cdot \left(\sqrt{\left(a + b\right) + c} \cdot \sqrt{a}\right)\right) + b\right|\right)\right) \cdot \left({b}^{2} - \left(4 \cdot a\right) \cdot c\right)}{8 \cdot {a}^{\left(\frac{\frac{3}{2}}{2}\right)}}}{{a}^{\left(\frac{\frac{3}{2}}{2}\right)}}}\]

  3. Final simplification51.4

    \[\leadsto \frac{\frac{\left(2 \cdot \left(\sqrt{a} \cdot \left(\left(2 \cdot a + b\right) \cdot \sqrt{\left(a + b\right) + c}\right)\right) + \left(-2 \cdot \left(\sqrt{a} \cdot \left(b \cdot \sqrt{c}\right)\right)\right)\right) + \left(\log \left(\left|b + 2 \cdot \left(\sqrt{a} \cdot \sqrt{c}\right)\right|\right) - \log \left(\left|\left(2 \cdot a + 2 \cdot \left(\sqrt{\left(a + b\right) + c} \cdot \sqrt{a}\right)\right) + b\right|\right)\right) \cdot \left({b}^{2} - \left(4 \cdot a\right) \cdot c\right)}{8 \cdot {a}^{\left(\frac{\frac{3}{2}}{2}\right)}}}{{a}^{\left(\frac{\frac{3}{2}}{2}\right)}}\]

Reproduce

herbie shell --seed 2020153 
(FPCore (b a c)
  :name "(/ (+ (- (- (* (pow b 2) (- (log (fabs (+ b (* 2 (* (sqrt a) (sqrt c)))))) (log (fabs (+ (+ (* 2 a) (* 2 (* (sqrt (+ (+ a b) c)) (sqrt a)))) b))))) (* 4 (* a (* c (- (log (fabs (+ b (* 2 (* (sqrt a) (sqrt c)))))) (log (fabs (+ (+ (* 2 a) (* 2 (* (sqrt (+ (+ a b) c)) (sqrt a)))) b)))))))) (* 2 (* (sqrt a) (* b (sqrt c))))) (* 2 (* (sqrt a) (* (+ (* 2 a) b) (sqrt (+ (+ a b) c)))))) (* 8 (pow a (/ 3 2))))"
  :precision binary64
  (/ (+ (- (- (* (pow b 2.0) (- (log (fabs (+ b (* 2.0 (* (sqrt a) (sqrt c)))))) (log (fabs (+ (+ (* 2.0 a) (* 2.0 (* (sqrt (+ (+ a b) c)) (sqrt a)))) b))))) (* 4.0 (* a (* c (- (log (fabs (+ b (* 2.0 (* (sqrt a) (sqrt c)))))) (log (fabs (+ (+ (* 2.0 a) (* 2.0 (* (sqrt (+ (+ a b) c)) (sqrt a)))) b)))))))) (* 2.0 (* (sqrt a) (* b (sqrt c))))) (* 2.0 (* (sqrt a) (* (+ (* 2.0 a) b) (sqrt (+ (+ a b) c)))))) (* 8.0 (pow a (/ 3.0 2.0)))))