Average Error: 33.5 → 23.5
Time: 16.3s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 5.8033656073298216 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot -3\right) \cdot c + \left(b \cdot b\right))_*} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot -3\right) \cdot c}{\left(b + \sqrt{(\left(a \cdot -3\right) \cdot c + \left(b \cdot b\right))_*}\right) \cdot \left(a \cdot 3\right)}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Split input into 2 regimes

  2. if b < 5.8033656073298216e-40

    1. Initial program 21.3

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

    2. Simplified21.3

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

    3. Taylor expanded around -inf 21.3

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

    4. Simplified21.3

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

    if 5.8033656073298216e-40 < b

    1. Initial program 54.4

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

    2. Simplified54.4

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

    3. Taylor expanded around -inf 54.4

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

    4. Simplified54.4

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

    6. Applied flip--54.5

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

    7. Applied associate-/l/55.4

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

    8. Simplified27.4

      \[\leadsto \frac{\color{blue}{\left(a \cdot -3\right) \cdot c}}{\left(3 \cdot a\right) \cdot \left(\sqrt{(\left(a \cdot -3\right) \cdot c + \left(b \cdot b\right))_*} + b\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification23.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 5.8033656073298216 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot -3\right) \cdot c + \left(b \cdot b\right))_*} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot -3\right) \cdot c}{\left(b + \sqrt{(\left(a \cdot -3\right) \cdot c + \left(b \cdot b\right))_*}\right) \cdot \left(a \cdot 3\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019010 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))

Details

Time bar (total: 14.5s)Debug log

sample132.0ms

Algorithm
intervals

simplify96.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
96.0ms
(/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a))

prune16.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 32.9b

localize24.0ms

Local error

Found 4 expressions with local error:

21.5b
(sqrt (fma -3 (* c a) (* b b)))
1.9b
(- (sqrt (fma -3 (* c a) (* b b))) b)
0.5b
(fma -3 (* c a) (* b b))
0.5b
(/ (- (sqrt (fma -3 (* c a) (* b b))) b) (* 3 a))

rewrite31.0ms

Algorithm
rewrite-expression-head
Rules
10×add-sqr-sqrt
9×add-cube-cbrt
9×*-un-lft-identity
6×sqrt-prod
6×add-exp-log
6×fma-neg
6×add-cbrt-cube
4×add-log-exp
4×log1p-expm1-u
4×pow1
4×expm1-log1p-u
3×associate-/l*
3×times-frac
2×associate-/l/
2×flip--
2×flip3--
1×div-inv
1×fma-udef
1×div-exp
1×div-sub
1×pow1/2
1×frac-2neg
1×sub-neg
1×associate-/r*
1×clear-num
1×rem-sqrt-square
1×cbrt-undiv
Counts
4 → 67
Calls
4 calls:
Slowest
23.0ms
(/ (- (sqrt (fma -3 (* c a) (* b b))) b) (* 3 a))
6.0ms
(- (sqrt (fma -3 (* c a) (* b b))) b)
1.0ms
(sqrt (fma -3 (* c a) (* b b)))
0.0ms
(fma -3 (* c a) (* b b))

series275.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
85.0ms
(/ (- (sqrt (fma -3 (* c a) (* b b))) b) (* 3 a))
73.0ms
(- (sqrt (fma -3 (* c a) (* b b))) b)
65.0ms
(sqrt (fma -3 (* c a) (* b b)))
52.0ms
(fma -3 (* c a) (* b b))

simplify1.4s

Counts
34 → 79
Calls
34 calls:
Slowest
732.0ms
(- (* (sqrt (fma -3 (* c a) (* b b))) (sqrt (fma -3 (* c a) (* b b)))) (* b b))
100.0ms
(- (pow (sqrt (fma -3 (* c a) (* b b))) 3) (pow b 3))
83.0ms
(/ (* (* (- (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)))
51.0ms
(- (pow b 2) (* 3 (* a c)))
47.0ms
(- (log (- (sqrt (fma -3 (* c a) (* b b))) b)) (log (* 3 a)))

prune1.1s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 21.4b

localize7.0ms

Local error

Found 4 expressions with local error:

21.5b
(sqrt (fma -3 (* c a) (* b b)))
1.9b
(- (sqrt (fma -3 (* c a) (* b b))) b)
0.5b
(fma -3 (* c a) (* b b))
0.1b
(/ (/ (- (sqrt (fma -3 (* c a) (* b b))) b) 3) a)

rewrite18.0ms

Algorithm
rewrite-expression-head
Rules
9×add-sqr-sqrt
8×add-cube-cbrt
8×*-un-lft-identity
6×sqrt-prod
6×fma-neg
4×add-log-exp
4×associate-/l*
4×log1p-expm1-u
4×add-exp-log
4×add-cbrt-cube
4×pow1
4×expm1-log1p-u
2×div-inv
2×div-sub
1×associate-/l/
1×flip--
1×fma-udef
1×pow1/2
1×flip3--
1×frac-2neg
1×sub-neg
1×clear-num
1×rem-sqrt-square
Counts
4 → 61
Calls
4 calls:
Slowest
11.0ms
(/ (/ (- (sqrt (fma -3 (* c a) (* b b))) b) 3) a)
6.0ms
(- (sqrt (fma -3 (* c a) (* b b))) b)
1.0ms
(sqrt (fma -3 (* c a) (* b b)))
0.0ms
(fma -3 (* c a) (* b b))

series218.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
61.0ms
(/ (/ (- (sqrt (fma -3 (* c a) (* b b))) b) 3) a)
58.0ms
(sqrt (fma -3 (* c a) (* b b)))
51.0ms
(fma -3 (* c a) (* b b))
48.0ms
(- (sqrt (fma -3 (* c a) (* b b))) b)

simplify231.0ms

Counts
26 → 73
Calls
26 calls:
Slowest
33.0ms
(- (pow b 2) (* 3 (* a c)))
25.0ms
(/ (/ (sqrt (fma -3 (* c a) (* b b))) 3) a)
25.0ms
(- (pow b 2) (* 3 (* a c)))
24.0ms
(- (pow b 2) (* 3 (* a c)))
15.0ms
(* (cbrt (/ (- (sqrt (fma -3 (* c a) (* b b))) b) 3)) (cbrt (/ (- (sqrt (fma -3 (* c a) (* b b))) b) 3)))

prune896.0ms

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 21.4b

localize16.0ms

Local error

Found 4 expressions with local error:

21.5b
(sqrt (fma (* a -3) c (* b b)))
1.9b
(- (sqrt (fma (* a -3) c (* b b))) b)
0.5b
(/ (- (sqrt (fma (* a -3) c (* b b))) b) (* 3 a))
0.3b
(fma (* a -3) c (* b b))

rewrite20.0ms

Algorithm
rewrite-expression-head
Rules
10×add-sqr-sqrt
9×add-cube-cbrt
9×*-un-lft-identity
6×sqrt-prod
6×add-exp-log
6×fma-neg
6×add-cbrt-cube
4×add-log-exp
4×log1p-expm1-u
4×pow1
4×expm1-log1p-u
3×associate-/l*
3×times-frac
2×associate-/l/
2×flip--
2×flip3--
1×div-inv
1×fma-udef
1×div-exp
1×div-sub
1×pow1/2
1×frac-2neg
1×sub-neg
1×associate-/r*
1×clear-num
1×rem-sqrt-square
1×cbrt-undiv
Counts
4 → 67
Calls
4 calls:
Slowest
12.0ms
(/ (- (sqrt (fma (* a -3) c (* b b))) b) (* 3 a))
6.0ms
(- (sqrt (fma (* a -3) c (* b b))) b)
1.0ms
(sqrt (fma (* a -3) c (* b b)))
0.0ms
(fma (* a -3) c (* b b))

series215.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
81.0ms
(/ (- (sqrt (fma (* a -3) c (* b b))) b) (* 3 a))
51.0ms
(sqrt (fma (* a -3) c (* b b)))
51.0ms
(- (sqrt (fma (* a -3) c (* b b))) b)
31.0ms
(fma (* a -3) c (* b b))

simplify1.5s

Counts
34 → 79
Calls
34 calls:
Slowest
780.0ms
(- (* (sqrt (fma (* a -3) c (* b b))) (sqrt (fma (* a -3) c (* b b)))) (* b b))
116.0ms
(- (pow (sqrt (fma (* a -3) c (* b b))) 3) (pow b 3))
71.0ms
(- (log (- (sqrt (fma (* a -3) c (* b b))) b)) (log (* 3 a)))
60.0ms
(/ (* (* (- (sqrt (fma (* a -3) c (* b b))) b) (- (sqrt (fma (* a -3) c (* b b))) b)) (- (sqrt (fma (* a -3) c (* b b))) b)) (* (* (* 3 a) (* 3 a)) (* 3 a)))
45.0ms
(- (pow b 2) (* 3 (* a c)))

prune1.2s

Pruning

13 alts after pruning (12 fresh and 1 done)

Merged error: 21.3b

localize17.0ms

Local error

Found 4 expressions with local error:

21.5b
(sqrt (fma -3 (* c a) (* b b)))
1.9b
(- (sqrt (fma -3 (* c a) (* b b))) b)
0.5b
(fma -3 (* c a) (* b b))
0.2b
(/ 1 (/ a (/ (- (sqrt (fma -3 (* c a) (* b b))) b) 3)))

rewrite15.0ms

Algorithm
rewrite-expression-head
Rules
9×add-sqr-sqrt
8×add-cube-cbrt
8×*-un-lft-identity
6×sqrt-prod
6×fma-neg
5×add-exp-log
5×associate-/r*
5×pow1
4×add-log-exp
4×log1p-expm1-u
4×add-cbrt-cube
4×expm1-log1p-u
2×div-inv
2×associate-/r/
1×inv-pow
1×flip--
1×pow-flip
1×fma-udef
1×pow1/2
1×flip3--
1×frac-2neg
1×sub-neg
1×clear-num
1×rec-exp
1×rem-sqrt-square
Counts
4 → 64
Calls
4 calls:
Slowest
11.0ms
(- (sqrt (fma -3 (* c a) (* b b))) b)
2.0ms
(sqrt (fma -3 (* c a) (* b b)))
1.0ms
(/ 1 (/ a (/ (- (sqrt (fma -3 (* c a) (* b b))) b) 3)))
0.0ms
(fma -3 (* c a) (* b b))

series209.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
79.0ms
(/ 1 (/ a (/ (- (sqrt (fma -3 (* c a) (* b b))) b) 3)))
54.0ms
(- (sqrt (fma -3 (* c a) (* b b))) b)
45.0ms
(sqrt (fma -3 (* c a) (* b b)))
31.0ms
(fma -3 (* c a) (* b b))

simplify387.0ms

Counts
27 → 76
Calls
27 calls:
Slowest
45.0ms
(- (pow b 2) (* 3 (* a c)))
45.0ms
(- (pow b 2) (* 3 (* a c)))
38.0ms
(/ 1 (/ (- (sqrt (fma -3 (* c a) (* b b))) b) 3))
34.0ms
(cbrt (/ a (/ (- (sqrt (fma -3 (* c a) (* b b))) b) 3)))
32.0ms
(/ a (/ (- (sqrt (fma -3 (* c a) (* b b))) b) 3))

prune942.0ms

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 21.3b

regimes432.0ms

Accuracy

89.7% (1.1b remaining)

Error of 23.5b against oracle of 22.4b and baseline of 33.5b

bsearch151.0ms

end0.0ms

sample5.0s

Algorithm
intervals