Average Error: 52.2 → 0.1
Time: 29.2s
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 52.2

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

  2. Simplified52.2

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

  4. Applied flip--52.2

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

  5. Applied associate-/l/52.2

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

  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 2019022 +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: 28.2s)Debug log

sample210.0ms

Algorithm
intervals
Results
53.0ms90×body160valid
48.0ms117×body320valid
36.0ms276×body80nan
33.0ms532×pre80true
10.0ms12×body640valid
5.0ms37×body80valid

simplify54.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
54.0ms
(/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a))

prune21.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 50.6b

localize391.0ms

Local error

Found 4 expressions with local error:

25.4b
(- (sqrt (fma -3 (* c a) (* b b))) b)
0.2b
(/ (- (sqrt (fma -3 (* c a) (* b b))) b) (* 3 a))
0.0b
(fma -3 (* c a) (* b b))
0.0b
(sqrt (fma -3 (* c a) (* b b)))

rewrite33.0ms

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))
11.0ms
(- (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))

series224.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
65.0ms
(/ (- (sqrt (fma -3 (* c a) (* b b))) b) (* 3 a))
59.0ms
(- (sqrt (fma -3 (* c a) (* b b))) b)
55.0ms
(sqrt (fma -3 (* c a) (* b b)))
45.0ms
(fma -3 (* c a) (* b b))

simplify1.3s

Counts
34 → 79
Calls
34 calls:
Slowest
692.0ms
(- (* (sqrt (fma -3 (* c a) (* b b))) (sqrt (fma -3 (* c a) (* b b)))) (* b b))
103.0ms
(- (pow (sqrt (fma -3 (* c a) (* b b))) 3) (pow b 3))
84.0ms
(/ (* (* (- (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)))
59.0ms
(- (log (- (sqrt (fma -3 (* c a) (* b b))) b)) (log (* 3 a)))
45.0ms
(- (pow b 2) (* 3 (* a c)))

prune907.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.5b

localize18.0ms

Local error

Found 4 expressions with local error:

0.4b
(/ (* (* a c) -3) (* (* 3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b)))
0.3b
(* (* 3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b))
0.3b
(* (* a c) -3)
0.1b
(+ (sqrt (fma -3 (* c a) (* b b))) b)

rewrite31.0ms

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
12.0ms
(/ (* (* a c) -3) (* (* 3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b)))
9.0ms
(* (* 3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b))
6.0ms
(+ (sqrt (fma -3 (* c a) (* b b))) b)
2.0ms
(* (* a c) -3)

series283.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
107.0ms
(/ (* (* a c) -3) (* (* 3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b)))
91.0ms
(* (* 3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b))
65.0ms
(+ (sqrt (fma -3 (* c a) (* b b))) b)
20.0ms
(* (* a c) -3)

simplify3.0s

Counts
37 → 83
Calls
37 calls:
Slowest
589.0ms
(/ (* (* (* (* 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))))
541.0ms
(/ (* (* (* (* 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))))
515.0ms
(- (log (* (* a c) -3)) (+ (log (* 3 a)) (log (+ (sqrt (fma -3 (* c a) (* b b))) b))))
325.0ms
(- (log (* (* a c) -3)) (log (* (* 3 a) (+ (sqrt (fma -3 (* c a) (* b b))) b))))
278.0ms
(+ (* (sqrt (fma -3 (* c a) (* b b))) (sqrt (fma -3 (* c a) (* b b)))) (- (* b b) (* (sqrt (fma -3 (* c a) (* b b))) b)))

prune929.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.0b

localize6.0ms

Local error

Found 4 expressions with local error:

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

rewrite31.0ms

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
18.0ms
(/ (/ (* (* 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)

series201.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
114.0ms
(/ (/ (* (* a c) -3) (* 3 a)) (+ (sqrt (fma -3 (* c a) (* b b))) b))
61.0ms
(+ (sqrt (fma -3 (* c a) (* b b))) b)
17.0ms
(* (* a c) -3)
9.0ms
(/ (* (* a c) -3) (* 3 a))

simplify10.5s

Counts
66 → 100
Calls
66 calls:
Slowest
668.0ms
(/ (/ (* (* (* (* 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)))
609.0ms
(/ (* (* (/ (* (* 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)))
456.0ms
(* (cbrt (/ (* (* a c) -3) (* 3 a))) (cbrt (/ (* (* a c) -3) (* 3 a))))
449.0ms
(/ (cbrt (/ (* (* a c) -3) (* 3 a))) (sqrt (+ (sqrt (fma -3 (* c a) (* b b))) b)))
446.0ms
(/ (cbrt (/ (* (* a c) -3) (* 3 a))) (+ (sqrt (fma -3 (* c a) (* b b))) b))

prune1.2s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0b

localize8.0ms

Local error

Found 4 expressions with local error:

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

rewrite23.0ms

Algorithm
rewrite-expression-head
Rules
17×add-sqr-sqrt
16×add-cube-cbrt
16×*-un-lft-identity
12×times-frac
6×sqrt-prod
6×add-exp-log
6×add-cbrt-cube
6×fma-def
4×add-log-exp
4×associate-/l*
4×log1p-expm1-u
4×neg-mul-1
4×pow1
4×expm1-log1p-u
3×associate-/r*
2×flip-+
2×associate-/r/
2×flip3-+
1×div-inv
1×fma-udef
1×distribute-frac-neg
1×div-exp
1×div-sub
1×pow1/2
1×frac-2neg
1×clear-num
1×rem-sqrt-square
1×+-commutative
1×cbrt-undiv
1×neg-sub0
Counts
4 → 79
Calls
4 calls:
Slowest
13.0ms
(/ (- c) (+ (sqrt (fma -3 (* c a) (* b b))) b))
8.0ms
(+ (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))

series314.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
163.0ms
(/ (- c) (+ (sqrt (fma -3 (* c a) (* b b))) b))
61.0ms
(sqrt (fma -3 (* c a) (* b b)))
54.0ms
(+ (sqrt (fma -3 (* c a) (* b b))) b)
36.0ms
(fma -3 (* c a) (* b b))

simplify1.4s

Counts
56 → 91
Calls
56 calls:
Slowest
356.0ms
(/ (* (* (- 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)))
260.0ms
(+ (* (sqrt (fma -3 (* c a) (* b b))) (sqrt (fma -3 (* c a) (* b b)))) (- (* b b) (* (sqrt (fma -3 (* c a) (* b b))) b)))
29.0ms
(/ (* (cbrt (- c)) (cbrt (- c))) (* (cbrt (+ (sqrt (fma -3 (* c a) (* b b))) b)) (cbrt (+ (sqrt (fma -3 (* c a) (* b b))) b))))
29.0ms
(- (pow b 2) (* 3 (* a c)))
28.0ms
(/ (* (cbrt (- c)) (cbrt (- c))) (sqrt (+ (sqrt (fma -3 (* c a) (* b b))) b)))

prune1.0s

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0b

regimes210.0ms

Accuracy

0% (0.1b remaining)

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

bsearch1.0ms

end0.0ms

sample6.0s

Algorithm
intervals
Results
1.5s3586×body320valid
1.4s16212×pre80true
1.0s8212×body80nan
865.0ms2918×body160valid
683.0ms515×body640valid
137.0ms981×body80valid