Average Error: 52.5 → 0.1
Time: 19.4s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[-2 \cdot \frac{c}{b + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 52.5

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

  2. Simplified52.5

    \[\leadsto \color{blue}{\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}{2 \cdot a}}\]
  3. Using strategy rm

  4. Applied flip--52.5

    \[\leadsto \frac{\color{blue}{\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} \cdot \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b \cdot b}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} + b}}}{2 \cdot a}\]

  5. Applied associate-/l/52.5

    \[\leadsto \color{blue}{\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} \cdot \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b \cdot b}{\left(2 \cdot a\right) \cdot \left(\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} + b\right)}}\]

  6. Simplified0.4

    \[\leadsto \frac{\color{blue}{\left(a \cdot -4\right) \cdot c}}{\left(2 \cdot a\right) \cdot \left(\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} + b\right)}\]
  7. Using strategy rm

  8. Applied times-frac0.1

    \[\leadsto \color{blue}{\frac{a \cdot -4}{2 \cdot a} \cdot \frac{c}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} + b}}\]

  9. Simplified0.1

    \[\leadsto \color{blue}{-2} \cdot \frac{c}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} + b}\]

  10. Final simplification0.1

    \[\leadsto -2 \cdot \frac{c}{b + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\]

Reproduce

herbie shell --seed 2019022 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4 a) c)))) (* 2 a)))

Details

Time bar (total: 18.5s)Debug log

sample357.0ms

Algorithm
intervals
Results
93.0ms276×body80nan
67.0ms532×pre80true
50.0ms100×body320valid
41.0ms112×body160valid
9.0ms13×body640valid
5.0ms31×body80valid

simplify99.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
98.0ms
(/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))

prune38.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 51.1b

localize31.0ms

Local error

Found 4 expressions with local error:

29.7b
(- (sqrt (fma (* a -4) c (* b b))) b)
0.1b
(/ (- (sqrt (fma (* a -4) c (* b b))) b) (* 2 a))
0.1b
(fma (* a -4) c (* b b))
0.1b
(sqrt (fma (* a -4) c (* b b)))

rewrite22.0ms

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 -4) c (* b b))) b) (* 2 a))
7.0ms
(- (sqrt (fma (* a -4) c (* b b))) b)
1.0ms
(sqrt (fma (* a -4) c (* b b)))
0.0ms
(fma (* a -4) c (* b b))

series282.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
96.0ms
(- (sqrt (fma (* a -4) c (* b b))) b)
84.0ms
(/ (- (sqrt (fma (* a -4) c (* b b))) b) (* 2 a))
57.0ms
(sqrt (fma (* a -4) c (* b b)))
44.0ms
(fma (* a -4) c (* b b))

simplify1.5s

Counts
34 → 79
Calls
34 calls:
Slowest
719.0ms
(- (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (* b b))
150.0ms
(- (pow (sqrt (fma (* a -4) c (* b b))) 3) (pow b 3))
81.0ms
(/ (* (* (- (sqrt (fma (* a -4) c (* b b))) b) (- (sqrt (fma (* a -4) c (* b b))) b)) (- (sqrt (fma (* a -4) c (* b b))) b)) (* (* (* 2 a) (* 2 a)) (* 2 a)))
77.0ms
(- (log (- (sqrt (fma (* a -4) c (* b b))) b)) (log (* 2 a)))
46.0ms
(- (pow b 2) (* 4 (* a c)))

prune641.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.4b

localize20.0ms

Local error

Found 4 expressions with local error:

0.4b
(/ (* (* a -4) c) (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)))
0.1b
(fma (* a -4) c (* b b))
0.1b
(* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b))
0.1b
(+ (sqrt (fma (* a -4) c (* b b))) b)

rewrite49.0ms

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×associate-/r/
2×prod-exp
2×div-exp
2×cbrt-undiv
1×associate-/l*
1×div-inv
1×fma-udef
1×*-commutative
1×pow-prod-down
1×associate-*l*
1×times-frac
1×distribute-lft-in
1×frac-2neg
1×associate-/r*
1×clear-num
1×+-commutative
1×distribute-rgt-in
Counts
4 → 70
Calls
4 calls:
Slowest
19.0ms
(/ (* (* a -4) c) (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)))
16.0ms
(* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b))
11.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)
0.0ms
(fma (* a -4) c (* b b))

series352.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
150.0ms
(/ (* (* a -4) c) (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)))
94.0ms
(* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b))
59.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)
48.0ms
(fma (* a -4) c (* b b))

simplify3.3s

Counts
37 → 82
Calls
37 calls:
Slowest
624.0ms
(- (log (* (* a -4) c)) (+ (log (* 2 a)) (log (+ (sqrt (fma (* a -4) c (* b b))) b))))
610.0ms
(/ (* (* (* (* a -4) c) (* (* a -4) c)) (* (* a -4) c)) (* (* (* (* 2 a) (* 2 a)) (* 2 a)) (* (* (+ (sqrt (fma (* a -4) c (* b b))) b) (+ (sqrt (fma (* a -4) c (* b b))) b)) (+ (sqrt (fma (* a -4) c (* b b))) b))))
587.0ms
(/ (* (* (* (* a -4) c) (* (* a -4) c)) (* (* a -4) c)) (* (* (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)) (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b))) (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b))))
388.0ms
(- (log (* (* a -4) c)) (log (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b))))
252.0ms
(+ (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (- (* b b) (* (sqrt (fma (* a -4) c (* b b))) b)))

prune946.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0b

localize12.0ms

Local error

Found 4 expressions with local error:

0.1b
(fma (* a -4) c (* b b))
0.1b
(+ (sqrt (fma (* a -4) c (* b b))) b)
0.1b
(/ c (+ (sqrt (fma (* a -4) c (* b b))) b))
0.1b
(sqrt (fma (* a -4) c (* b b)))

rewrite15.0ms

Algorithm
rewrite-expression-head
Rules
9×add-sqr-sqrt
8×add-cube-cbrt
8×*-un-lft-identity
6×sqrt-prod
6×fma-def
4×add-log-exp
4×log1p-expm1-u
4×add-exp-log
4×add-cbrt-cube
4×pow1
4×expm1-log1p-u
3×associate-/r*
2×flip-+
2×associate-/r/
2×flip3-+
1×div-inv
1×fma-udef
1×pow1/2
1×frac-2neg
1×clear-num
1×rem-sqrt-square
1×+-commutative
Counts
4 → 59
Calls
4 calls:
Slowest
7.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)
5.0ms
(/ c (+ (sqrt (fma (* a -4) c (* b b))) b))
2.0ms
(sqrt (fma (* a -4) c (* b b)))
0.0ms
(fma (* a -4) c (* b b))

series285.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
147.0ms
(/ c (+ (sqrt (fma (* a -4) c (* b b))) b))
62.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)
41.0ms
(sqrt (fma (* a -4) c (* b b)))
34.0ms
(fma (* a -4) c (* b b))

simplify527.0ms

Counts
26 → 71
Calls
26 calls:
Slowest
238.0ms
(+ (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (- (* b b) (* (sqrt (fma (* a -4) c (* b b))) b)))
35.0ms
(- (pow b 2) (* 4 (* a c)))
32.0ms
(- (pow b 2) (* 4 (* a c)))
25.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)
25.0ms
(sqrt (sqrt (fma (* a -4) c (* b b))))

prune753.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0b

localize13.0ms

Local error

Found 4 expressions with local error:

0.1b
(fma (sqrt (sqrt (fma (* a -4) c (* b b)))) (sqrt (sqrt (fma (* a -4) c (* b b)))) b)
0.1b
(fma (* a -4) c (* b b))
0.1b
(fma (* a -4) c (* b b))
0.1b
(/ c (fma (sqrt (sqrt (fma (* a -4) c (* b b)))) (sqrt (sqrt (fma (* a -4) c (* b b)))) b))

rewrite1.0ms

Algorithm
rewrite-expression-head
Rules
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
4×add-log-exp
4×log1p-expm1-u
4×add-exp-log
4×add-cbrt-cube
4×pow1
4×expm1-log1p-u
3×fma-udef
3×associate-/r*
1×div-inv
1×frac-2neg
1×clear-num
Counts
4 → 45
Calls
4 calls:
Slowest
0.0ms
(/ c (fma (sqrt (sqrt (fma (* a -4) c (* b b)))) (sqrt (sqrt (fma (* a -4) c (* b b)))) b))
0.0ms
(fma (sqrt (sqrt (fma (* a -4) c (* b b)))) (sqrt (sqrt (fma (* a -4) c (* b b)))) b)
0.0ms
(fma (* a -4) c (* b b))
0.0ms
(fma (* a -4) c (* b b))

series705.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
328.0ms
(/ c (fma (sqrt (sqrt (fma (* a -4) c (* b b)))) (sqrt (sqrt (fma (* a -4) c (* b b)))) b))
281.0ms
(fma (sqrt (sqrt (fma (* a -4) c (* b b)))) (sqrt (sqrt (fma (* a -4) c (* b b)))) b)
48.0ms
(fma (* a -4) c (* b b))
48.0ms
(fma (* a -4) c (* b b))

simplify1.0s

Counts
15 → 57
Calls
15 calls:
Slowest
265.0ms
(+ (pow (exp (* 1/4 (- (log -4) (+ (log (/ 1 a)) (log (/ 1 c)))))) 2) b)
235.0ms
(- (/ c b) (+ (* 1/4 (/ (pow (exp (* 1/4 (- (log -4) (+ (log (/ -1 c)) (log (/ -1 a)))))) 4) (* a b))) (/ (* c (pow (exp (* 1/4 (- (log -4) (+ (log (/ -1 c)) (log (/ -1 a)))))) 2)) (pow b 2))))
175.0ms
(- (/ c b) (+ (/ (* (pow (exp (* 1/4 (- (log -4) (+ (log (/ 1 a)) (log (/ 1 c)))))) 2) c) (pow b 2)) (* 1/4 (/ (pow (exp (* 1/4 (- (log -4) (+ (log (/ 1 a)) (log (/ 1 c)))))) 4) (* a b)))))
91.0ms
(+ (pow (exp (* 1/4 (- (log -4) (+ (log (/ -1 c)) (log (/ -1 a)))))) 2) b)
42.0ms
(- (pow b 2) (* 4 (* a c)))

prune791.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0b

regimes194.0ms

Accuracy

0% (0.1b remaining)

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

bsearch3.0ms

end0.0ms

sample6.5s

Algorithm
intervals
Results
1.7s3675×body320valid
1.3s2942×body160valid
1.2s8168×body80nan
1.0s16168×pre80true
704.0ms474×body640valid
156.0ms909×body80valid