Average Error: 3.6 → 1.7
Time: 51.9s
Precision: 64
Internal Precision: 128
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\frac{x}{x + y \cdot e^{(\left(\left(a + \frac{5.0}{6.0}\right) - \frac{\frac{2.0}{t}}{3.0}\right) \cdot \left(c - b\right) + \left(\frac{z}{t} \cdot \sqrt{a + t}\right))_* \cdot 2.0}}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 3.6

    \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt3.6

    \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\color{blue}{\left(\sqrt[3]{\frac{z \cdot \sqrt{t + a}}{t}} \cdot \sqrt[3]{\frac{z \cdot \sqrt{t + a}}{t}}\right) \cdot \sqrt[3]{\frac{z \cdot \sqrt{t + a}}{t}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]

  4. Applied prod-diff22.5

    \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \color{blue}{\left((\left(\sqrt[3]{\frac{z \cdot \sqrt{t + a}}{t}} \cdot \sqrt[3]{\frac{z \cdot \sqrt{t + a}}{t}}\right) \cdot \left(\sqrt[3]{\frac{z \cdot \sqrt{t + a}}{t}}\right) + \left(-\left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right) \cdot \left(b - c\right)\right))_* + (\left(-\left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right) \cdot \left(b - c\right) + \left(\left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right) \cdot \left(b - c\right)\right))_*\right)}}}\]

  5. Simplified20.7

    \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\color{blue}{(\left(\left(a + \frac{5.0}{6.0}\right) - \frac{\frac{2.0}{t}}{3.0}\right) \cdot \left(c - b\right) + \left(\frac{z}{t} \cdot \sqrt{t + a}\right))_*} + (\left(-\left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right) \cdot \left(b - c\right) + \left(\left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right) \cdot \left(b - c\right)\right))_*\right)}}\]

  6. Simplified1.7

    \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left((\left(\left(a + \frac{5.0}{6.0}\right) - \frac{\frac{2.0}{t}}{3.0}\right) \cdot \left(c - b\right) + \left(\frac{z}{t} \cdot \sqrt{t + a}\right))_* + \color{blue}{0}\right)}}\]

  7. Final simplification1.7

    \[\leadsto \frac{x}{x + y \cdot e^{(\left(\left(a + \frac{5.0}{6.0}\right) - \frac{\frac{2.0}{t}}{3.0}\right) \cdot \left(c - b\right) + \left(\frac{z}{t} \cdot \sqrt{a + t}\right))_* \cdot 2.0}}\]

Reproduce

herbie shell --seed 2019008 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))

Details

Time bar (total: 51.4s)Debug log

sample289.0ms

Algorithm
intervals

simplify292.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

292.0ms
(/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))))))

prune25.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 2.7b

localize80.0ms

Local error

Found 4 expressions with local error:

4.5b
(/ (* z (sqrt (+ t a))) t)
1.9b
(- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))
0.3b
(/ 2.0 (* t 3.0))
0.2b
(* z (sqrt (+ t a)))

rewrite23.0ms

Algorithm
rewrite-expression-head
Counts
4 → 112
Calls

4 calls. Slowest were:

11.0ms
(- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))
4.0ms
(* z (sqrt (+ t a)))
4.0ms
(/ (* z (sqrt (+ t a))) t)

series282.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

153.0ms
(- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))
78.0ms
(/ (* z (sqrt (+ t a))) t)
43.0ms
(* z (sqrt (+ t a)))
8.0ms
(/ 2.0 (* t 3.0))

simplify16.2s

Counts
92 → 124
Calls

92 calls. Slowest were:

595.0ms
(* t (* (+ b c) (* (- a (/ 5.0 6.0)) (* t 3.0))))
540.0ms
(- (* (* z (sqrt (+ t a))) (+ (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))) (* t (* (- b c) (- (* (+ a (/ 5.0 6.0)) (+ a (/ 5.0 6.0))) (* (/ 2.0 (* t 3.0)) (/ 2.0 (* t 3.0)))))))
507.0ms
(* (fma (- (sqrt (/ 2.0 (* t 3.0)))) (sqrt (/ 2.0 (* t 3.0))) (* (sqrt (/ 2.0 (* t 3.0))) (sqrt (/ 2.0 (* t 3.0))))) (- b c))

prune2.7s

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize44.0ms

Local error

Found 4 expressions with local error:

4.6b
(* (/ z t) (sqrt (+ t a)))
1.6b
(fma (- (+ a (/ 5.0 6.0)) (/ (/ 2.0 t) 3.0)) (- c b) (* (/ z t) (sqrt (+ t a))))
0.5b
(/ (/ 2.0 t) 3.0)
0.0b
(+ a (/ 5.0 6.0))

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
4 → 67
Calls

4 calls. Slowest were:

9.0ms
(* (/ z t) (sqrt (+ t a)))
5.0ms
(/ (/ 2.0 t) 3.0)
2.0ms
(+ a (/ 5.0 6.0))

series228.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

147.0ms
(fma (- (+ a (/ 5.0 6.0)) (/ (/ 2.0 t) 3.0)) (- c b) (* (/ z t) (sqrt (+ t a))))
53.0ms
(* (/ z t) (sqrt (+ t a)))
15.0ms
(+ a (/ 5.0 6.0))
12.0ms
(/ (/ 2.0 t) 3.0)

simplify1.3s

Counts
32 → 79
Calls

32 calls. Slowest were:

598.0ms
(* (* (* (/ z t) (/ z t)) (/ z t)) (* (* (sqrt (+ t a)) (sqrt (+ t a))) (sqrt (+ t a))))
139.0ms
(- (* a c) (+ (* a b) (* 0.8333333333333334 b)))
134.0ms
(- (+ (* 0.6666666666666666 (/ b t)) (* 0.8333333333333334 c)) (* 0.6666666666666666 (/ c t)))

prune1.9s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize44.0ms

Local error

Found 4 expressions with local error:

3.6b
(fma (* z (sqrt (+ t a))) (/ 1 t) (- (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))
0.3b
(/ 2.0 (* t 3.0))
0.2b
(* z (sqrt (+ t a)))
0.1b
(* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))

rewrite30.0ms

Algorithm
rewrite-expression-head
Counts
4 → 106
Calls

4 calls. Slowest were:

23.0ms
(* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))
4.0ms
(* z (sqrt (+ t a)))
1.0ms
(/ 2.0 (* t 3.0))

series276.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

170.0ms
(fma (* z (sqrt (+ t a))) (/ 1 t) (- (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))
53.0ms
(* z (sqrt (+ t a)))
40.0ms
(* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))
13.0ms
(/ 2.0 (* t 3.0))

simplify12.6s

Counts
82 → 118
Calls

82 calls. Slowest were:

731.0ms
(+ (* (+ a (/ 5.0 6.0)) (+ a (/ 5.0 6.0))) (+ (* (/ 2.0 (* t 3.0)) (/ 2.0 (* t 3.0))) (* (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))
666.0ms
(* (+ b c) (* (+ (* a a) (- (* (/ 5.0 6.0) (/ 5.0 6.0)) (* a (/ 5.0 6.0)))) (* t 3.0)))
652.0ms
(* (+ (* b b) (+ (* c c) (* b c))) (* (- a (/ 5.0 6.0)) (* t 3.0)))

prune2.9s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

localize36.0ms

Local error

Found 4 expressions with local error:

0.0b
(* y (exp (* 2.0 (fma (- c b) a (* 0.8333333333333334 c)))))
0.0b
(fma (- c b) a (* 0.8333333333333334 c))
0.0b
(exp (* 2.0 (fma (- c b) a (* 0.8333333333333334 c))))
0.0b
(/ x (+ x (* y (exp (* 2.0 (fma (- c b) a (* 0.8333333333333334 c)))))))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
4 → 55
Calls

4 calls. Slowest were:

3.0ms
(* y (exp (* 2.0 (fma (- c b) a (* 0.8333333333333334 c)))))
2.0ms
(exp (* 2.0 (fma (- c b) a (* 0.8333333333333334 c))))
0.0ms
(/ x (+ x (* y (exp (* 2.0 (fma (- c b) a (* 0.8333333333333334 c)))))))

series255.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

107.0ms
(/ x (+ x (* y (exp (* 2.0 (fma (- c b) a (* 0.8333333333333334 c)))))))
71.0ms
(* y (exp (* 2.0 (fma (- c b) a (* 0.8333333333333334 c)))))
40.0ms
(exp (* 2.0 (fma (- c b) a (* 0.8333333333333334 c))))
38.0ms
(fma (- c b) a (* 0.8333333333333334 c))

simplify4.1s

Counts
26 → 67
Calls

26 calls. Slowest were:

475.0ms
(exp (* 2.0 (* (- c b) a)))
464.0ms
(exp (* (* (- c b) a) 2.0))
326.0ms
(+ (* x x) (- (* (* y (exp (* 2.0 (fma (- c b) a (* 0.8333333333333334 c))))) (* y (exp (* 2.0 (fma (- c b) a (* 0.8333333333333334 c)))))) (* x (* y (exp (* 2.0 (fma (- c b) a (* 0.8333333333333334 c))))))))

prune988.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

regimes657.0ms

Accuracy

0% (1.6b remaining)

Error of 1.7b against oracle of 0.0b and baseline of 1.7b

bsearch5.0ms

end0.0ms

sample6.1s

Algorithm
intervals