Average Error: 43.5 → 0.5
Time: 30.3s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{c}{\frac{3}{a}} \cdot \left(\frac{1}{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} + b} \cdot \frac{-3}{a}\right)\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Initial program 43.5

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

  2. Simplified43.5

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

  4. Applied flip--43.5

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

  5. Applied associate-/l/43.5

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

  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*0.5

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

  10. Applied *-un-lft-identity0.5

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

  11. Applied times-frac0.4

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

  12. Applied times-frac0.5

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

  13. Simplified0.4

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

  15. Applied div-inv0.5

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

  16. Final simplification0.5

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

Reproduce

herbie shell --seed 2019022 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))

Details

Time bar (total: 29.3s)Debug log

sample174.0ms

Algorithm
intervals
Results
55.0ms266×body80nan
36.0ms135×body160valid
30.0ms522×pre80true
17.0ms46×body320valid
10.0ms75×body80valid

simplify71.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
71.0ms
(/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a))

prune41.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 39.9b

localize40.0ms

Local error

Found 4 expressions with local error:

36.2b
(- (sqrt (fma -3 (* c a) (* b b))) b)
0.3b
(/ (- (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)))

rewrite32.0ms

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
20.0ms
(/ (- (sqrt (fma -3 (* c a) (* b b))) b) (* 3 a))
10.0ms
(- (sqrt (fma -3 (* c a) (* b b))) b)
2.0ms
(sqrt (fma -3 (* c a) (* b b)))
0.0ms
(fma -3 (* c a) (* b b))

series232.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
76.0ms
(- (sqrt (fma -3 (* c a) (* b b))) b)
60.0ms
(/ (- (sqrt (fma -3 (* c a) (* b b))) b) (* 3 a))
51.0ms
(sqrt (fma -3 (* c a) (* b b)))
45.0ms
(fma -3 (* c a) (* b b))

simplify1.4s

Counts
34 → 79
Calls
34 calls:
Slowest
690.0ms
(- (* (sqrt (fma -3 (* c a) (* b b))) (sqrt (fma -3 (* c a) (* b b)))) (* b b))
107.0ms
(- (pow (sqrt (fma -3 (* c a) (* b b))) 3) (pow b 3))
77.0ms
(/ (* (* (- (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)))
77.0ms
(- (log (- (sqrt (fma -3 (* c a) (* b b))) b)) (log (* 3 a)))
47.0ms
(- (pow b 2) (* 3 (* a c)))

prune795.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.5b

localize17.0ms

Local error

Found 4 expressions with local error:

0.3b
(* (* 3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b))
0.3b
(/ (* (* a c) -3) (* (* 3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b)))
0.2b
(* (* a c) -3)
0.2b
(+ (sqrt (fma -3 (* c a) (* b b))) b)

rewrite32.0ms

Algorithm
rewrite-expression-head
Rules
11×add-exp-log
11×add-cbrt-cube
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
6×fma-def
6×pow1
4×add-log-exp
4×log1p-expm1-u
4×associate-*r/
4×expm1-log1p-u
3×flip-+
3×sqrt-prod
3×associate-*r*
3×flip3-+
2×cbrt-unprod
2×*-commutative
2×associate-/r/
2×prod-exp
2×associate-*l*
2×div-exp
2×cbrt-undiv
1×associate-/l*
1×div-inv
1×pow-prod-down
1×times-frac
1×distribute-lft-in
1×frac-2neg
1×associate-/r*
1×clear-num
1×+-commutative
1×distribute-rgt-in
Counts
4 → 71
Calls
4 calls:
Slowest
11.0ms
(/ (* (* a c) -3) (* (* 3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b)))
8.0ms
(* (* 3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b))
7.0ms
(+ (sqrt (fma -3 (* c a) (* b b))) b)
4.0ms
(* (* a c) -3)

series295.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
117.0ms
(/ (* (* a c) -3) (* (* 3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b)))
103.0ms
(* (* 3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b))
63.0ms
(+ (sqrt (fma -3 (* c a) (* b b))) b)
11.0ms
(* (* a c) -3)

simplify3.4s

Counts
37 → 83
Calls
37 calls:
Slowest
631.0ms
(/ (* (* (* (* 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))))
594.0ms
(- (log (* (* a c) -3)) (+ (log (* 3 a)) (log (+ (sqrt (fma -3 (* c a) (* b b))) b))))
562.0ms
(/ (* (* (* (* 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))))
356.0ms
(+ (sqrt (fma -3 (* c a) (* b b))) b)
352.0ms
(- (log (* (* a c) -3)) (log (* (* 3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b))))

prune1.0s

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

rewrite36.0ms

Algorithm
rewrite-expression-head
Rules
20×times-frac
16×add-cube-cbrt
16×*-un-lft-identity
16×add-sqr-sqrt
11×add-exp-log
11×add-cbrt-cube
6×associate-/l*
6×div-inv
6×fma-def
4×add-log-exp
4×log1p-expm1-u
4×div-exp
4×associate-/r*
4×cbrt-undiv
4×pow1
4×expm1-log1p-u
3×sqrt-prod
2×flip-+
2×associate-/r/
2×flip3-+
2×frac-2neg
2×clear-num
1×associate-/l/
1×*-commutative
1×associate-*l*
1×+-commutative
Counts
4 → 88
Calls
4 calls:
Slowest
23.0ms
(/ (/ (* (* a c) -3) (* 3 a)) (+ (sqrt (fma -3 (* c a) (* b b))) b))
5.0ms
(/ (* (* a c) -3) (* 3 a))
4.0ms
(+ (sqrt (fma -3 (* c a) (* b b))) b)
2.0ms
(* (* a c) -3)

series191.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
110.0ms
(/ (/ (* (* a c) -3) (* 3 a)) (+ (sqrt (fma -3 (* c a) (* b b))) b))
60.0ms
(+ (sqrt (fma -3 (* c a) (* b b))) b)
11.0ms
(* (* a c) -3)
10.0ms
(/ (* (* a c) -3) (* 3 a))

simplify9.6s

Counts
66 → 100
Calls
66 calls:
Slowest
535.0ms
(/ (/ (* (* (* (* 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)))
483.0ms
(/ (* (* (/ (* (* a c) -3) (* 3 a)) (/ (* (* a c) -3) (* 3 a))) (/ (* (* a c) -3) (* 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)))
462.0ms
(/ (* (cbrt (/ (* (* a c) -3) (* 3 a))) (cbrt (/ (* (* a c) -3) (* 3 a)))) (* (cbrt (+ (sqrt (fma -3 (* c a) (* b b))) b)) (cbrt (+ (sqrt (fma -3 (* c a) (* b b))) b))))
456.0ms
(/ (* (* (* (* a c) -3) (* (* a c) -3)) (* (* a c) -3)) (* (* (* 3 a) (* 3 a)) (* 3 a)))
405.0ms
(/ (sqrt (/ (* (* a c) -3) (* 3 a))) (sqrt (+ (sqrt (fma -3 (* c a) (* b b))) b)))

prune1.2s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0b

localize36.0ms

Local error

Found 4 expressions with local error:

0.3b
(* (/ c (/ 3 a)) (/ (/ -3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b)))
0.3b
(/ c (/ 3 a))
0.2b
(/ (/ -3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b))
0.2b
(+ (sqrt (fma -3 (* c a) (* b b))) b)

rewrite42.0ms

Algorithm
rewrite-expression-head
Rules
25×add-cube-cbrt
25×*-un-lft-identity
25×add-sqr-sqrt
24×times-frac
18×associate-*r*
12×div-inv
11×add-exp-log
11×add-cbrt-cube
7×associate-/r*
6×associate-/r/
6×fma-def
6×pow1
5×associate-*l*
4×add-log-exp
4×associate-/l*
4×log1p-expm1-u
4×expm1-log1p-u
3×flip-+
3×sqrt-prod
3×flip3-+
2×cbrt-unprod
2×prod-exp
2×div-exp
2×frac-2neg
2×clear-num
2×cbrt-undiv
1×associate-/l/
1×*-commutative
1×associate-*r/
1×associate-*l/
1×pow-prod-down
1×+-commutative
1×frac-times
Counts
4 → 112
Calls
4 calls:
Slowest
18.0ms
(* (/ c (/ 3 a)) (/ (/ -3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b)))
14.0ms
(/ (/ -3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b))
7.0ms
(+ (sqrt (fma -3 (* c a) (* b b))) b)
1.0ms
(/ c (/ 3 a))

series374.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
147.0ms
(/ (/ -3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b))
144.0ms
(* (/ c (/ 3 a)) (/ (/ -3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b)))
65.0ms
(+ (sqrt (fma -3 (* c a) (* b b))) b)
18.0ms
(/ c (/ 3 a))

simplify2.9s

Counts
84 → 124
Calls
84 calls:
Slowest
385.0ms
(/ (* (* (/ -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)))
275.0ms
(+ (log (/ c (/ 3 a))) (log (/ (/ -3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b))))
238.0ms
(+ (* (sqrt (fma -3 (* c a) (* b b))) (sqrt (fma -3 (* c a) (* b b)))) (- (* b b) (* (sqrt (fma -3 (* c a) (* b b))) b)))
223.0ms
(+ (* (sqrt (fma -3 (* c a) (* b b))) (sqrt (fma -3 (* c a) (* b b)))) (- (* b b) (* (sqrt (fma -3 (* c a) (* b b))) b)))
213.0ms
(* (* (* (/ c (/ 3 a)) (/ c (/ 3 a))) (/ c (/ 3 a))) (/ (* (* (/ -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))))

prune1.6s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0b

regimes257.0ms

Accuracy

0% (0.5b remaining)

Error of 0.5b against oracle of 0.0b and baseline of 0.5b

bsearch4.0ms

end0.0ms

sample5.5s

Algorithm
intervals
Results
1.9s4780×body160valid
1.3s16250×pre80true
1.1s8250×body80nan
549.0ms1358×body320valid
254.0ms1862×body80valid