Average Error: 19.1 → 13.1
Time: 22.7s
Precision: 64
Internal Precision: 128
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 4.534645027012494 \cdot 10^{+77}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\sqrt{\sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 4.534645027012494e+77

    1. Initial program 16.3

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

    2. Initial simplification16.2

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

    4. Applied add-sqr-sqrt16.2

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

    5. Applied sqrt-prod16.4

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

    6. Applied fma-neg16.3

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

    if 4.534645027012494e+77 < b

    1. Initial program 27.5

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

    2. Initial simplification27.5

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

    3. Taylor expanded around 0 3.6

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

  4. Final simplification13.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 4.534645027012494 \cdot 10^{+77}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\sqrt{\sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2018362 +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: 15.2s)Debug log

start220.0ms

Algorithm
intervals

setup131.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 18.7b

localize157.0ms

Local error

Found 4 expressions with local error:

21.8b
(sqrt (fma -4 (* c a) (* b b)))
21.8b
(sqrt (fma -4 (* c a) (* b b)))
1.8b
(- (sqrt (fma -4 (* c a) (* b b))) b)
0.9b
(- (- b) (sqrt (fma -4 (* c a) (* b b))))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
4 → 93
Calls

4 calls. Slowest were:

5.0ms
(- (sqrt (fma -4 (* c a) (* b b))) b)
5.0ms
(- (- b) (sqrt (fma -4 (* c a) (* b b))))
2.0ms
(sqrt (fma -4 (* c a) (* b b)))

series243.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

68.0ms
(sqrt (fma -4 (* c a) (* b b)))
64.0ms
(- (sqrt (fma -4 (* c a) (* b b))) b)
58.0ms
(- (- b) (sqrt (fma -4 (* c a) (* b b))))
53.0ms
(sqrt (fma -4 (* c a) (* b b)))

simplify2.3s

Counts
81 → 105
Calls

81 calls. Slowest were:

100.0ms
(- (* 2 (/ (* a c) b)) (* 2 b))
80.0ms
(fma 1 (- b) (- (* (sqrt (fma -4 (* c a) (* b b))) 1)))
69.0ms
(fma -1 b (- (* (sqrt (fma -4 (* c a) (* b b))) (sqrt 1))))

prune2.2s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 13.1b

localize138.0ms

Local error

Found 4 expressions with local error:

21.8b
(sqrt (fma -4 (* c a) (* b b)))
21.8b
(sqrt (fma -4 (* c a) (* b b)))
21.8b
(sqrt (fma -4 (* c a) (* b b)))
19.1b
(fma (sqrt (sqrt (fma -4 (* c a) (* b b)))) (sqrt (sqrt (fma -4 (* c a) (* b b)))) (- b))

rewrite6.0ms

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

series511.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

329.0ms
(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)))
62.0ms
(sqrt (fma -4 (* c a) (* b b)))
53.0ms
(sqrt (fma -4 (* c a) (* b b)))

simplify580.0ms

Counts
24 → 64
Calls

24 calls. Slowest were:

378.0ms
(- (pow (exp (* 1/4 (- (log -4) (+ (log (/ 1 a)) (log (/ 1 c)))))) 2) b)
90.0ms
(- (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

8 alts after pruning (7 fresh and 1 done)

Merged error: 13.0b

localize170.0ms

Local error

Found 4 expressions with local error:

22.6b
(cbrt (fma -4 (* c a) (* b b)))
22.6b
(cbrt (fma (* -4 a) c (* b b)))
21.8b
(sqrt (fma -4 (* c a) (* b b)))
21.8b
(sqrt (fma -4 (* c a) (* b b)))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
4 → 54
Calls

4 calls. Slowest were:

4.0ms
(cbrt (fma -4 (* c a) (* b b)))
2.0ms
(sqrt (fma -4 (* c a) (* b b)))
2.0ms
(sqrt (fma -4 (* c a) (* b b)))

series491.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

232.0ms
(cbrt (fma -4 (* c a) (* b b)))
174.0ms
(cbrt (fma (* -4 a) c (* b b)))
45.0ms
(sqrt (fma -4 (* c a) (* b b)))
40.0ms
(sqrt (fma -4 (* c a) (* b b)))

simplify1.1s

Counts
26 → 66
Calls

26 calls. Slowest were:

211.0ms
(exp (* 1/3 (- (log -4) (+ (log (/ 1 a)) (log (/ 1 c))))))
206.0ms
(- (pow b 2/3) (* 4/3 (* (* a c) (pow (/ 1 (pow b 4)) 1/3))))
189.0ms
(exp (* 1/3 (- (log -4) (+ (log (/ 1 a)) (log (/ 1 c))))))

prune2.1s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 13.0b

localize122.0ms

Local error

Found 4 expressions with local error:

21.8b
(sqrt (fma -4 (* c a) (* b b)))
21.8b
(sqrt (fma -4 (* c a) (* b b)))
21.8b
(sqrt (fma -4 (* c a) (* b b)))
6.5b
(- (- b) (* (sqrt (sqrt (fma -4 (* c a) (* b b)))) (sqrt (sqrt (fma -4 (* c a) (* b b))))))

rewrite7.0ms

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)))

series221.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

70.0ms
(sqrt (fma -4 (* c a) (* b b)))
60.0ms
(sqrt (fma -4 (* c a) (* b b)))
48.0ms
(sqrt (fma -4 (* c a) (* b b)))
43.0ms
(- (- b) (* (sqrt (sqrt (fma -4 (* c a) (* b b)))) (sqrt (sqrt (fma -4 (* c a) (* b b))))))

simplify695.0ms

Counts
37 → 78
Calls

37 calls. Slowest were:

71.0ms
(/ (exp (- b)) (exp (* (sqrt (sqrt (fma -4 (* c a) (* b b)))) (sqrt (sqrt (fma -4 (* c a) (* b b)))))))
70.0ms
(* -2 (/ (* a c) b))
67.0ms
(- (* 2 (/ (* a c) b)) (* 2 b))

prune1.6s

Pruning

10 alts after pruning (8 fresh and 2 done)

Merged error: 13.0b

regimes467.0ms

Accuracy

88.9% (0.8b remaining)

Error of 13.1b against oracle of 12.3b and baseline of 19.2b

bsearch185.0ms