Average Error: 10.9 → 5.4
Time: 7.8s
Precision: 64
Internal Precision: 128
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 = -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -2.632195272802239 \cdot 10^{-178}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 0.0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.4431123399596268 \cdot 10^{+161}:\\ \;\;\;\;\frac{a1 \cdot a2}{b2 \cdot b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.9
Target11.2
Herbie5.4
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (* a1 a2) < -inf.0 or -2.632195272802239e-178 < (* a1 a2) < 0.0 or 1.4431123399596268e+161 < (* a1 a2)

    1. Initial program 22.6

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
    2. Using strategy rm

    3. Applied times-frac6.4

      \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}}\]

    if -inf.0 < (* a1 a2) < -2.632195272802239e-178

    1. Initial program 5.2

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
    2. Using strategy rm

    3. Applied div-inv5.4

      \[\leadsto \color{blue}{\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}}\]
    4. Using strategy rm

    5. Applied associate-/r*5.1

      \[\leadsto \left(a1 \cdot a2\right) \cdot \color{blue}{\frac{\frac{1}{b1}}{b2}}\]

    if 0.0 < (* a1 a2) < 1.4431123399596268e+161

    1. Initial program 4.8

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
    2. Using strategy rm

    3. Applied div-inv5.1

      \[\leadsto \color{blue}{\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}}\]
    4. Using strategy rm

    5. Applied associate-/r*4.9

      \[\leadsto \left(a1 \cdot a2\right) \cdot \color{blue}{\frac{\frac{1}{b1}}{b2}}\]

    6. Taylor expanded around inf 4.8

      \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b2 \cdot b1}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification5.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 = -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -2.632195272802239 \cdot 10^{-178}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 0.0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.4431123399596268 \cdot 10^{+161}:\\ \;\;\;\;\frac{a1 \cdot a2}{b2 \cdot b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019010 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

  :herbie-target
  (* (/ a1 b1) (/ a2 b2))

  (/ (* a1 a2) (* b1 b2)))

Details

Time bar (total: 6.6s)Debug log

sample31.0ms

Algorithm
intervals

simplify24.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
24.0ms
(/ (* a1 a2) (* b1 b2))

prune6.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 9.3b

localize11.0ms

Local error

Found 1 expressions with local error:

9.3b
(/ (* a1 a2) (* b1 b2))

rewrite3.0ms

Algorithm
rewrite-expression-head
Rules
3×add-exp-log
3×add-cbrt-cube
1×add-log-exp
1×associate-/l*
1×div-inv
1×log1p-expm1-u
1×add-cube-cbrt
1×div-exp
1×times-frac
1×*-un-lft-identity
1×frac-2neg
1×associate-/r*
1×clear-num
1×cbrt-undiv
1×pow1
1×expm1-log1p-u
1×add-sqr-sqrt
Counts
1 → 17
Calls
1 calls:
Slowest
3.0ms
(/ (* a1 a2) (* b1 b2))

series20.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
20.0ms
(/ (* a1 a2) (* b1 b2))

simplify357.0ms

Counts
7 → 20
Calls
7 calls:
Slowest
188.0ms
(/ (* (* (* a1 a2) (* a1 a2)) (* a1 a2)) (* (* (* b1 b2) (* b1 b2)) (* b1 b2)))
62.0ms
(- (log (* a1 a2)) (log (* b1 b2)))
36.0ms
(/ (* a1 a2) (* b2 b1))
34.0ms
(/ (* a1 a2) (* b2 b1))
34.0ms
(/ (* a1 a2) (* b2 b1))

prune284.0ms

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 2 expressions with local error:

9.1b
(* (* a1 a2) (/ 1 (* b1 b2)))
0.7b
(/ 1 (* b1 b2))

rewrite10.0ms

Algorithm
rewrite-expression-head
Rules
7×add-exp-log
5×pow1
4×associate-*r*
4×add-cbrt-cube
3×add-cube-cbrt
3×*-un-lft-identity
3×add-sqr-sqrt
2×add-log-exp
2×div-inv
2×log1p-expm1-u
2×prod-exp
2×rec-exp
2×expm1-log1p-u
1×inv-pow
1×pow-flip
1×cbrt-unprod
1×*-commutative
1×un-div-inv
1×associate-*r/
1×pow-prod-down
1×associate-*l*
1×frac-2neg
1×associate-/r*
1×clear-num
Counts
2 → 37
Calls
2 calls:
Slowest
8.0ms
(* (* a1 a2) (/ 1 (* b1 b2)))
1.0ms
(/ 1 (* b1 b2))

series44.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
33.0ms
(* (* a1 a2) (/ 1 (* b1 b2)))
12.0ms
(/ 1 (* b1 b2))

simplify996.0ms

Counts
17 → 43
Calls
17 calls:
Slowest
421.0ms
(* (* a1 a2) (/ 1 (* b1 b2)))
274.0ms
(+ (log (* a1 a2)) (log (/ 1 (* b1 b2))))
121.0ms
(+ (log (* a1 a2)) (- (log (* b1 b2))))
99.0ms
(* (* (* (* a1 a2) (* a1 a2)) (* a1 a2)) (* (* (/ 1 (* b1 b2)) (/ 1 (* b1 b2))) (/ 1 (* b1 b2))))
20.0ms
(/ (* a1 a2) (* b2 b1))

prune486.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.0b

localize19.0ms

Local error

Found 2 expressions with local error:

9.1b
(* (* a1 a2) (/ (/ 1 b1) b2))
0.2b
(/ (/ 1 b1) b2)

rewrite13.0ms

Algorithm
rewrite-expression-head
Rules
4×associate-/l*
4×add-cube-cbrt
4×add-exp-log
4×associate-*r*
4×add-cbrt-cube
4×*-un-lft-identity
4×pow1
4×add-sqr-sqrt
3×div-inv
2×add-log-exp
2×log1p-expm1-u
2×expm1-log1p-u
1×associate-/l/
1×cbrt-unprod
1×*-commutative
1×associate-*r/
1×prod-exp
1×pow-prod-down
1×associate-*l*
1×frac-2neg
1×clear-num
Counts
2 → 36
Calls
2 calls:
Slowest
8.0ms
(* (* a1 a2) (/ (/ 1 b1) b2))
5.0ms
(/ (/ 1 b1) b2)

series47.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
35.0ms
(* (* a1 a2) (/ (/ 1 b1) b2))
11.0ms
(/ (/ 1 b1) b2)

simplify990.0ms

Counts
15 → 42
Calls
15 calls:
Slowest
649.0ms
(+ (log (* a1 a2)) (log (/ (/ 1 b1) b2)))
147.0ms
(* (* a1 a2) (/ (/ 1 b1) b2))
84.0ms
(* (* (* (* a1 a2) (* a1 a2)) (* a1 a2)) (* (* (/ (/ 1 b1) b2) (/ (/ 1 b1) b2)) (/ (/ 1 b1) b2)))
30.0ms
(/ (* a1 a2) (* b2 b1))
22.0ms
(/ (* a1 a2) (* b2 b1))

prune438.0ms

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 1 expressions with local error:

9.3b
(/ (* a1 a2) (* b2 b1))

rewrite6.0ms

Algorithm
rewrite-expression-head
Rules
3×add-exp-log
3×add-cbrt-cube
1×add-log-exp
1×associate-/l*
1×div-inv
1×log1p-expm1-u
1×add-cube-cbrt
1×div-exp
1×times-frac
1×*-un-lft-identity
1×frac-2neg
1×associate-/r*
1×clear-num
1×cbrt-undiv
1×pow1
1×expm1-log1p-u
1×add-sqr-sqrt
Counts
1 → 17
Calls
1 calls:
Slowest
6.0ms
(/ (* a1 a2) (* b2 b1))

series25.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
25.0ms
(/ (* a1 a2) (* b2 b1))

simplify359.0ms

Counts
7 → 20
Calls
7 calls:
Slowest
221.0ms
(/ (* (* (* a1 a2) (* a1 a2)) (* a1 a2)) (* (* (* b2 b1) (* b2 b1)) (* b2 b1)))
71.0ms
(- (log (* a1 a2)) (log (* b2 b1)))
23.0ms
(/ (* a1 a2) (* b2 b1))
23.0ms
(/ (* a1 a2) (* b2 b1))
17.0ms
(/ (* a1 a2) (* b2 b1))

prune222.0ms

Pruning

9 alts after pruning (7 fresh and 2 done)

Merged error: 0.0b

regimes781.0ms

Accuracy

50.3% (5.4b remaining)

Error of 5.4b against oracle of 0.0b and baseline of 10.9b

bsearch4.0ms

end0.0ms

sample1.4s

Algorithm
intervals