Average Error: 3.8 → 1.9
Time: 27.3s
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 3.8

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

  2. Initial simplification1.9

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

  3. Final simplification1.9

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

Reproduce

herbie shell --seed 2018365 +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: 14.2s)Debug log

start325.0ms

Algorithm
intervals

setup367.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 1.0b

localize74.0ms

Local error

Found 4 expressions with local error:

4.9b
(* (/ (* 2.0 z) t) (sqrt (+ a t)))
1.1b
(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))

rewrite34.0ms

Algorithm
rewrite-expression-head
Counts
4 → 89
Calls

4 calls. Slowest were:

18.0ms
(- (+ (/ 5.0 6.0) a) (/ (/ 2.0 t) 3.0))
9.0ms
(/ (/ 2.0 t) 3.0)
6.0ms
(* (/ (* 2.0 z) t) (sqrt (+ a t)))

series220.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

148.0ms
(fma (- (+ (/ 5.0 6.0) a) (/ (/ 2.0 t) 3.0)) (* (- c b) 2.0) (* (/ (* 2.0 z) t) (sqrt (+ a t))))
50.0ms
(* (/ (* 2.0 z) t) (sqrt (+ a t)))
14.0ms
(- (+ (/ 5.0 6.0) a) (/ (/ 2.0 t) 3.0))
8.0ms
(/ (/ 2.0 t) 3.0)

simplify6.2s

Counts
68 → 101
Calls

68 calls. Slowest were:

406.0ms
(fma (sqrt (+ (/ 5.0 6.0) a)) (sqrt (+ (/ 5.0 6.0) a)) (- (* (/ 1 3.0) (/ 2.0 t))))
377.0ms
(fma (* (cbrt (+ (/ 5.0 6.0) a)) (cbrt (+ (/ 5.0 6.0) a))) (cbrt (+ (/ 5.0 6.0) a)) (- (* (/ 1 3.0) (/ 2.0 t))))
351.0ms
(fma 1 (+ (/ 5.0 6.0) a) (- (* (/ 1 3.0) (/ 2.0 t))))

prune2.6s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize21.0ms

Local error

Found 3 expressions with local error:

0.0b
(- (* (* 2.0 a) (- c b)) (* 1.6666666666666667 b))
0.0b
(/ x (fma y (exp (- (* (* 2.0 a) (- c b)) (* 1.6666666666666667 b))) x))
0.0b
(* (* 2.0 a) (- c b))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
3 → 58
Calls

3 calls. Slowest were:

6.0ms
(- (* (* 2.0 a) (- c b)) (* 1.6666666666666667 b))
4.0ms
(* (* 2.0 a) (- c b))
0.0ms
(/ x (fma y (exp (- (* (* 2.0 a) (- c b)) (* 1.6666666666666667 b))) x))

series239.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

161.0ms
(/ x (fma y (exp (- (* (* 2.0 a) (- c b)) (* 1.6666666666666667 b))) x))
46.0ms
(- (* (* 2.0 a) (- c b)) (* 1.6666666666666667 b))
32.0ms
(* (* 2.0 a) (- c b))

simplify2.5s

Counts
32 → 67
Calls

32 calls. Slowest were:

320.0ms
(/ (exp (* (* 2.0 a) (- c b))) (exp (* 1.6666666666666667 b)))
296.0ms
(- (* 2.0 (* a c)) (+ (* 2.0 (* a b)) (* 1.6666666666666667 b)))
287.0ms
(- (* 2.0 (* a c)) (+ (* 2.0 (* a b)) (* 1.6666666666666667 b)))

prune1.2s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes461.0ms

Accuracy

0% (1.0b remaining)

Error of 1.9b against oracle of 0.8b and baseline of 1.9b

bsearch5.0ms