Average Error: 19.9 → 13.2
Time: 21.7s
Precision: 64
Internal Precision: 128
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 1.0425090330492645 \cdot 10^{+112}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 1.0425090330492645e+112

    1. Initial program 16.5

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    2. Initial simplification16.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \end{array}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt16.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{\sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} \cdot \sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \end{array}\]

    5. Applied sqrt-prod16.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\sqrt{\sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \end{array}\]

    if 1.0425090330492645e+112 < b

    1. Initial program 31.4

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    2. Initial simplification31.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \end{array}\]

    3. Taylor expanded around 0 1.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \end{array}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification13.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1.0425090330492645 \cdot 10^{+112}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2018360 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 2"
  (if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))))

Details

Time bar (total: 14.4s)Debug log

start211.0ms

Algorithm
intervals

setup156.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 18.7b

localize187.0ms

Local error

Found 4 expressions with local error:

22.5b
(sqrt (fma -4 (* c a) (* b b)))
22.5b
(sqrt (fma -4 (* c a) (* b b)))
0.9b
(- (sqrt (fma -4 (* c a) (* b b))) b)
0.8b
(- (- b) (sqrt (fma -4 (* c a) (* b b))))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 93
Calls

4 calls. Slowest were:

9.0ms
(- (- b) (sqrt (fma -4 (* c a) (* b b))))
6.0ms
(- (sqrt (fma -4 (* c a) (* b b))) b)
3.0ms
(sqrt (fma -4 (* c a) (* b b)))

series250.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

72.0ms
(- (- b) (sqrt (fma -4 (* c a) (* b b))))
62.0ms
(- (sqrt (fma -4 (* c a) (* b b))) b)
60.0ms
(sqrt (fma -4 (* c a) (* b b)))
56.0ms
(sqrt (fma -4 (* c a) (* b b)))

simplify2.2s

Counts
81 → 105
Calls

81 calls. Slowest were:

96.0ms
(- (* 2 (/ (* a c) b)) (* 2 b))
62.0ms
(* -2 (/ (* a c) b))
59.0ms
(fma 1 (- b) (- (* (sqrt (fma -4 (* c a) (* b b))) (sqrt 1))))

prune2.2s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 12.1b

localize117.0ms

Local error

Found 4 expressions with local error:

22.5b
(sqrt (fma -4 (* c a) (* b b)))
22.5b
(sqrt (fma -4 (* c a) (* b b)))
22.5b
(sqrt (fma -4 (* c a) (* b b)))
8.6b
(- (- b) (* (sqrt (sqrt (fma -4 (* c a) (* b b)))) (sqrt (sqrt (fma -4 (* c a) (* b b))))))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
4 → 66
Calls

4 calls. Slowest were:

4.0ms
(- (- b) (* (sqrt (sqrt (fma -4 (* c a) (* b b)))) (sqrt (sqrt (fma -4 (* c a) (* b b))))))
1.0ms
(sqrt (fma -4 (* c a) (* b b)))
1.0ms
(sqrt (fma -4 (* c a) (* b b)))

series250.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

74.0ms
(- (- b) (* (sqrt (sqrt (fma -4 (* c a) (* b b)))) (sqrt (sqrt (fma -4 (* c a) (* b b))))))
69.0ms
(sqrt (fma -4 (* c a) (* b b)))
60.0ms
(sqrt (fma -4 (* c a) (* b b)))
46.0ms
(sqrt (fma -4 (* c a) (* b b)))

simplify653.0ms

Counts
37 → 78
Calls

37 calls. Slowest were:

101.0ms
(- (* 2 (/ (* a c) b)) (* 2 b))
60.0ms
(/ (exp (- b)) (exp (* (sqrt (sqrt (fma -4 (* c a) (* b b)))) (sqrt (sqrt (fma -4 (* c a) (* b b)))))))
53.0ms
(fma (sqrt (- b)) (sqrt (- b)) (- (* (sqrt (sqrt (fma -4 (* c a) (* b b)))) (sqrt (sqrt (fma -4 (* c a) (* b b)))))))

prune1.7s

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 12.1b

localize159.0ms

Local error

Found 4 expressions with local error:

23.6b
(fma (sqrt (sqrt (fma -4 (* c a) (* b b)))) (sqrt (sqrt (fma -4 (* c a) (* b b)))) (- b))
22.5b
(sqrt (fma -4 (* c a) (* b b)))
22.5b
(sqrt (fma -4 (* c a) (* b b)))
22.5b
(sqrt (fma -4 (* c a) (* b b)))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 52
Calls

4 calls. Slowest were:

2.0ms
(sqrt (fma -4 (* c a) (* b b)))
2.0ms
(sqrt (fma -4 (* c a) (* b b)))
2.0ms
(sqrt (fma -4 (* c a) (* b b)))

series502.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

315.0ms
(fma (sqrt (sqrt (fma -4 (* c a) (* b b)))) (sqrt (sqrt (fma -4 (* c a) (* b b)))) (- b))
67.0ms
(sqrt (fma -4 (* c a) (* b b)))
67.0ms
(sqrt (fma -4 (* c a) (* b b)))
54.0ms
(sqrt (fma -4 (* c a) (* b b)))

simplify511.0ms

Counts
24 → 64
Calls

24 calls. Slowest were:

206.0ms
(- (pow (exp (* 1/4 (- (log -4) (+ (log (/ 1 a)) (log (/ 1 c)))))) 2) b)
155.0ms
(- (pow (exp (* 1/4 (- (log -4) (+ (log (/ -1 c)) (log (/ -1 a)))))) 2) b)
21.0ms
(sqrt (* (cbrt (fma -4 (* c a) (* b b))) (cbrt (fma -4 (* c a) (* b b)))))

prune1.5s

Pruning

11 alts after pruning (9 fresh and 2 done)

Merged error: 12.1b

localize173.0ms

Local error

Found 4 expressions with local error:

23.6b
(fma (sqrt (* (* (cbrt (sqrt (fma -4 (* c a) (* b b)))) (cbrt (sqrt (fma -4 (* c a) (* b b))))) (cbrt (sqrt (fma -4 (* c a) (* b b)))))) (sqrt (sqrt (fma -4 (* c a) (* b b)))) (- b))
22.5b
(sqrt (fma -4 (* c a) (* b b)))
22.5b
(sqrt (fma -4 (* c a) (* b b)))
22.5b
(sqrt (fma -4 (* c a) (* b b)))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 52
Calls

4 calls. Slowest were:

2.0ms
(sqrt (fma -4 (* c a) (* b b)))
2.0ms
(sqrt (fma -4 (* c a) (* b b)))
2.0ms
(sqrt (fma -4 (* c a) (* b b)))

series507.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

328.0ms
(fma (sqrt (* (* (cbrt (sqrt (fma -4 (* c a) (* b b)))) (cbrt (sqrt (fma -4 (* c a) (* b b))))) (cbrt (sqrt (fma -4 (* c a) (* b b)))))) (sqrt (sqrt (fma -4 (* c a) (* b b)))) (- b))
67.0ms
(sqrt (fma -4 (* c a) (* b b)))
60.0ms
(sqrt (fma -4 (* c a) (* b b)))
53.0ms
(sqrt (fma -4 (* c a) (* b b)))

simplify579.0ms

Counts
24 → 64
Calls

24 calls. Slowest were:

273.0ms
(- (pow (exp (* 1/4 (- (log -4) (+ (log (/ 1 a)) (log (/ 1 c)))))) 2) b)
144.0ms
(- (pow (exp (* 1/4 (- (log -4) (+ (log (/ -1 c)) (log (/ -1 a)))))) 2) b)
21.0ms
(sqrt (* (cbrt (fma -4 (* c a) (* b b))) (cbrt (fma -4 (* c a) (* b b)))))

prune1.7s

Pruning

11 alts after pruning (8 fresh and 3 done)

Merged error: 12.1b

regimes543.0ms

Accuracy

92.8% (0.5b remaining)

Error of 13.2b against oracle of 12.7b and baseline of 19.9b

bsearch219.0ms