Average Error: 54.3 → 54.3
Time: 1.1m
Precision: binary64
\[\left(\frac{-\left(\frac{\left(-3\right) \cdot c}{a} + \frac{{b}^{2}}{{a}^{2}}\right)}{3 \cdot {\left(\left(\left(\frac{\sqrt{\left(-4\right) \cdot {\left(\frac{\left(-3\right) \cdot c}{a} + \frac{{b}^{2}}{{a}^{2}}\right)}^{3} + {\left(\left(\frac{27 \cdot d}{a} - \frac{\left(9 \cdot b\right) \cdot c}{{a}^{2}}\right) + \frac{2 \cdot {b}^{3}}{{a}^{3}}\right)}^{2}}}{2} + \frac{27 \cdot d}{2 \cdot a}\right) - \frac{\left(9 \cdot b\right) \cdot c}{2 \cdot {a}^{2}}\right) + \frac{{b}^{3}}{{a}^{3}}\right)}^{\left(\frac{1}{3}\right)}} - \frac{{\left(\left(\left(\frac{\sqrt{\left(-4\right) \cdot {\left(\frac{\left(-3\right) \cdot c}{a} + \frac{{b}^{2}}{{a}^{2}}\right)}^{3} + {\left(\left(\frac{27 \cdot d}{a} - \frac{\left(9 \cdot b\right) \cdot c}{{a}^{2}}\right) + \frac{2 \cdot {b}^{3}}{{a}^{3}}\right)}^{2}}}{2} + \frac{27 \cdot d}{2 \cdot a}\right) - \frac{\left(9 \cdot b\right) \cdot c}{2 \cdot {a}^{2}}\right) + \frac{{b}^{3}}{{a}^{3}}\right)}^{\left(\frac{1}{3}\right)}}{3}\right) - \frac{b}{3 \cdot a}\]
↓
\[\left(\frac{-\left(\frac{\left(-3\right) \cdot c}{a} + \frac{{b}^{2}}{{a}^{2}}\right)}{3 \cdot {\left(\left(\left(\frac{\sqrt{\left(-4\right) \cdot {\left(\frac{\left(-3\right) \cdot c}{a} + \frac{{b}^{2}}{{a}^{2}}\right)}^{3} + {\left(\left(\frac{27 \cdot d}{a} - \frac{\left(9 \cdot b\right) \cdot c}{{a}^{2}}\right) + \frac{2 \cdot {b}^{3}}{{a}^{3}}\right)}^{2}}}{2} + \frac{27 \cdot d}{2 \cdot a}\right) - \frac{\left(9 \cdot b\right) \cdot c}{2 \cdot {a}^{2}}\right) + \frac{{b}^{3}}{{a}^{3}}\right)}^{\left(\frac{1}{3}\right)}} - \frac{{\left(\left(\left(\frac{\sqrt{\left(-4\right) \cdot {\left(\frac{\left(-3\right) \cdot c}{a} + \frac{{b}^{2}}{{a}^{2}}\right)}^{3} + {\left(\left(\frac{27 \cdot d}{a} - \frac{\left(9 \cdot b\right) \cdot c}{{a}^{2}}\right) + \frac{2 \cdot {b}^{3}}{{a}^{3}}\right)}^{2}}}{2} + \frac{27 \cdot d}{2 \cdot a}\right) - \frac{\left(9 \cdot b\right) \cdot c}{2 \cdot {a}^{2}}\right) + \frac{{b}^{3}}{{a}^{3}}\right)}^{\left(\frac{1}{3}\right)}}{3}\right) - \frac{b}{3 \cdot a}\]

Error

Bits error versus c

Bits error versus a

Bits error versus b

Bits error versus d

Derivation

  1. Initial program 54.3

    \[\left(\frac{-\left(\frac{\left(-3\right) \cdot c}{a} + \frac{{b}^{2}}{{a}^{2}}\right)}{3 \cdot {\left(\left(\left(\frac{\sqrt{\left(-4\right) \cdot {\left(\frac{\left(-3\right) \cdot c}{a} + \frac{{b}^{2}}{{a}^{2}}\right)}^{3} + {\left(\left(\frac{27 \cdot d}{a} - \frac{\left(9 \cdot b\right) \cdot c}{{a}^{2}}\right) + \frac{2 \cdot {b}^{3}}{{a}^{3}}\right)}^{2}}}{2} + \frac{27 \cdot d}{2 \cdot a}\right) - \frac{\left(9 \cdot b\right) \cdot c}{2 \cdot {a}^{2}}\right) + \frac{{b}^{3}}{{a}^{3}}\right)}^{\left(\frac{1}{3}\right)}} - \frac{{\left(\left(\left(\frac{\sqrt{\left(-4\right) \cdot {\left(\frac{\left(-3\right) \cdot c}{a} + \frac{{b}^{2}}{{a}^{2}}\right)}^{3} + {\left(\left(\frac{27 \cdot d}{a} - \frac{\left(9 \cdot b\right) \cdot c}{{a}^{2}}\right) + \frac{2 \cdot {b}^{3}}{{a}^{3}}\right)}^{2}}}{2} + \frac{27 \cdot d}{2 \cdot a}\right) - \frac{\left(9 \cdot b\right) \cdot c}{2 \cdot {a}^{2}}\right) + \frac{{b}^{3}}{{a}^{3}}\right)}^{\left(\frac{1}{3}\right)}}{3}\right) - \frac{b}{3 \cdot a}\]

  2. Final simplification54.3

    \[\leadsto \left(\frac{-\left(\frac{\left(-3\right) \cdot c}{a} + \frac{{b}^{2}}{{a}^{2}}\right)}{3 \cdot {\left(\left(\left(\frac{\sqrt{\left(-4\right) \cdot {\left(\frac{\left(-3\right) \cdot c}{a} + \frac{{b}^{2}}{{a}^{2}}\right)}^{3} + {\left(\left(\frac{27 \cdot d}{a} - \frac{\left(9 \cdot b\right) \cdot c}{{a}^{2}}\right) + \frac{2 \cdot {b}^{3}}{{a}^{3}}\right)}^{2}}}{2} + \frac{27 \cdot d}{2 \cdot a}\right) - \frac{\left(9 \cdot b\right) \cdot c}{2 \cdot {a}^{2}}\right) + \frac{{b}^{3}}{{a}^{3}}\right)}^{\left(\frac{1}{3}\right)}} - \frac{{\left(\left(\left(\frac{\sqrt{\left(-4\right) \cdot {\left(\frac{\left(-3\right) \cdot c}{a} + \frac{{b}^{2}}{{a}^{2}}\right)}^{3} + {\left(\left(\frac{27 \cdot d}{a} - \frac{\left(9 \cdot b\right) \cdot c}{{a}^{2}}\right) + \frac{2 \cdot {b}^{3}}{{a}^{3}}\right)}^{2}}}{2} + \frac{27 \cdot d}{2 \cdot a}\right) - \frac{\left(9 \cdot b\right) \cdot c}{2 \cdot {a}^{2}}\right) + \frac{{b}^{3}}{{a}^{3}}\right)}^{\left(\frac{1}{3}\right)}}{3}\right) - \frac{b}{3 \cdot a}\]

Reproduce

herbie shell --seed 2020153 
(FPCore (c a b d)
  :name "(- (- (/ (- (+ (/ (* (- 3) c) a) (/ (pow b 2) (pow a 2)))) (* 3 (pow (+ (- (+ (/ (sqrt (+ (* (- 4) (pow (+ (/ (* (- 3) c) a) (/ (pow b 2) (pow a 2))) 3)) (pow (+ (- (/ (* 27 d) a) (/ (* (* 9 b) c) (pow a 2))) (/ (* 2 (pow b 3)) (pow a 3))) 2))) 2) (/ (* 27 d) (* 2 a))) (/ (* (* 9 b) c) (* 2 (pow a 2)))) (/ (pow b 3) (pow a 3))) (/ 1 3)))) (/ (pow (+ (- (+ (/ (sqrt (+ (* (- 4) (pow (+ (/ (* (- 3) c) a) (/ (pow b 2) (pow a 2))) 3)) (pow (+ (- (/ (* 27 d) a) (/ (* (* 9 b) c) (pow a 2))) (/ (* 2 (pow b 3)) (pow a 3))) 2))) 2) (/ (* 27 d) (* 2 a))) (/ (* (* 9 b) c) (* 2 (pow a 2)))) (/ (pow b 3) (pow a 3))) (/ 1 3)) 3)) (/ b (* 3 a)))"
  :precision binary64
  (- (- (/ (neg (+ (/ (* (neg 3.0) c) a) (/ (pow b 2.0) (pow a 2.0)))) (* 3.0 (pow (+ (- (+ (/ (sqrt (+ (* (neg 4.0) (pow (+ (/ (* (neg 3.0) c) a) (/ (pow b 2.0) (pow a 2.0))) 3.0)) (pow (+ (- (/ (* 27.0 d) a) (/ (* (* 9.0 b) c) (pow a 2.0))) (/ (* 2.0 (pow b 3.0)) (pow a 3.0))) 2.0))) 2.0) (/ (* 27.0 d) (* 2.0 a))) (/ (* (* 9.0 b) c) (* 2.0 (pow a 2.0)))) (/ (pow b 3.0) (pow a 3.0))) (/ 1.0 3.0)))) (/ (pow (+ (- (+ (/ (sqrt (+ (* (neg 4.0) (pow (+ (/ (* (neg 3.0) c) a) (/ (pow b 2.0) (pow a 2.0))) 3.0)) (pow (+ (- (/ (* 27.0 d) a) (/ (* (* 9.0 b) c) (pow a 2.0))) (/ (* 2.0 (pow b 3.0)) (pow a 3.0))) 2.0))) 2.0) (/ (* 27.0 d) (* 2.0 a))) (/ (* (* 9.0 b) c) (* 2.0 (pow a 2.0)))) (/ (pow b 3.0) (pow a 3.0))) (/ 1.0 3.0)) 3.0)) (/ b (* 3.0 a))))