Average Error: 33.9 → 11.2
Time: 20.1s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.122414463296767 \cdot 10^{+147}:\\ \;\;\;\;\frac{\frac{(-2 \cdot b + \left(\left(a \cdot c\right) \cdot \frac{\frac{3}{2}}{b}\right))_*}{3}}{a}\\ \mathbf{elif}\;b \le 1.1212619524677665 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot c\right) \cdot -3 + \left(b \cdot b\right))_*}}{3 \cdot a} - \frac{b}{\frac{a}{\frac{1}{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Split input into 3 regimes

  2. if b < -7.122414463296767e+147

    1. Initial program 58.2

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

    2. Taylor expanded around -inf 58.2

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

    3. Simplified58.1

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

    5. Applied associate-/r*58.1

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

    6. Taylor expanded around -inf 8.9

      \[\leadsto \frac{\frac{\color{blue}{\frac{3}{2} \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{3}}{a}\]

    7. Simplified8.9

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

    if -7.122414463296767e+147 < b < 1.1212619524677665e-29

    1. Initial program 14.3

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

    2. Taylor expanded around -inf 14.4

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

    3. Simplified14.3

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

    5. Applied associate-/r*14.3

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

    7. Applied div-inv14.4

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

    8. Applied associate-/l*14.4

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

    9. Simplified14.4

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

    11. Applied div-sub14.4

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

    12. Simplified14.4

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

    if 1.1212619524677665e-29 < b

    1. Initial program 54.3

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

    2. Taylor expanded around inf 7.3

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.122414463296767 \cdot 10^{+147}:\\ \;\;\;\;\frac{\frac{(-2 \cdot b + \left(\left(a \cdot c\right) \cdot \frac{\frac{3}{2}}{b}\right))_*}{3}}{a}\\ \mathbf{elif}\;b \le 1.1212619524677665 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot c\right) \cdot -3 + \left(b \cdot b\right))_*}}{3 \cdot a} - \frac{b}{\frac{a}{\frac{1}{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019016 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))

Details

Time bar (total: 18.6s)Debug log

sample141.0ms

Algorithm
intervals
Results
393×(pre true 80)
190×(body real 80)
113×(body nan 80)
37×(body real 2560)
26×(body real 1280)
14×(body real 640)
6×(body real 160)
4×(body real 320)
3×(body real 5120)

simplify97.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
97.0ms
(/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a))

prune26.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 33.4b

localize40.0ms

Local error

Found 4 expressions with local error:

24.1b
(sqrt (- (* b b) (* (* 3 a) c)))
1.3b
(+ (- b) (sqrt (- (* b b) (* (* 3 a) c))))
0.6b
(- (* b b) (* (* 3 a) c))
0.2b
(/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a))

rewrite47.0ms

Algorithm
rewrite-expression-head
Rules
14×*-un-lft-identity
9×add-sqr-sqrt
8×add-log-exp
8×add-cube-cbrt
6×add-exp-log
6×add-cbrt-cube
4×associate-/l*
4×log1p-expm1-u
4×times-frac
4×fma-def
4×pow1
4×expm1-log1p-u
3×distribute-lft-out
3×sqrt-prod
2×associate-/l/
2×flip--
2×flip-+
2×associate-+l-
2×sqrt-div
2×flip3--
2×flip3-+
2×neg-sub0
1×div-inv
1×prod-diff
1×div-exp
1×div-sub
1×diff-log
1×fma-neg
1×pow1/2
1×frac-2neg
1×sub-neg
1×sum-log
1×neg-mul-1
1×associate-/r*
1×clear-num
1×rem-sqrt-square
1×+-commutative
1×cbrt-undiv
Counts
4 → 77
Calls
4 calls:
Slowest
34.0ms
(/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a))
6.0ms
(+ (- b) (sqrt (- (* b b) (* (* 3 a) c))))
3.0ms
(sqrt (- (* b b) (* (* 3 a) c)))
2.0ms
(- (* b b) (* (* 3 a) c))

series263.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
79.0ms
(/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a))
72.0ms
(+ (- b) (sqrt (- (* b b) (* (* 3 a) c))))
67.0ms
(sqrt (- (* b b) (* (* 3 a) c)))
45.0ms
(- (* b b) (* (* 3 a) c))

simplify2.6s

Counts
40 → 89
Calls
40 calls:
Slowest
413.0ms
(- (* (- b) (- b)) (* (sqrt (- (* b b) (* (* 3 a) c))) (sqrt (- (* b b) (* (* 3 a) c)))))
345.0ms
(sqrt (- (pow (* b b) 3) (pow (* (* 3 a) c) 3)))
169.0ms
(/ (* (* (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (+ (- b) (sqrt (- (* b b) (* (* 3 a) c))))) (+ (- b) (sqrt (- (* b b) (* (* 3 a) c))))) (* (* (* 3 a) (* 3 a)) (* 3 a)))
164.0ms
(- b (* 3/2 (/ (* a c) b)))
148.0ms
(sqrt (- (* (* b b) (* b b)) (* (* (* 3 a) c) (* (* 3 a) c))))

prune1.1s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 5.1b

localize17.0ms

Local error

Found 4 expressions with local error:

24.1b
(sqrt (fma (* a -3) c (* b b)))
1.3b
(+ (- b) (sqrt (fma (* a -3) c (* b b))))
0.2b
(/ (+ (- b) (sqrt (fma (* a -3) c (* b b)))) (* 3 a))
0.0b
(fma (* a -3) c (* b b))

rewrite27.0ms

Algorithm
rewrite-expression-head
Rules
14×*-un-lft-identity
9×add-sqr-sqrt
8×add-cube-cbrt
6×add-log-exp
6×add-exp-log
6×add-cbrt-cube
4×associate-/l*
4×log1p-expm1-u
4×times-frac
4×fma-def
4×pow1
4×expm1-log1p-u
3×distribute-lft-out
3×sqrt-prod
2×associate-/l/
2×flip-+
2×associate-+l-
2×flip3-+
2×neg-sub0
1×div-inv
1×fma-udef
1×div-exp
1×div-sub
1×pow1/2
1×frac-2neg
1×sum-log
1×neg-mul-1
1×associate-/r*
1×clear-num
1×rem-sqrt-square
1×+-commutative
1×cbrt-undiv
Counts
4 → 69
Calls
4 calls:
Slowest
21.0ms
(/ (+ (- b) (sqrt (fma (* a -3) c (* b b)))) (* 3 a))
4.0ms
(+ (- b) (sqrt (fma (* a -3) c (* 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
67.0ms
(/ (+ (- b) (sqrt (fma (* a -3) c (* b b)))) (* 3 a))
60.0ms
(sqrt (fma (* a -3) c (* b b)))
55.0ms
(+ (- b) (sqrt (fma (* a -3) c (* b b))))
33.0ms
(fma (* a -3) c (* b b))

simplify1.3s

Counts
35 → 81
Calls
35 calls:
Slowest
335.0ms
(- (* (- b) (- b)) (* (sqrt (fma (* a -3) c (* b b))) (sqrt (fma (* a -3) c (* b b)))))
165.0ms
(+ (pow (- b) 3) (pow (sqrt (fma (* a -3) c (* b b))) 3))
121.0ms
(/ (* (* (+ (- b) (sqrt (fma (* a -3) c (* b b)))) (+ (- b) (sqrt (fma (* a -3) c (* b b))))) (+ (- b) (sqrt (fma (* a -3) c (* b b))))) (* (* (* 3 a) (* 3 a)) (* 3 a)))
67.0ms
(- (* 1/2 (/ c b)) (* 2/3 (/ b a)))
60.0ms
(- (* 3/2 (/ (* a c) b)) (* 2 b))

prune987.0ms

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 5.1b

localize7.0ms

Local error

Found 4 expressions with local error:

24.1b
(sqrt (fma (* a -3) c (* b b)))
1.3b
(+ (- b) (sqrt (fma (* a -3) c (* b b))))
0.2b
(/ (/ (+ (- b) (sqrt (fma (* a -3) c (* b b)))) 3) a)
0.1b
(/ (+ (- b) (sqrt (fma (* a -3) c (* b b)))) 3)

rewrite45.0ms

Algorithm
rewrite-expression-head
Rules
12×*-un-lft-identity
9×add-sqr-sqrt
8×associate-/l*
8×add-cube-cbrt
6×add-log-exp
4×log1p-expm1-u
4×add-exp-log
4×add-cbrt-cube
4×fma-def
4×pow1
4×expm1-log1p-u
3×associate-/l/
3×div-inv
3×sqrt-prod
3×associate-+l-
3×div-sub
3×neg-sub0
2×distribute-lft-out
2×flip-+
2×flip3-+
2×frac-2neg
2×clear-num
1×pow1/2
1×sum-log
1×neg-mul-1
1×rem-sqrt-square
1×+-commutative
Counts
4 → 70
Calls
4 calls:
Slowest
22.0ms
(/ (+ (- b) (sqrt (fma (* a -3) c (* b b)))) 3)
16.0ms
(/ (/ (+ (- b) (sqrt (fma (* a -3) c (* b b)))) 3) a)
4.0ms
(+ (- b) (sqrt (fma (* a -3) c (* b b))))
1.0ms
(sqrt (fma (* a -3) c (* b b)))

series232.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
71.0ms
(/ (+ (- b) (sqrt (fma (* a -3) c (* b b)))) 3)
69.0ms
(/ (/ (+ (- b) (sqrt (fma (* a -3) c (* b b)))) 3) a)
49.0ms
(sqrt (fma (* a -3) c (* b b)))
43.0ms
(+ (- b) (sqrt (fma (* a -3) c (* b b))))

simplify1.1s

Counts
30 → 82
Calls
30 calls:
Slowest
355.0ms
(- (* (- b) (- b)) (* (sqrt (fma (* a -3) c (* b b))) (sqrt (fma (* a -3) c (* b b)))))
205.0ms
(+ (pow (- b) 3) (pow (sqrt (fma (* a -3) c (* b b))) 3))
59.0ms
(- (* 1/2 (/ (* a c) b)) (* 2/3 b))
54.0ms
(* (exp (- b)) (exp (sqrt (fma (* a -3) c (* b b)))))
53.0ms
(- (* 3/2 (/ (* a c) b)) (* 2 b))

prune1.1s

Pruning

13 alts after pruning (12 fresh and 1 done)

Merged error: 5.1b

localize26.0ms

Local error

Found 4 expressions with local error:

24.1b
(sqrt (fma a (* -3 c) (* b b)))
1.3b
(- (sqrt (fma a (* -3 c) (* b b))) b)
0.3b
(/ a (/ 1 3))
0.2b
(/ (- (sqrt (fma a (* -3 c) (* b b))) b) (/ a (/ 1 3)))

rewrite25.0ms

Algorithm
rewrite-expression-head
Rules
19×add-sqr-sqrt
18×add-cube-cbrt
18×*-un-lft-identity
15×times-frac
9×associate-/r*
7×div-inv
6×sqrt-prod
6×associate-/r/
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*
2×associate-/l/
2×flip--
2×flip3--
2×frac-2neg
2×clear-num
1×div-exp
1×div-sub
1×pow1/2
1×sub-neg
1×rem-sqrt-square
1×cbrt-undiv
Counts
4 → 91
Calls
4 calls:
Slowest
17.0ms
(/ (- (sqrt (fma a (* -3 c) (* b b))) b) (/ a (/ 1 3)))
5.0ms
(- (sqrt (fma a (* -3 c) (* b b))) b)
1.0ms
(sqrt (fma a (* -3 c) (* b b)))
0.0ms
(/ a (/ 1 3))

series221.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
99.0ms
(/ (- (sqrt (fma a (* -3 c) (* b b))) b) (/ a (/ 1 3)))
68.0ms
(- (sqrt (fma a (* -3 c) (* b b))) b)
47.0ms
(sqrt (fma a (* -3 c) (* b b)))
7.0ms
(/ a (/ 1 3))

simplify2.1s

Counts
67 → 103
Calls
67 calls:
Slowest
675.0ms
(- (* (sqrt (fma a (* -3 c) (* b b))) (sqrt (fma a (* -3 c) (* b b)))) (* b b))
148.0ms
(- (pow (sqrt (fma a (* -3 c) (* b b))) 3) (pow b 3))
121.0ms
(- (log (- (sqrt (fma a (* -3 c) (* b b))) b)) (log (/ a (/ 1 3))))
104.0ms
(/ (- (sqrt (fma a (* -3 c) (* b b))) b) (/ a (/ 1 3)))
73.0ms
(/ (* (* (- (sqrt (fma a (* -3 c) (* b b))) b) (- (sqrt (fma a (* -3 c) (* b b))) b)) (- (sqrt (fma a (* -3 c) (* b b))) b)) (* (* (/ a (/ 1 3)) (/ a (/ 1 3))) (/ a (/ 1 3))))

prune1.3s

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 5.1b

regimes544.0ms

Accuracy

80.7% (5.4b remaining)

Error of 11.2b against oracle of 5.7b and baseline of 34.0b

bsearch298.0ms

end0.0ms

sample4.9s

Algorithm
intervals
Results
11582×(pre true 80)
5896×(body real 80)
2852×(body nan 80)
1063×(body real 2560)
886×(body real 1280)
431×(body real 640)
237×(body real 320)
109×(body real 160)
108×(body real 5120)