Average Error: 11.7 → 11.4

Time: 24.2s

Precision: 64

Internal Precision: 128

\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le 1.3099434334785528 \cdot 10^{+80}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + \left(\left(\left(x \cdot t\right) \cdot \left(-a\right) + \sqrt[3]{\left(y \cdot z\right) \cdot x} \cdot \left(\sqrt[3]{\left(y \cdot z\right) \cdot x} \cdot \sqrt[3]{\left(y \cdot z\right) \cdot x}\right)\right) - b \cdot \left(z \cdot c - a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z - a \cdot t\right) \cdot x - \sqrt[3]{z \cdot c - a \cdot i} \cdot \left(b \cdot \left(\sqrt[3]{z \cdot c - a \cdot i} \cdot \sqrt[3]{z \cdot c - a \cdot i}\right)\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \end{array}\]

Error

The X axis uses an exponential scale — Bits error versus `x`

Try it out

In
Out

Enter valid numbers for all inputs

Derivation

Split input into 2 regimes
if x < 1.3099434334785528e+80
1. Initial program 12.3
  \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
2. Using strategy rm
3. Applied sub-neg12.3
  \[\leadsto \left(x \cdot \color{blue}{\left(y \cdot z + \left(-t \cdot a\right)\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
4. Applied distribute-lft-in12.3
  \[\leadsto \left(\color{blue}{\left(x \cdot \left(y \cdot z\right) + x \cdot \left(-t \cdot a\right)\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
5. Taylor expanded around -inf 11.8
  \[\leadsto \left(\left(x \cdot \left(y \cdot z\right) + \color{blue}{-1 \cdot \left(a \cdot \left(x \cdot t\right)\right)}\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
6. Simplified11.8
  \[\leadsto \left(\left(x \cdot \left(y \cdot z\right) + \color{blue}{\left(x \cdot t\right) \cdot \left(-a\right)}\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
7. Using strategy rm
8. Applied add-cube-cbrt11.9
  \[\leadsto \left(\left(\color{blue}{\left(\sqrt[3]{x \cdot \left(y \cdot z\right)} \cdot \sqrt[3]{x \cdot \left(y \cdot z\right)}\right) \cdot \sqrt[3]{x \cdot \left(y \cdot z\right)}} + \left(x \cdot t\right) \cdot \left(-a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
if 1.3099434334785528e+80 < x
1. Initial program 7.4
  \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
2. Using strategy rm
3. Applied add-cube-cbrt7.6
  \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(\left(\sqrt[3]{c \cdot z - i \cdot a} \cdot \sqrt[3]{c \cdot z - i \cdot a}\right) \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
4. Applied associate-*r*7.6
  \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\left(b \cdot \left(\sqrt[3]{c \cdot z - i \cdot a} \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)\right) \cdot \sqrt[3]{c \cdot z - i \cdot a}}\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
Recombined 2 regimes into one program.
Final simplification11.4
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.3099434334785528 \cdot 10^{+80}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + \left(\left(\left(x \cdot t\right) \cdot \left(-a\right) + \sqrt[3]{\left(y \cdot z\right) \cdot x} \cdot \left(\sqrt[3]{\left(y \cdot z\right) \cdot x} \cdot \sqrt[3]{\left(y \cdot z\right) \cdot x}\right)\right) - b \cdot \left(z \cdot c - a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z - a \cdot t\right) \cdot x - \sqrt[3]{z \cdot c - a \cdot i} \cdot \left(b \cdot \left(\sqrt[3]{z \cdot c - a \cdot i} \cdot \sqrt[3]{z \cdot c - a \cdot i}\right)\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019022 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))

Details

sample229.0ms

Algorithm

intervals

Results

185.0ms

623×

body

valid

simplify144.0ms

Counts

1 → 1

Calls

1 calls:

Slowest

144.0ms

(+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y))))

prune12.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 10.3b

localize57.0ms

Local error

Found 4 expressions with local error:

5.5b	(* b (- (* c z) (* i a)))
4.7b	(* j (- (* c t) (* i y)))
3.8b	(* x (- (* y z) (* t a)))
0.2b	(- (* y z) (* t a))

rewrite12.0ms

Algorithm

rewrite-expression-head

Rules

9×	`associate-r`
7×	`add-cube-cbrt`
7×	`*-un-lft-identity`
7×	`sub-neg`
7×	`add-sqr-sqrt`
6×	`add-log-exp`
6×	`associate-*r/`
6×	`distribute-lft-in`
6×	`distribute-rgt-in`
4×	`flip--`
4×	`add-exp-log`
4×	`add-cbrt-cube`
4×	`flip3--`
4×	`pow1`
3×	`*-commutative`
1×	`diff-log`

Counts

4 → 63

Calls

4 calls:

Slowest

3.0ms	(* b (- (* c z) (* i a)))
3.0ms	(* j (- (* c t) (* i y)))
3.0ms	(* x (- (* y z) (* t a)))
2.0ms	(- (* y z) (* t a))

series219.0ms

Counts

4 → 12

Calls

4 calls:

Slowest

72.0ms	(* x (- (* y z) (* t a)))
69.0ms	(* j (- (* c t) (* i y)))
50.0ms	(* b (- (* c z) (* i a)))
28.0ms	(- (* y z) (* t a))

simplify1.2s

Counts

40 → 75

Calls

40 calls:

Slowest

128.0ms	(+ (* (* y z) (* y z)) (+ (* (* t a) (* t a)) (* (* y z) (* t a))))
121.0ms	(+ (* (* c t) (* c t)) (+ (* (* i y) (* i y)) (* (* c t) (* i y))))
102.0ms	(+ (* (* c z) (* c z)) (+ (* (* i a) (* i a)) (* (* c z) (* i a))))
88.0ms	(- (* z (* b c)) (* a (* i b)))
70.0ms	(- (* t (* j c)) (* i (* y j)))

prune1.4s

Pruning

18 alts after pruning (18 fresh and 0 done)

Merged error: 7.1b

localize19.0ms

Local error

Found 4 expressions with local error:

5.5b	(* b (- (* c z) (* i a)))
4.8b	(* x (- (* t a)))
4.7b	(* j (- (* c t) (* i y)))
3.1b	(* x (* y z))

rewrite9.0ms

Algorithm

rewrite-expression-head

Rules

13×	`associate-r`
7×	`add-cube-cbrt`
7×	`*-un-lft-identity`
7×	`add-sqr-sqrt`
4×	`add-log-exp`
4×	`*-commutative`
4×	`add-exp-log`
4×	`associate-*r/`
4×	`add-cbrt-cube`
4×	`distribute-lft-in`
4×	`sub-neg`
4×	`pow1`
4×	`distribute-rgt-in`
2×	`flip--`
2×	`flip3--`
1×	`distribute-rgt-neg-out`
1×	`distribute-rgt-neg-in`
1×	`neg-mul-1`
1×	`distribute-lft-neg-in`

Counts

4 → 58

Calls

4 calls:

Slowest

3.0ms	(* b (- (* c z) (* i a)))
3.0ms	(* j (- (* c t) (* i y)))
1.0ms	(* x (- (* t a)))
1.0ms	(* x (* y z))

series547.0ms

Counts

4 → 12

Calls

4 calls:

Slowest

444.0ms	(* j (- (* c t) (* i y)))
58.0ms	(* b (- (* c z) (* i a)))
29.0ms	(* x (- (* t a)))
16.0ms	(* x (* y z))

simplify886.0ms

Counts

35 → 70

Calls

35 calls:

Slowest

112.0ms	(+ (* (* c t) (* c t)) (+ (* (* i y) (* i y)) (* (* c t) (* i y))))
104.0ms	(+ (* (* c z) (* c z)) (+ (* (* i a) (* i a)) (* (* c z) (* i a))))
87.0ms	(- (* z (* b c)) (* a (* i b)))
61.0ms	(- (* t (* j c)) (* i (* y j)))
56.0ms	(- (* z (* b c)) (* a (* i b)))

prune1.6s

Pruning

17 alts after pruning (17 fresh and 0 done)

Merged error: 5.7b

localize22.0ms

Local error

Found 4 expressions with local error:

5.5b	(* b (- (* c z) (* i a)))
4.7b	(* j (- (* c t) (* i y)))
3.5b	(* (* x t) (- a))
3.1b	(* x (* y z))

rewrite14.0ms

Algorithm

rewrite-expression-head

Rules

11×	`associate-r`
7×	`add-cube-cbrt`
7×	`*-un-lft-identity`
7×	`add-sqr-sqrt`
6×	`add-exp-log`
6×	`add-cbrt-cube`
6×	`pow1`
4×	`add-log-exp`
4×	`*-commutative`
4×	`associate-*r/`
4×	`distribute-lft-in`
4×	`sub-neg`
4×	`distribute-rgt-in`
2×	`flip--`
2×	`flip3--`
1×	`cbrt-unprod`
1×	`distribute-rgt-neg-out`
1×	`prod-exp`
1×	`pow-prod-down`
1×	`associate-l`
1×	`neg-mul-1`

Counts

4 → 60

Calls

4 calls:

Slowest

6.0ms	(* b (- (* c z) (* i a)))
3.0ms	(* j (- (* c t) (* i y)))
3.0ms	(* (* x t) (- a))
1.0ms	(* x (* y z))

series156.0ms

Counts

4 → 12

Calls

4 calls:

Slowest

64.0ms	(* j (- (* c t) (* i y)))
47.0ms	(* b (- (* c z) (* i a)))
27.0ms	(* (* x t) (- a))
17.0ms	(* x (* y z))

simplify1.4s

Counts

36 → 72

Calls

36 calls:

Slowest

546.0ms	(* (* (* (* x t) (* x t)) (* x t)) (* (* (- a) (- a)) (- a)))
150.0ms	(+ (* (* c t) (* c t)) (+ (* (* i y) (* i y)) (* (* c t) (* i y))))
124.0ms	(+ (* (* c z) (* c z)) (+ (* (* i a) (* i a)) (* (* c z) (* i a))))
67.0ms	(- (* t (* j c)) (* i (* y j)))
56.0ms	(- (* z (* b c)) (* a (* i b)))

prune1.4s

Pruning

17 alts after pruning (17 fresh and 0 done)

Merged error: 5.2b

localize38.0ms

Local error

Found 4 expressions with local error:

17.1b	(cbrt (* x (* y z)))
17.1b	(cbrt (* x (* y z)))
17.1b	(cbrt (* x (* y z)))
5.5b	(* b (- (* c z) (* i a)))

rewrite10.0ms

Algorithm

rewrite-expression-head

Rules

5×	`add-cube-cbrt`
5×	`*-un-lft-identity`
5×	`add-sqr-sqrt`
4×	`add-log-exp`
4×	`add-exp-log`
4×	`add-cbrt-cube`
4×	`pow1`
3×	`associate-r`
3×	`pow1/3`
3×	`cbrt-prod`
2×	`associate-*r/`
2×	`distribute-lft-in`
2×	`sub-neg`
2×	`distribute-rgt-in`
1×	`flip--`
1×	`*-commutative`
1×	`flip3--`

Counts

4 → 44

Calls

4 calls:

Slowest

6.0ms	(* b (- (* c z) (* i a)))
1.0ms	(cbrt (* x (* y z)))
1.0ms	(cbrt (* x (* y z)))
1.0ms	(cbrt (* x (* y z)))

series386.0ms

Counts

4 → 12

Calls

4 calls:

Slowest

126.0ms	(cbrt (* x (* y z)))
110.0ms	(cbrt (* x (* y z)))
94.0ms	(cbrt (* x (* y z)))
55.0ms	(* b (- (* c z) (* i a)))

simplify1.5s

Counts

24 → 56

Calls

24 calls:

Slowest

207.0ms	(* (cbrt -1) (exp (* -1/3 (+ (log (/ -1 x)) (+ (log (/ -1 z)) (log (/ -1 y)))))))
190.0ms	(* (cbrt -1) (exp (* -1/3 (+ (log (/ -1 x)) (+ (log (/ -1 z)) (log (/ -1 y)))))))
144.0ms	(* (cbrt -1) (exp (* -1/3 (+ (log (/ -1 x)) (+ (log (/ -1 z)) (log (/ -1 y)))))))
123.0ms	(exp (* -1/3 (+ (log (/ 1 z)) (+ (log (/ 1 x)) (log (/ 1 y))))))
118.0ms	(+ (* (* c z) (* c z)) (+ (* (* i a) (* i a)) (* (* c z) (* i a))))

prune1.3s

Pruning

17 alts after pruning (16 fresh and 1 done)

Merged error: 5.2b

regimes2.0s

Accuracy

13.8% (5.1b remaining)

Error of 11.4b against oracle of 6.3b and baseline of 12.2b

bsearch302.0ms

end0.0ms

sample6.5s

Algorithm

intervals

Results

5.6s

18959×

body

valid

Error

Try it out

Derivation

`if x < 1.3099434334785528e+80`

`if 1.3099434334785528e+80 < x`

Reproduce

Details

sample229.0ms

simplify144.0ms

prune12.0ms

localize57.0ms

rewrite12.0ms

series219.0ms

simplify1.2s

prune1.4s

localize19.0ms

rewrite9.0ms

series547.0ms

simplify886.0ms

prune1.6s

localize22.0ms

rewrite14.0ms

series156.0ms

simplify1.4s

prune1.4s

localize38.0ms

rewrite10.0ms

series386.0ms

simplify1.5s

prune1.3s

regimes2.0s

bsearch302.0ms

end0.0ms

sample6.5s