Average Error: 19.1 → 13.1
Time: 21.1s
Precision: 64
Internal Precision: 128
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 5.001839899625071 \cdot 10^{+77}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{(\left(\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*} - b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 5.001839899625071e+77

    1. Initial program 14.7

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

    2. Initial simplification14.7

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

    4. Applied add-sqr-sqrt14.7

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

    5. Applied sqrt-prod14.8

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

    6. Applied fma-neg14.7

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

    if 5.001839899625071e+77 < b

    1. Initial program 40.3

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

    2. Initial simplification40.3

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

    3. Taylor expanded around 0 5.3

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

  4. Final simplification13.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 5.001839899625071 \cdot 10^{+77}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{(\left(\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*} - b}\\ \end{array}\]

Reproduce

herbie shell --seed 2018362 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 1"
  (if (>= b 0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))))))

Details

Time bar (total: 14.4s)Debug log

start208.0ms

Algorithm
intervals

setup130.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 17.0b

localize170.0ms

Local error

Found 4 expressions with local error:

21.0b
(sqrt (fma c (* a -4) (* b b)))
21.0b
(sqrt (fma c (* a -4) (* b b)))
1.8b
(- (sqrt (fma c (* a -4) (* b b))) b)
0.9b
(- (- b) (sqrt (fma c (* a -4) (* b b))))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
4 → 93
Calls

4 calls. Slowest were:

5.0ms
(- (sqrt (fma c (* a -4) (* b b))) b)
5.0ms
(- (- b) (sqrt (fma c (* a -4) (* b b))))
1.0ms
(sqrt (fma c (* a -4) (* b b)))

series214.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

67.0ms
(- (sqrt (fma c (* a -4) (* b b))) b)
57.0ms
(sqrt (fma c (* a -4) (* b b)))
45.0ms
(sqrt (fma c (* a -4) (* b b)))
44.0ms
(- (- b) (sqrt (fma c (* a -4) (* b b))))

simplify2.3s

Counts
81 → 105
Calls

81 calls. Slowest were:

86.0ms
(- (* 2 (/ (* a c) b)) (* 2 b))
79.0ms
(fma 1 (- b) (- (* (sqrt (fma c (* a -4) (* b b))) 1)))
78.0ms
(fma 1 (- b) (- (* (sqrt (fma c (* a -4) (* b b))) (sqrt 1))))

prune2.0s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 11.8b

localize132.0ms

Local error

Found 4 expressions with local error:

21.0b
(sqrt (fma c (* a -4) (* b b)))
21.0b
(sqrt (fma c (* a -4) (* b b)))
21.0b
(sqrt (fma c (* a -4) (* b b)))
13.8b
(fma (sqrt (sqrt (fma c (* a -4) (* b b)))) (sqrt (sqrt (fma c (* a -4) (* b b)))) (- b))

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
4 → 52
Calls

4 calls. Slowest were:

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

series443.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

278.0ms
(fma (sqrt (sqrt (fma c (* a -4) (* b b)))) (sqrt (sqrt (fma c (* a -4) (* b b)))) (- b))
64.0ms
(sqrt (fma c (* a -4) (* b b)))
51.0ms
(sqrt (fma c (* a -4) (* b b)))
50.0ms
(sqrt (fma c (* a -4) (* b b)))

simplify643.0ms

Counts
24 → 64
Calls

24 calls. Slowest were:

366.0ms
(- (pow (exp (* 1/4 (- (log -4) (+ (log (/ 1 a)) (log (/ 1 c)))))) 2) b)
137.0ms
(- (pow (exp (* 1/4 (- (log -4) (+ (log (/ -1 c)) (log (/ -1 a)))))) 2) b)
21.0ms
(sqrt (* (cbrt (fma c (* a -4) (* b b))) (cbrt (fma c (* a -4) (* b b)))))

prune1.4s

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 11.8b

localize104.0ms

Local error

Found 4 expressions with local error:

21.0b
(sqrt (fma c (* a -4) (* b b)))
21.0b
(sqrt (fma c (* a -4) (* b b)))
21.0b
(sqrt (fma c (* a -4) (* b b)))
8.9b
(- (- b) (* (sqrt (sqrt (fma c (* a -4) (* b b)))) (sqrt (sqrt (fma c (* a -4) (* b b))))))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 66
Calls

4 calls. Slowest were:

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

series190.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

53.0ms
(sqrt (fma c (* a -4) (* b b)))
49.0ms
(sqrt (fma c (* a -4) (* b b)))
44.0ms
(sqrt (fma c (* a -4) (* b b)))
43.0ms
(- (- b) (* (sqrt (sqrt (fma c (* a -4) (* b b)))) (sqrt (sqrt (fma c (* a -4) (* b b))))))

simplify620.0ms

Counts
37 → 78
Calls

37 calls. Slowest were:

74.0ms
(- (* 2 (/ (* a c) b)) (* 2 b))
72.0ms
(* -2 (/ (* a c) b))
56.0ms
(/ (exp (- b)) (exp (* (sqrt (sqrt (fma c (* a -4) (* b b)))) (sqrt (sqrt (fma c (* a -4) (* b b)))))))

prune1.5s

Pruning

12 alts after pruning (10 fresh and 2 done)

Merged error: 11.8b

localize168.0ms

Local error

Found 4 expressions with local error:

21.8b
(cbrt (fma c (* a -4) (* b b)))
21.8b
(cbrt (fma (* c -4) a (* b b)))
21.0b
(sqrt (fma c (* a -4) (* b b)))
21.0b
(sqrt (fma c (* a -4) (* b b)))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
4 → 54
Calls

4 calls. Slowest were:

2.0ms
(sqrt (fma c (* a -4) (* b b)))
2.0ms
(sqrt (fma c (* a -4) (* b b)))
1.0ms
(cbrt (fma c (* a -4) (* b b)))

series504.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

204.0ms
(cbrt (fma c (* a -4) (* b b)))
197.0ms
(cbrt (fma (* c -4) a (* b b)))
59.0ms
(sqrt (fma c (* a -4) (* b b)))
45.0ms
(sqrt (fma c (* a -4) (* b b)))

simplify1.1s

Counts
26 → 66
Calls

26 calls. Slowest were:

228.0ms
(- (pow b 2/3) (* 4/3 (* (* a c) (pow (/ 1 (pow b 4)) 1/3))))
194.0ms
(exp (* 1/3 (- (log -4) (+ (log (/ 1 a)) (log (/ 1 c))))))
171.0ms
(exp (* 1/3 (- (log -4) (+ (log (/ 1 a)) (log (/ 1 c))))))

prune1.9s

Pruning

12 alts after pruning (9 fresh and 3 done)

Merged error: 11.8b

regimes498.0ms

Accuracy

88.9% (0.8b remaining)

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

bsearch198.0ms