Original	25.7
Target	0.4
Herbie	12.7

Derivation

Split input into 3 regimes
if c < -1.3845764181634926e+184
1. Initial program 43.8
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Simplified43.8
  \[\leadsto \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt43.8
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied *-un-lft-identity43.8
  \[\leadsto \frac{\color{blue}{1 \cdot (a \cdot c + \left(b \cdot d\right))_*}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
6. Applied times-frac43.8
  \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
7. Simplified43.8
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Simplified30.4
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied associate-*l/30.4
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]
11. Simplified30.4
  \[\leadsto \frac{\color{blue}{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
12. Taylor expanded around -inf 13.1
  \[\leadsto \frac{\color{blue}{-1 \cdot a}}{\sqrt{d^2 + c^2}^*}\]
13. Simplified13.1
  \[\leadsto \frac{\color{blue}{-a}}{\sqrt{d^2 + c^2}^*}\]
if -1.3845764181634926e+184 < c < 3.0299220901443166e+165
1. Initial program 20.3
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Simplified20.3
  \[\leadsto \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt20.3
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied *-un-lft-identity20.3
  \[\leadsto \frac{\color{blue}{1 \cdot (a \cdot c + \left(b \cdot d\right))_*}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
6. Applied times-frac20.3
  \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
7. Simplified20.3
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Simplified12.7
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied associate-*l/12.6
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]
11. Simplified12.6
  \[\leadsto \frac{\color{blue}{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
if 3.0299220901443166e+165 < c
1. Initial program 43.2
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Simplified43.2
  \[\leadsto \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt43.2
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied *-un-lft-identity43.2
  \[\leadsto \frac{\color{blue}{1 \cdot (a \cdot c + \left(b \cdot d\right))_*}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
6. Applied times-frac43.2
  \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
7. Simplified43.2
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Simplified29.4
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied associate-*l/29.4
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]
11. Simplified29.4
  \[\leadsto \frac{\color{blue}{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
12. Using strategy rm
13. Applied *-un-lft-identity29.4
  \[\leadsto \frac{\frac{\color{blue}{1 \cdot (c \cdot a + \left(b \cdot d\right))_*}}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\]
14. Applied associate-/l*29.4
  \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sqrt{d^2 + c^2}^*}{(c \cdot a + \left(b \cdot d\right))_*}}}}{\sqrt{d^2 + c^2}^*}\]
15. Taylor expanded around 0 13.1
  \[\leadsto \frac{\color{blue}{a}}{\sqrt{d^2 + c^2}^*}\]
Recombined 3 regimes into one program.
Final simplification12.7
\[\leadsto \begin{array}{l} \mathbf{if}\;c \le -1.3845764181634926 \cdot 10^{+184}:\\ \;\;\;\;\frac{-a}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;c \le 3.0299220901443166 \cdot 10^{+165}:\\ \;\;\;\;\frac{\frac{(c \cdot a + \left(d \cdot b\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Details

sample50.0ms

Algorithm

intervals

Results

272×	(pre true 80)
272×	(body real 80)

simplify9.0ms

Counts

1 → 1

Calls

1 calls:

Slowest

9.0ms

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

prune22.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 23.8b

localize36.0ms

Local error

Found 3 expressions with local error:

23.8b	(/ (fma a c (* b d)) (fma d d (* c c)))
0.0b	(fma d d (* c c))
0.0b	(fma a c (* b d))

rewrite9.0ms

Algorithm

rewrite-expression-head

Rules

11×	`add-cube-cbrt`
11×	`*-un-lft-identity`
11×	`add-sqr-sqrt`
9×	`times-frac`
5×	`add-exp-log`
5×	`add-cbrt-cube`
3×	`add-log-exp`
3×	`associate-/l*`
3×	`log1p-expm1-u`
3×	`associate-/r*`
3×	`pow1`
3×	`expm1-log1p-u`
2×	`fma-udef`
1×	`div-inv`
1×	`div-exp`
1×	`frac-2neg`
1×	`clear-num`
1×	`cbrt-undiv`

Counts

3 → 49

Calls

3 calls:

Slowest

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

series163.0ms

Counts

3 → 9

Calls

3 calls:

Slowest

112.0ms	(/ (fma a c (* b d)) (fma d d (* c c)))
27.0ms	(fma a c (* b d))
23.0ms	(fma d d (* c c))

simplify626.0ms

Counts

34 → 58

Calls

34 calls:

Slowest

227.0ms	(/ (* (* (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))))
42.0ms	(/ (* (cbrt (fma a c (* b d))) (cbrt (fma a c (* b d)))) (* (cbrt (fma d d (* c c))) (cbrt (fma d d (* c c)))))
31.0ms	(/ (* (cbrt (fma a c (* b d))) (cbrt (fma a c (* b d)))) 1)
24.0ms	(/ (cbrt (fma a c (* b d))) (sqrt (fma d d (* c c))))
22.0ms	(/ (sqrt (fma a c (* b d))) (sqrt (fma d d (* c c))))

prune633.0ms

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 13.9b

localize17.0ms

Local error

Found 4 expressions with local error:

13.7b	(/ (fma d b (* a c)) (hypot d c))
0.3b	(fma d b (* a c))
0.2b	(* (/ 1 (hypot d c)) (/ (fma d b (* a c)) (hypot d c)))
0.0b	(/ 1 (hypot d c))

rewrite12.0ms

Algorithm

rewrite-expression-head

Rules

21×	`add-cube-cbrt`
21×	`*-un-lft-identity`
21×	`add-sqr-sqrt`
18×	`times-frac`
17×	`add-exp-log`
13×	`associate-r`
11×	`add-cbrt-cube`
7×	`pow1`
6×	`associate-/r*`
4×	`add-log-exp`
4×	`div-inv`
4×	`log1p-expm1-u`
4×	`prod-exp`
4×	`associate-l`
4×	`expm1-log1p-u`
3×	`associate-/l*`
3×	`div-exp`
3×	`rec-exp`
2×	`cbrt-unprod`
2×	`frac-2neg`
2×	`clear-num`
2×	`cbrt-undiv`
1×	`inv-pow`
1×	`pow-flip`
1×	`fma-udef`
1×	`*-commutative`
1×	`associate-*r/`
1×	`associate-*l/`
1×	`pow-prod-down`
1×	`frac-times`

Counts

4 → 94

Calls

4 calls:

Slowest

6.0ms	(* (/ 1 (hypot d c)) (/ (fma d b (* a c)) (hypot d c)))
3.0ms	(/ (fma d b (* a c)) (hypot d c))
1.0ms	(/ 1 (hypot d c))
0.0ms	(fma d b (* a c))

series277.0ms

Counts

4 → 12

Calls

4 calls:

Slowest

132.0ms	(* (/ 1 (hypot d c)) (/ (fma d b (* a c)) (hypot d c)))
78.0ms	(/ (fma d b (* a c)) (hypot d c))
40.0ms	(/ 1 (hypot d c))
26.0ms	(fma d b (* a c))

simplify2.2s

Counts

67 → 106

Calls

67 calls:

Slowest

699.0ms	(* (* (* (/ 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))))
299.0ms	(* (/ 1 (hypot d c)) (/ (fma d b (* a c)) (hypot d c)))
186.0ms	(* (* (* (/ 1 (hypot d c)) (/ 1 (hypot d c))) (/ 1 (hypot d c))) (/ (* (* (fma d b (* a c)) (fma d b (* a c))) (fma d b (* a c))) (* (* (hypot d c) (hypot d c)) (hypot d c))))
181.0ms	(/ (* (* (fma d b (* a c)) (fma d b (* a c))) (fma d b (* a c))) (* (* (hypot d c) (hypot d c)) (hypot d c)))
106.0ms	(+ (log (/ 1 (hypot d c))) (log (/ (fma d b (* a c)) (hypot d c))))

prune1.4s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 6.1b

localize12.0ms

Local error

Found 3 expressions with local error:

13.7b	(/ (fma c a (* b d)) (hypot d c))
0.0b	(/ (/ (fma c a (* b d)) (hypot d c)) (hypot d c))
0.0b	(fma c a (* b d))

rewrite18.0ms

Algorithm

rewrite-expression-head

Rules

84×	`times-frac`
53×	`add-cube-cbrt`
53×	`*-un-lft-identity`
53×	`add-sqr-sqrt`
16×	`associate-/l*`
10×	`add-exp-log`
10×	`add-cbrt-cube`
6×	`div-inv`
6×	`associate-/r*`
4×	`div-exp`
4×	`cbrt-undiv`
3×	`add-log-exp`
3×	`log1p-expm1-u`
3×	`pow1`
3×	`expm1-log1p-u`
2×	`frac-2neg`
2×	`clear-num`
1×	`associate-/l/`
1×	`fma-udef`

Counts

3 → 111

Calls

3 calls:

Slowest

11.0ms	(/ (/ (fma c a (* b d)) (hypot d c)) (hypot d c))
3.0ms	(/ (fma c a (* b d)) (hypot d c))
0.0ms	(fma c a (* b d))

series181.0ms

Counts

3 → 9

Calls

3 calls:

Slowest

84.0ms	(/ (fma c a (* b d)) (hypot d c))
67.0ms	(/ (/ (fma c a (* b d)) (hypot d c)) (hypot d c))
30.0ms	(fma c a (* b d))

simplify3.3s

Counts

132 → 120

Calls

132 calls:

Slowest

374.0ms	(/ (/ (* (* (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)))
270.0ms	(/ (* (* (/ (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)))
208.0ms	(/ (/ (* (cbrt (fma c a (* b d))) (cbrt (fma c a (* b d)))) (* (cbrt (hypot d c)) (cbrt (hypot d c)))) 1)
182.0ms	(/ (* (* (fma c a (* b d)) (fma c a (* b d))) (fma c a (* b d))) (* (* (hypot d c) (hypot d c)) (hypot d c)))
149.0ms	(/ (/ (* (cbrt (fma c a (* b d))) (cbrt (fma c a (* b d)))) 1) (* (cbrt (hypot d c)) (cbrt (hypot d c))))

prune1.6s

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 1.7b

localize8.0ms

Local error

Found 4 expressions with local error:

13.7b	(/ (hypot d c) (fma c a (* b d)))
0.1b	(/ 1 (/ (hypot d c) (fma c a (* b d))))
0.0b	(/ (/ 1 (/ (hypot d c) (fma c a (* b d)))) (hypot d c))
0.0b	(fma c a (* b d))

rewrite21.0ms

Algorithm

rewrite-expression-head

Rules

33×	`times-frac`
29×	`add-cube-cbrt`
29×	`*-un-lft-identity`
29×	`add-sqr-sqrt`
19×	`associate-/r*`
16×	`add-exp-log`
8×	`associate-/l*`
8×	`div-inv`
8×	`add-cbrt-cube`
6×	`div-exp`
5×	`associate-/r/`
5×	`pow1`
4×	`add-log-exp`
4×	`log1p-expm1-u`
4×	`rec-exp`
4×	`expm1-log1p-u`
3×	`frac-2neg`
3×	`clear-num`
2×	`cbrt-undiv`
1×	`associate-/l/`
1×	`inv-pow`
1×	`pow-flip`
1×	`fma-udef`

Counts

4 → 109

Calls

4 calls:

Slowest

8.0ms	(/ (/ 1 (/ (hypot d c) (fma c a (* b d)))) (hypot d c))
7.0ms	(/ (hypot d c) (fma c a (* b d)))
3.0ms	(/ 1 (/ (hypot d c) (fma c a (* b d))))
0.0ms	(fma c a (* b d))

series356.0ms

Counts

4 → 12

Calls

4 calls:

Slowest

127.0ms	(/ (/ 1 (/ (hypot d c) (fma c a (* b d)))) (hypot d c))
111.0ms	(/ 1 (/ (hypot d c) (fma c a (* b d))))
73.0ms	(/ (hypot d c) (fma c a (* b d)))
46.0ms	(fma c a (* b d))

simplify6.1s

Counts

92 → 121

Calls

92 calls:

Slowest

517.0ms	(/ (/ 1 (/ (hypot d c) (fma c a (* b d)))) (cbrt (hypot d c)))
494.0ms	(/ (sqrt (/ 1 (/ (hypot d c) (fma c a (* b d))))) (cbrt (hypot d c)))
416.0ms	(/ (/ 1 (/ (hypot d c) (fma c a (* b d)))) (sqrt (hypot d c)))
411.0ms	(/ (/ 1 (/ (hypot d c) (fma c a (* b d)))) (sqrt (hypot d c)))
383.0ms	(/ (/ 1 (/ (hypot d c) (fma c a (* b d)))) (cbrt (hypot d c)))

prune1.5s

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 1.7b

regimes434.0ms

Accuracy

27.9% (10.2b remaining)

Error of 12.7b against oracle of 2.5b and baseline of 16.6b

bsearch150.0ms

end0.0ms

sample1.5s

Algorithm

intervals

Results

8493×	(pre true 80)
8493×	(body real 80)

Error

Target

Derivation

`if c < -1.3845764181634926e+184`

`if -1.3845764181634926e+184 < c < 3.0299220901443166e+165`

`if 3.0299220901443166e+165 < c`

Reproduce

Details

sample50.0ms

simplify9.0ms

prune22.0ms

localize36.0ms

rewrite9.0ms

series163.0ms

simplify626.0ms

prune633.0ms

localize17.0ms

rewrite12.0ms

series277.0ms

simplify2.2s

prune1.4s

localize12.0ms

rewrite18.0ms

series181.0ms

simplify3.3s

prune1.6s

localize8.0ms

rewrite21.0ms

series356.0ms

simplify6.1s

prune1.5s

regimes434.0ms

bsearch150.0ms

end0.0ms

sample1.5s