Average Error: 3.9 → 2.0
Time: 25.4s
Precision: 64
Internal Precision: 128
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\frac{x}{(y \cdot \left(e^{(\left(\left(a + \frac{5.0}{6.0}\right) - \frac{\frac{2.0}{t}}{3.0}\right) \cdot \left(\left(c - b\right) \cdot 2.0\right) + \left(\sqrt{a + t} \cdot \frac{z \cdot 2.0}{t}\right))_*}\right) + x)_*}\]

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 3.9

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

  2. Initial simplification2.0

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

  3. Final simplification2.0

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

Reproduce

herbie shell --seed 2018360 +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: 11.7s)Debug log

start409.0ms

Algorithm
intervals

setup303.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.2b

localize77.0ms

Local error

Found 4 expressions with local error:

4.7b
(* (/ (* 2.0 z) t) (sqrt (+ a t)))
0.3b
(fma (- (+ (/ 5.0 6.0) a) (/ (/ 2.0 t) 3.0)) (* (- c b) 2.0) (* (/ (* 2.0 z) t) (sqrt (+ a t))))
0.2b
(/ (/ 2.0 t) 3.0)
0.0b
(- (+ (/ 5.0 6.0) a) (/ (/ 2.0 t) 3.0))

rewrite54.0ms

Algorithm
rewrite-expression-head
Counts
4 → 89
Calls

4 calls. Slowest were:

36.0ms
(- (+ (/ 5.0 6.0) a) (/ (/ 2.0 t) 3.0))
11.0ms
(* (/ (* 2.0 z) t) (sqrt (+ a t)))
5.0ms
(/ (/ 2.0 t) 3.0)

series286.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

195.0ms
(fma (- (+ (/ 5.0 6.0) a) (/ (/ 2.0 t) 3.0)) (* (- c b) 2.0) (* (/ (* 2.0 z) t) (sqrt (+ a t))))
58.0ms
(* (/ (* 2.0 z) t) (sqrt (+ a t)))
19.0ms
(- (+ (/ 5.0 6.0) a) (/ (/ 2.0 t) 3.0))
13.0ms
(/ (/ 2.0 t) 3.0)

simplify5.8s

Counts
68 → 101
Calls

68 calls. Slowest were:

673.0ms
(- (* (+ (pow (/ 5.0 6.0) 3) (pow a 3)) 3.0) (* (+ (* (/ 5.0 6.0) (/ 5.0 6.0)) (- (* a a) (* (/ 5.0 6.0) a))) (/ 2.0 t)))
331.0ms
(- (* 2.0 (* a c)) (+ (* 2.0 (* a b)) (* 1.6666666666666667 b)))
319.0ms
(fma 1 (+ (/ 5.0 6.0) a) (- (* (/ (/ 2.0 t) 3.0) 1)))

prune2.7s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize41.0ms

Local error

Found 2 expressions with local error:

0.1b
(fma (/ 1.3333333333333333 t) (- b c) (* 1.6666666666666667 c))
0.0b
(/ x (fma y (exp (fma (/ 1.3333333333333333 t) (- b c) (* 1.6666666666666667 c))) x))

rewrite2.0ms

Algorithm
rewrite-expression-head
Counts
2 → 25
Calls

2 calls. Slowest were:

1.0ms
(/ x (fma y (exp (fma (/ 1.3333333333333333 t) (- b c) (* 1.6666666666666667 c))) x))
0.0ms
(fma (/ 1.3333333333333333 t) (- b c) (* 1.6666666666666667 c))

series196.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

159.0ms
(/ x (fma y (exp (fma (/ 1.3333333333333333 t) (- b c) (* 1.6666666666666667 c))) x))
37.0ms
(fma (/ 1.3333333333333333 t) (- b c) (* 1.6666666666666667 c))

simplify746.0ms

Counts
9 → 31
Calls

9 calls. Slowest were:

177.0ms
(cbrt (fma y (exp (fma (/ 1.3333333333333333 t) (- b c) (* 1.6666666666666667 c))) x))
167.0ms
(- (+ (* 1.3333333333333333 (/ b t)) (* 1.6666666666666667 c)) (* 1.3333333333333333 (/ c t)))
127.0ms
(- (+ (* 1.6666666666666667 c) (* 1.3333333333333333 (/ b t))) (* 1.3333333333333333 (/ c t)))

prune541.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes475.0ms

Accuracy

0% (1.7b remaining)

Error of 2.0b against oracle of 0.3b and baseline of 2.0b

bsearch4.0ms