Original	26.3
Target	0.4
Herbie	13.2

Derivation

Split input into 3 regimes
if d < -8.346460948837449e+220
1. Initial program 42.5
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Simplified42.5
  \[\leadsto \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt42.5
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied *-un-lft-identity42.5
  \[\leadsto \frac{\color{blue}{1 \cdot (a \cdot c + \left(b \cdot d\right))_*}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
6. Applied times-frac42.5
  \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
7. Simplified42.5
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Simplified32.6
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied associate-*l/32.6
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]
11. Simplified32.6
  \[\leadsto \frac{\color{blue}{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
12. Using strategy rm
13. Applied *-un-lft-identity32.6
  \[\leadsto \frac{\frac{\color{blue}{1 \cdot (c \cdot a + \left(b \cdot d\right))_*}}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\]
14. Applied associate-/l*32.6
  \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sqrt{d^2 + c^2}^*}{(c \cdot a + \left(b \cdot d\right))_*}}}}{\sqrt{d^2 + c^2}^*}\]
15. Taylor expanded around -inf 11.2
  \[\leadsto \frac{\color{blue}{-1 \cdot b}}{\sqrt{d^2 + c^2}^*}\]
16. Simplified11.2
  \[\leadsto \frac{\color{blue}{-b}}{\sqrt{d^2 + c^2}^*}\]
if -8.346460948837449e+220 < d < 1.0043357683224773e+119
1. Initial program 21.3
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Simplified21.3
  \[\leadsto \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt21.3
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied *-un-lft-identity21.3
  \[\leadsto \frac{\color{blue}{1 \cdot (a \cdot c + \left(b \cdot d\right))_*}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
6. Applied times-frac21.3
  \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
7. Simplified21.3
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Simplified12.9
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied associate-*l/12.8
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]
11. Simplified12.8
  \[\leadsto \frac{\color{blue}{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
if 1.0043357683224773e+119 < d
1. Initial program 40.6
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Simplified40.6
  \[\leadsto \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt40.6
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied *-un-lft-identity40.6
  \[\leadsto \frac{\color{blue}{1 \cdot (a \cdot c + \left(b \cdot d\right))_*}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
6. Applied times-frac40.6
  \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
7. Simplified40.6
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Simplified26.2
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied associate-*l/26.2
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]
11. Simplified26.2
  \[\leadsto \frac{\color{blue}{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
12. Using strategy rm
13. Applied *-un-lft-identity26.2
  \[\leadsto \frac{\frac{\color{blue}{1 \cdot (c \cdot a + \left(b \cdot d\right))_*}}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\]
14. Applied associate-/l*26.2
  \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sqrt{d^2 + c^2}^*}{(c \cdot a + \left(b \cdot d\right))_*}}}}{\sqrt{d^2 + c^2}^*}\]
15. Taylor expanded around inf 15.8
  \[\leadsto \frac{\color{blue}{b}}{\sqrt{d^2 + c^2}^*}\]
Recombined 3 regimes into one program.
Final simplification13.2
\[\leadsto \begin{array}{l} \mathbf{if}\;d \le -8.346460948837449 \cdot 10^{+220}:\\ \;\;\;\;\frac{-b}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;d \le 1.0043357683224773 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Details

sample61.0ms

Algorithm: intervals

simplify13.0ms

Counts

1 → 1

Calls

1 calls. Slowest were:

12.0ms

(/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))

prune23.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 23.6b

localize17.0ms

Local error

Found 2 expressions with local error:

23.6b	(/ (fma a c (* b d)) (fma d d (* c c)))
0.0b	(fma a c (* b d))

rewrite4.0ms

Algorithm

rewrite-expression-head

Counts

2 → 39

Calls

2 calls. Slowest were:

3.0ms	(/ (fma a c (* b d)) (fma d d (* c c)))
0.0ms	(fma a c (* b d))

series119.0ms

Counts

2 → 6

Calls

2 calls. Slowest were:

72.0ms	(/ (fma a c (* b d)) (fma d d (* c c)))
47.0ms	(fma a c (* b d))

simplify603.0ms

Counts

31 → 45

Calls

31 calls. Slowest were:

241.0ms

(/ (* (* (fma a c (* b d)) (fma a c (* b d))) (fma a c (* b d))) (* (* (fma d d (* c c)) (fma d d (* c c))) (fma d d (* c c))))

28.0ms

(/ (* (cbrt (fma a c (* b d))) (cbrt (fma a c (* b d)))) (* (cbrt (fma d d (* c c))) (cbrt (fma d d (* c c)))))

27.0ms

(/ (* (cbrt (fma a c (* b d))) (cbrt (fma a c (* b d)))) 1)

prune529.0ms

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 14.7b

localize34.0ms

Local error

Found 4 expressions with local error:

14.4b	(/ (fma d b (* a c)) (hypot d c))
0.2b	(* (/ 1 (hypot d c)) (/ (fma d b (* a c)) (hypot d c)))
0.2b	(fma d b (* a c))
0.0b	(hypot d c)

rewrite12.0ms

Algorithm

rewrite-expression-head

Counts

4 → 86

Calls

4 calls. Slowest were:

7.0ms	(* (/ 1 (hypot d c)) (/ (fma d b (* a c)) (hypot d c)))
3.0ms	(/ (fma d b (* a c)) (hypot d c))
0.0ms	(fma d b (* a c))

series271.0ms

Counts

4 → 12

Calls

4 calls. Slowest were:

125.0ms	(* (/ 1 (hypot d c)) (/ (fma d b (* a c)) (hypot d c)))
79.0ms	(/ (fma d b (* a c)) (hypot d c))
40.0ms	(fma d b (* a c))
26.0ms	(hypot d c)

simplify2.1s

Counts

63 → 98

Calls

63 calls. Slowest were:

519.0ms

(* (* (* (/ 1 (hypot d c)) (/ 1 (hypot d c))) (/ 1 (hypot d c))) (* (* (/ (fma d b (* a c)) (hypot d c)) (/ (fma d b (* a c)) (hypot d c))) (/ (fma d b (* a c)) (hypot d c))))

331.0ms

(* (/ 1 (hypot d c)) (/ (fma d b (* a c)) (hypot d c)))

201.0ms

(/ (* (* (fma d b (* a c)) (fma d b (* a c))) (fma d b (* a c))) (* (* (hypot d c) (hypot d c)) (hypot d c)))

prune1.1s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 5.5b

localize11.0ms

Local error

Found 4 expressions with local error:

14.4b	(/ (fma c a (* b d)) (hypot d c))
0.1b	(/ (/ (fma c a (* b d)) (hypot d c)) (hypot d c))
0.0b	(hypot d c)
0.0b	(hypot d c)

rewrite23.0ms

Algorithm

rewrite-expression-head

Counts

4 → 121

Calls

4 calls. Slowest were:

12.0ms	(/ (/ (fma c a (* b d)) (hypot d c)) (hypot d c))
7.0ms	(/ (fma c a (* b d)) (hypot d c))
0.0ms	(hypot d c)

series270.0ms

Counts

4 → 12

Calls

4 calls. Slowest were:

97.0ms	(/ (/ (fma c a (* b d)) (hypot d c)) (hypot d c))
91.0ms	(/ (fma c a (* b d)) (hypot d c))
41.0ms	(hypot d c)
40.0ms	(hypot d c)

simplify3.3s

Counts

135 → 133

Calls

135 calls. Slowest were:

304.0ms

(/ (/ (* (* (fma c a (* b d)) (fma c a (* b d))) (fma c a (* b d))) (* (* (hypot d c) (hypot d c)) (hypot d c))) (* (* (hypot d c) (hypot d c)) (hypot d c)))

258.0ms

(/ (* (* (/ (fma c a (* b d)) (hypot d c)) (/ (fma c a (* b d)) (hypot d c))) (/ (fma c a (* b d)) (hypot d c))) (* (* (hypot d c) (hypot d c)) (hypot d c)))

226.0ms

(/ (/ (* (cbrt (fma c a (* b d))) (cbrt (fma c a (* b d)))) (* (cbrt (hypot d c)) (cbrt (hypot d c)))) 1)

prune1.6s

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 3.0b

localize8.0ms

Local error

Found 4 expressions with local error:

14.5b	(/ (hypot d c) (fma c a (* b d)))
0.2b	(/ 1 (/ (hypot d c) (fma c a (* b d))))
0.1b	(/ (/ 1 (/ (hypot d c) (fma c a (* b d)))) (hypot d c))
0.0b	(hypot d c)

rewrite12.0ms

Algorithm

rewrite-expression-head

Counts

4 → 109

Calls

4 calls. Slowest were:

6.0ms	(/ (hypot d c) (fma c a (* b d)))
4.0ms	(/ (/ 1 (/ (hypot d c) (fma c a (* b d)))) (hypot d c))
2.0ms	(/ 1 (/ (hypot d c) (fma c a (* b d))))

series361.0ms

Counts

4 → 12

Calls

4 calls. Slowest were:

136.0ms	(/ (/ 1 (/ (hypot d c) (fma c a (* b d)))) (hypot d c))
117.0ms	(/ 1 (/ (hypot d c) (fma c a (* b d))))
69.0ms	(/ (hypot d c) (fma c a (* b d)))
38.0ms	(hypot d c)

simplify4.7s

Counts

92 → 121

Calls

92 calls. Slowest were:

495.0ms	(- (log (/ 1 (/ (hypot d c) (fma c a (* b d))))) (log (hypot d c)))
351.0ms	(/ (/ 1 (/ (hypot d c) (fma c a (* b d)))) (cbrt (hypot d c)))
348.0ms	(/ (/ 1 (/ (hypot d c) (fma c a (* b d)))) (sqrt (hypot d c)))

prune1.2s

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 3.0b

regimes251.0ms

Accuracy

24.7% (10.5b remaining)

Error of 13.2b against oracle of 2.7b and baseline of 16.6b

bsearch61.0ms

end0.0ms

sample1.1s

Algorithm: intervals

Error

Target

Derivation

`if d < -8.346460948837449e+220`

`if -8.346460948837449e+220 < d < 1.0043357683224773e+119`

`if 1.0043357683224773e+119 < d`

Reproduce

Details

sample61.0ms

simplify13.0ms

prune23.0ms

localize17.0ms

rewrite4.0ms

series119.0ms

simplify603.0ms

prune529.0ms

localize34.0ms

rewrite12.0ms

series271.0ms

simplify2.1s

prune1.1s

localize11.0ms

rewrite23.0ms

series270.0ms

simplify3.3s

prune1.6s

localize8.0ms

rewrite12.0ms

series361.0ms

simplify4.7s

prune1.2s

regimes251.0ms

bsearch61.0ms

end0.0ms

sample1.1s