Average Error: 33.6 → 7.1
Time: 25.3s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.7665482787645588 \cdot 10^{+108}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 2.566361082313029 \cdot 10^{-307}:\\ \;\;\;\;\frac{c}{2} \cdot \frac{4}{\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} - b}\\ \mathbf{elif}\;b \le 8.095749828230367 \cdot 10^{+95}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.6
Target20.6
Herbie7.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.7665482787645588e+108

    1. Initial program 59.6

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

    2. Simplified59.6

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

    3. Taylor expanded around -inf 2.4

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

    4. Simplified2.4

      \[\leadsto \color{blue}{\frac{-c}{b}}\]

    if -1.7665482787645588e+108 < b < 2.566361082313029e-307

    1. Initial program 33.1

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

    2. Simplified33.1

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

    4. Applied flip--33.2

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

    5. Applied associate-/l/37.5

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

    6. Simplified21.1

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

    8. Applied times-frac15.9

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

    9. Simplified9.4

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

    10. Simplified9.4

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

    if 2.566361082313029e-307 < b < 8.095749828230367e+95

    1. Initial program 9.6

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

    2. Simplified9.6

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

    4. Applied div-inv9.8

      \[\leadsto \color{blue}{\left(\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}\right) \cdot \frac{1}{2 \cdot a}}\]

    if 8.095749828230367e+95 < b

    1. Initial program 42.9

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

    2. Simplified42.8

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

    4. Applied flip--61.1

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

    5. Applied associate-/l/61.4

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

    6. Simplified61.6

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

    8. Applied times-frac61.4

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

    9. Simplified61.3

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

    10. Simplified61.3

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

    12. Applied add-sqr-sqrt61.2

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

    13. Applied fma-neg61.2

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

    14. Taylor expanded around 0 3.9

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}}\]

    15. Simplified3.9

      \[\leadsto \color{blue}{\frac{-b}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification7.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.7665482787645588 \cdot 10^{+108}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 2.566361082313029 \cdot 10^{-307}:\\ \;\;\;\;\frac{c}{2} \cdot \frac{4}{\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} - b}\\ \mathbf{elif}\;b \le 8.095749828230367 \cdot 10^{+95}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :herbie-target
  (if (< b 0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

Details

Time bar (total: 24.3s)Debug log

sample159.0ms

Algorithm
intervals

simplify67.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

67.0ms
(/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a))

prune20.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 34.4b

localize32.0ms

Local error

Found 4 expressions with local error:

24.3b
(sqrt (fma (* -4 a) c (* b b)))
1.1b
(- (- b) (sqrt (fma (* -4 a) c (* b b))))
0.1b
(/ (- (- b) (sqrt (fma (* -4 a) c (* b b)))) (* 2 a))
0.0b
(fma (* -4 a) c (* b b))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 100
Calls

4 calls. Slowest were:

15.0ms
(/ (- (- b) (sqrt (fma (* -4 a) c (* b b)))) (* 2 a))
5.0ms
(- (- b) (sqrt (fma (* -4 a) c (* b b))))
1.0ms
(sqrt (fma (* -4 a) c (* b b)))

series236.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

89.0ms
(/ (- (- b) (sqrt (fma (* -4 a) c (* b b)))) (* 2 a))
51.0ms
(- (- b) (sqrt (fma (* -4 a) c (* b b))))
49.0ms
(sqrt (fma (* -4 a) c (* b b)))
47.0ms
(fma (* -4 a) c (* b b))

simplify3.3s

Counts
93 → 112
Calls

93 calls. Slowest were:

262.0ms
(- (* (- b) (- b)) (* (sqrt (fma (* -4 a) c (* b b))) (sqrt (fma (* -4 a) c (* b b)))))
168.0ms
(- (pow (- b) 3) (pow (sqrt (fma (* -4 a) c (* b b))) 3))
103.0ms
(/ (* (* (- (- b) (sqrt (fma (* -4 a) c (* b b)))) (- (- b) (sqrt (fma (* -4 a) c (* b b))))) (- (- b) (sqrt (fma (* -4 a) c (* b b))))) (* (* (* 2 a) (* 2 a)) (* 2 a)))

prune1.4s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 6.4b

localize26.0ms

Local error

Found 4 expressions with local error:

24.3b
(sqrt (fma (* -4 a) c (* b b)))
16.9b
(/ (* (* c a) 4) (* (* 2 a) (+ (- b) (sqrt (fma (* -4 a) c (* b b))))))
1.1b
(+ (- b) (sqrt (fma (* -4 a) c (* b b))))
0.1b
(* (* 2 a) (+ (- b) (sqrt (fma (* -4 a) c (* b b)))))

rewrite47.0ms

Algorithm
rewrite-expression-head
Counts
4 → 76
Calls

4 calls. Slowest were:

19.0ms
(/ (* (* c a) 4) (* (* 2 a) (+ (- b) (sqrt (fma (* -4 a) c (* b b))))))
16.0ms
(* (* 2 a) (+ (- b) (sqrt (fma (* -4 a) c (* b b)))))
9.0ms
(+ (- b) (sqrt (fma (* -4 a) c (* b b))))

series373.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

158.0ms
(/ (* (* c a) 4) (* (* 2 a) (+ (- b) (sqrt (fma (* -4 a) c (* b b))))))
73.0ms
(+ (- b) (sqrt (fma (* -4 a) c (* b b))))
72.0ms
(sqrt (fma (* -4 a) c (* b b)))
70.0ms
(* (* 2 a) (+ (- b) (sqrt (fma (* -4 a) c (* b b)))))

simplify5.1s

Counts
41 → 88
Calls

41 calls. Slowest were:

1.2s
(/ (* (* (* (* c a) 4) (* (* c a) 4)) (* (* c a) 4)) (* (* (* (* 2 a) (* 2 a)) (* 2 a)) (* (* (+ (- b) (sqrt (fma (* -4 a) c (* b b)))) (+ (- b) (sqrt (fma (* -4 a) c (* b b))))) (+ (- b) (sqrt (fma (* -4 a) c (* b b)))))))
890.0ms
(/ (* (* (* (* c a) 4) (* (* c a) 4)) (* (* c a) 4)) (* (* (* (* 2 a) (+ (- b) (sqrt (fma (* -4 a) c (* b b))))) (* (* 2 a) (+ (- b) (sqrt (fma (* -4 a) c (* b b)))))) (* (* 2 a) (+ (- b) (sqrt (fma (* -4 a) c (* b b)))))))
672.0ms
(- (log (* (* c a) 4)) (+ (log (* 2 a)) (log (+ (- b) (sqrt (fma (* -4 a) c (* b b)))))))

prune1.1s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 6.4b

localize10.0ms

Local error

Found 4 expressions with local error:

24.3b
(sqrt (fma (* -4 a) c (* b b)))
1.1b
(- (sqrt (fma (* -4 a) c (* b b))) b)
0.2b
(* (/ c 2) (/ 4 (- (sqrt (fma (* -4 a) c (* b b))) b)))
0.1b
(/ 4 (- (sqrt (fma (* -4 a) c (* b b))) b))

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
4 → 76
Calls

4 calls. Slowest were:

5.0ms
(- (sqrt (fma (* -4 a) c (* b b))) b)
4.0ms
(* (/ c 2) (/ 4 (- (sqrt (fma (* -4 a) c (* b b))) b)))
3.0ms
(/ 4 (- (sqrt (fma (* -4 a) c (* b b))) b))

series407.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

172.0ms
(* (/ c 2) (/ 4 (- (sqrt (fma (* -4 a) c (* b b))) b)))
121.0ms
(/ 4 (- (sqrt (fma (* -4 a) c (* b b))) b))
58.0ms
(sqrt (fma (* -4 a) c (* b b)))
55.0ms
(- (sqrt (fma (* -4 a) c (* b b))) b)

simplify1.4s

Counts
41 → 88
Calls

41 calls. Slowest were:

488.0ms
(* (* (* (/ c 2) (/ c 2)) (/ c 2)) (* (* (/ 4 (- (sqrt (fma (* -4 a) c (* b b))) b)) (/ 4 (- (sqrt (fma (* -4 a) c (* b b))) b))) (/ 4 (- (sqrt (fma (* -4 a) c (* b b))) b))))
208.0ms
(+ (log (/ c 2)) (log (/ 4 (- (sqrt (fma (* -4 a) c (* b b))) b))))
87.0ms
(* (/ c 2) (/ 4 (- (sqrt (fma (* -4 a) c (* b b))) b)))

prune1.0s

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 6.4b

localize15.0ms

Local error

Found 4 expressions with local error:

24.3b
(sqrt (fma (* -4 a) c (* b b)))
24.3b
(sqrt (fma (* -4 a) c (* b b)))
14.1b
(fma (sqrt (sqrt (fma (* -4 a) c (* b b)))) (sqrt (sqrt (fma (* -4 a) c (* b b)))) (- b))
0.2b
(* (/ c 2) (/ 4 (fma (sqrt (sqrt (fma (* -4 a) c (* b b)))) (sqrt (sqrt (fma (* -4 a) c (* b b)))) (- b))))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
4 → 62
Calls

4 calls. Slowest were:

3.0ms
(* (/ c 2) (/ 4 (fma (sqrt (sqrt (fma (* -4 a) c (* b b)))) (sqrt (sqrt (fma (* -4 a) c (* b b)))) (- b))))
1.0ms
(sqrt (fma (* -4 a) c (* b b)))
1.0ms
(sqrt (fma (* -4 a) c (* b b)))

series720.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

314.0ms
(* (/ c 2) (/ 4 (fma (sqrt (sqrt (fma (* -4 a) c (* b b)))) (sqrt (sqrt (fma (* -4 a) c (* b b)))) (- b))))
308.0ms
(fma (sqrt (sqrt (fma (* -4 a) c (* b b)))) (sqrt (sqrt (fma (* -4 a) c (* b b)))) (- b))
55.0ms
(sqrt (fma (* -4 a) c (* b b)))
43.0ms
(sqrt (fma (* -4 a) c (* b b)))

simplify2.2s

Counts
33 → 74
Calls

33 calls. Slowest were:

490.0ms
(* (* (* (/ c 2) (/ c 2)) (/ c 2)) (* (* (/ 4 (fma (sqrt (sqrt (fma (* -4 a) c (* b b)))) (sqrt (sqrt (fma (* -4 a) c (* b b)))) (- b))) (/ 4 (fma (sqrt (sqrt (fma (* -4 a) c (* b b)))) (sqrt (sqrt (fma (* -4 a) c (* b b)))) (- b)))) (/ 4 (fma (sqrt (sqrt (fma (* -4 a) c (* b b)))) (sqrt (sqrt (fma (* -4 a) c (* b b)))) (- b)))))
364.0ms
(- (* 1/2 (/ (pow (exp (* 1/4 (- (log -4) (+ (log (/ -1 c)) (log (/ -1 a)))))) 4) (* a b))) (+ (* 2 (/ c b)) (* 2 (/ (* c (pow (exp (* 1/4 (- (log -4) (+ (log (/ -1 c)) (log (/ -1 a)))))) 2)) (pow b 2)))))
319.0ms
(- (* 1/2 (/ (pow (exp (* 1/4 (- (log -4) (+ (log (/ 1 a)) (log (/ 1 c)))))) 4) (* a b))) (+ (* 2 (/ c b)) (* 2 (/ (* (pow (exp (* 1/4 (- (log -4) (+ (log (/ 1 a)) (log (/ 1 c)))))) 2) c) (pow b 2)))))

prune879.0ms

Pruning

8 alts after pruning (5 fresh and 3 done)

Merged error: 6.4b

regimes265.0ms

Accuracy

95.2% (1.1b remaining)

Error of 7.1b against oracle of 6.0b and baseline of 29.3b

bsearch510.0ms

end0.0ms

sample5.1s

Algorithm
intervals