Average Error: 52.7 → 0.1
Time: 19.5s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{-c}{b + \sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Initial program 52.7

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

  2. Simplified52.7

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

  4. Applied flip--52.7

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

  5. Applied associate-/l/52.7

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

  6. Simplified0.5

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

  8. Applied associate-/r*0.4

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

  9. Taylor expanded around -inf 0.1

    \[\leadsto \frac{\color{blue}{-1 \cdot c}}{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} + b}\]

  10. Simplified0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2019002 +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)))

Details

Time bar (total: 18.7s)Debug log

sample224.0ms

Algorithm
intervals

simplify74.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

73.0ms
(/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a))

prune11.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 52.8b

localize20.0ms

Local error

Found 4 expressions with local error:

28.7b
(- (sqrt (fma -3 (* c a) (* b b))) b)
0.2b
(/ (- (sqrt (fma -3 (* c a) (* b b))) b) (* 3 a))
0.1b
(fma -3 (* c a) (* b b))
0.0b
(sqrt (fma -3 (* c a) (* b b)))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 67
Calls

4 calls. Slowest were:

11.0ms
(/ (- (sqrt (fma -3 (* c a) (* b b))) b) (* 3 a))
9.0ms
(- (sqrt (fma -3 (* c a) (* b b))) b)
1.0ms
(sqrt (fma -3 (* c a) (* b b)))

series226.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

75.0ms
(- (sqrt (fma -3 (* c a) (* b b))) b)
67.0ms
(/ (- (sqrt (fma -3 (* c a) (* b b))) b) (* 3 a))
50.0ms
(sqrt (fma -3 (* c a) (* b b)))
35.0ms
(fma -3 (* c a) (* b b))

simplify1.4s

Counts
34 → 79
Calls

34 calls. Slowest were:

777.0ms
(- (* (sqrt (fma -3 (* c a) (* b b))) (sqrt (fma -3 (* c a) (* b b)))) (* b b))
95.0ms
(- (pow (sqrt (fma -3 (* c a) (* b b))) 3) (pow b 3))
76.0ms
(/ (* (* (- (sqrt (fma -3 (* c a) (* b b))) b) (- (sqrt (fma -3 (* c a) (* b b))) b)) (- (sqrt (fma -3 (* c a) (* b b))) b)) (* (* (* 3 a) (* 3 a)) (* 3 a)))

prune746.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.5b

localize15.0ms

Local error

Found 4 expressions with local error:

0.3b
(/ (* (* a c) -3) (* (* 3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b)))
0.3b
(* (* a c) -3)
0.3b
(* (* 3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b))
0.1b
(fma -3 (* c a) (* b b))

rewrite27.0ms

Algorithm
rewrite-expression-head
Counts
4 → 63
Calls

4 calls. Slowest were:

15.0ms
(* (* 3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b))
9.0ms
(/ (* (* a c) -3) (* (* 3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b)))
2.0ms
(* (* a c) -3)

series332.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

152.0ms
(/ (* (* a c) -3) (* (* 3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b)))
122.0ms
(* (* 3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b))
39.0ms
(fma -3 (* c a) (* b b))
19.0ms
(* (* a c) -3)

simplify3.9s

Counts
32 → 75
Calls

32 calls. Slowest were:

1.0s
(- (log (* (* a c) -3)) (+ (log (* 3 a)) (log (+ (sqrt (fma -3 (* c a) (* b b))) b))))
636.0ms
(/ (* (* (* (* a c) -3) (* (* a c) -3)) (* (* a c) -3)) (* (* (* (* 3 a) (* 3 a)) (* 3 a)) (* (* (+ (sqrt (fma -3 (* c a) (* b b))) b) (+ (sqrt (fma -3 (* c a) (* b b))) b)) (+ (sqrt (fma -3 (* c a) (* b b))) b))))
621.0ms
(/ (* (* (* (* a c) -3) (* (* a c) -3)) (* (* a c) -3)) (* (* (* (* 3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b)) (* (* 3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b))) (* (* 3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b))))

prune867.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0b

localize6.0ms

Local error

Found 4 expressions with local error:

0.3b
(/ (* (* a c) -3) (* 3 a))
0.3b
(* (* a c) -3)
0.1b
(fma -3 (* c a) (* b b))
0.1b
(+ (sqrt (fma -3 (* c a) (* b b))) b)

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
4 → 56
Calls

4 calls. Slowest were:

5.0ms
(/ (* (* a c) -3) (* 3 a))
5.0ms
(+ (sqrt (fma -3 (* c a) (* b b))) b)
2.0ms
(* (* a c) -3)

series142.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

81.0ms
(+ (sqrt (fma -3 (* c a) (* b b))) b)
43.0ms
(fma -3 (* c a) (* b b))
11.0ms
(* (* a c) -3)
6.0ms
(/ (* (* a c) -3) (* 3 a))

simplify1.3s

Counts
22 → 68
Calls

22 calls. Slowest were:

539.0ms
(/ (* (* (* (* a c) -3) (* (* a c) -3)) (* (* a c) -3)) (* (* (* 3 a) (* 3 a)) (* 3 a)))
467.0ms
(- (log (* (* a c) -3)) (log (* 3 a)))
50.0ms
(- (pow b 2) (* 3 (* a c)))

prune741.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0b

localize5.0ms

Local error

Found 4 expressions with local error:

0.1b
(fma -3 (* c a) (* b b))
0.1b
(+ (sqrt (fma -3 (* c a) (* b b))) b)
0.0b
(/ (- c) (+ (sqrt (fma -3 (* c a) (* b b))) b))
0.0b
(sqrt (fma -3 (* c a) (* b b)))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
4 → 79
Calls

4 calls. Slowest were:

8.0ms
(/ (- c) (+ (sqrt (fma -3 (* c a) (* b b))) b))
4.0ms
(+ (sqrt (fma -3 (* c a) (* b b))) b)
1.0ms
(sqrt (fma -3 (* c a) (* b b)))

series334.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

171.0ms
(/ (- c) (+ (sqrt (fma -3 (* c a) (* b b))) b))
65.0ms
(sqrt (fma -3 (* c a) (* b b)))
63.0ms
(+ (sqrt (fma -3 (* c a) (* b b))) b)
35.0ms
(fma -3 (* c a) (* b b))

simplify1.3s

Counts
56 → 91
Calls

56 calls. Slowest were:

332.0ms
(/ (* (* (- c) (- c)) (- c)) (* (* (+ (sqrt (fma -3 (* c a) (* b b))) b) (+ (sqrt (fma -3 (* c a) (* b b))) b)) (+ (sqrt (fma -3 (* c a) (* b b))) b)))
245.0ms
(+ (* (sqrt (fma -3 (* c a) (* b b))) (sqrt (fma -3 (* c a) (* b b)))) (- (* b b) (* (sqrt (fma -3 (* c a) (* b b))) b)))
53.0ms
(/ (* (cbrt (- c)) (cbrt (- c))) (* (cbrt (+ (sqrt (fma -3 (* c a) (* b b))) b)) (cbrt (+ (sqrt (fma -3 (* c a) (* b b))) b))))

prune1.0s

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0b

regimes197.0ms

Accuracy

0% (0.1b remaining)

Error of 0.1b against oracle of 0.0b and baseline of 0.1b

bsearch2.0ms

end0.0ms

sample5.8s

Algorithm
intervals