Original	26.1
Target	0.4
Herbie	12.5

Derivation

Split input into 3 regimes
if c < -1.8506614466369815e+150
1. Initial program 44.1
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Simplified44.1
  \[\leadsto \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt44.1
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied *-un-lft-identity44.1
  \[\leadsto \frac{\color{blue}{1 \cdot (a \cdot c + \left(b \cdot d\right))_*}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
6. Applied times-frac44.1
  \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
7. Simplified44.1
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Simplified28.2
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied associate-*l/28.1
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]
11. Simplified28.1
  \[\leadsto \frac{\color{blue}{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
12. Taylor expanded around -inf 13.3
  \[\leadsto \frac{\color{blue}{-1 \cdot a}}{\sqrt{d^2 + c^2}^*}\]
13. Simplified13.3
  \[\leadsto \frac{\color{blue}{-a}}{\sqrt{d^2 + c^2}^*}\]
if -1.8506614466369815e+150 < c < 6.560820932670249e+185
1. Initial program 20.1
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Simplified20.1
  \[\leadsto \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt20.1
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied *-un-lft-identity20.1
  \[\leadsto \frac{\color{blue}{1 \cdot (a \cdot c + \left(b \cdot d\right))_*}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
6. Applied times-frac20.2
  \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
7. Simplified20.1
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Simplified12.4
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied associate-*l/12.2
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]
11. Simplified12.2
  \[\leadsto \frac{\color{blue}{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
12. Using strategy rm
13. Applied *-un-lft-identity12.2
  \[\leadsto \frac{\frac{\color{blue}{1 \cdot (c \cdot a + \left(b \cdot d\right))_*}}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\]
14. Applied associate-/l*12.3
  \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sqrt{d^2 + c^2}^*}{(c \cdot a + \left(b \cdot d\right))_*}}}}{\sqrt{d^2 + c^2}^*}\]
if 6.560820932670249e+185 < c
1. Initial program 44.6
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Simplified44.6
  \[\leadsto \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt44.6
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied *-un-lft-identity44.6
  \[\leadsto \frac{\color{blue}{1 \cdot (a \cdot c + \left(b \cdot d\right))_*}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
6. Applied times-frac44.6
  \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
7. Simplified44.6
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Simplified30.7
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied associate-*l/30.7
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]
11. Simplified30.7
  \[\leadsto \frac{\color{blue}{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
12. Taylor expanded around inf 12.2
  \[\leadsto \frac{\color{blue}{a}}{\sqrt{d^2 + c^2}^*}\]
Recombined 3 regimes into one program.
Final simplification12.5
\[\leadsto \begin{array}{l} \mathbf{if}\;c \le -1.8506614466369815 \cdot 10^{+150}:\\ \;\;\;\;\frac{-a}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;c \le 6.560820932670249 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{d^2 + c^2}^*}{(c \cdot a + \left(d \cdot b\right))_*}}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Details

sample64.0ms

Algorithm

intervals

Results

49.0ms

270×

body

valid

simplify10.0ms

Counts

1 → 1

Calls

1 calls:

Slowest

10.0ms

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

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 26.8b

localize16.0ms

Local error

Found 3 expressions with local error:

26.8b	(/ (fma a c (* b d)) (fma d d (* c c)))
0.3b	(fma a c (* b d))
0.0b	(fma d d (* c c))

rewrite4.0ms

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

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

series144.0ms

Counts

3 → 9

Calls

3 calls:

Slowest

80.0ms	(/ (fma a c (* b d)) (fma d d (* c c)))
39.0ms	(fma a c (* b d))
26.0ms	(fma d d (* c c))

simplify471.0ms

Counts

34 → 58

Calls

34 calls:

Slowest

203.0ms	(/ (* (* (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))))
33.0ms	(/ (* (cbrt (fma a c (* b d))) (cbrt (fma a c (* b d)))) (* (cbrt (fma d d (* c c))) (cbrt (fma d d (* c c)))))
15.0ms	(/ (sqrt (fma a c (* b d))) (cbrt (fma d d (* c c))))
15.0ms	(/ (sqrt (fma a c (* b d))) (sqrt (fma d d (* c c))))
15.0ms	(/ (sqrt (fma a c (* b d))) (sqrt (fma d d (* c c))))

prune616.0ms

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 16.4b

localize30.0ms

Local error

Found 4 expressions with local error:

16.2b	(/ (fma d b (* a c)) (hypot d c))
0.5b	(fma d b (* a c))
0.2b	(* (/ 1 (hypot d c)) (/ (fma d b (* a c)) (hypot d c)))
0.0b	(hypot d c)

rewrite10.0ms

Algorithm

rewrite-expression-head

Rules

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

Counts

4 → 86

Calls

4 calls:

Slowest

5.0ms	(* (/ 1 (hypot d c)) (/ (fma d b (* a c)) (hypot d c)))
3.0ms	(/ (fma d b (* a c)) (hypot d c))
0.0ms	(hypot d c)
0.0ms	(fma d b (* a c))

series275.0ms

Counts

4 → 12

Calls

4 calls:

Slowest

121.0ms	(* (/ 1 (hypot d c)) (/ (fma d b (* a c)) (hypot d c)))
84.0ms	(/ (fma d b (* a c)) (hypot d c))
43.0ms	(fma d b (* a c))
26.0ms	(hypot d c)

simplify1.7s

Counts

63 → 98

Calls

63 calls:

Slowest

472.0ms	(* (* (* (/ 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))))
224.0ms	(* (/ 1 (hypot d c)) (/ (fma d b (* a c)) (hypot d c)))
137.0ms	(/ (* (* (fma d b (* a c)) (fma d b (* a c))) (fma d b (* a c))) (* (* (hypot d c) (hypot d c)) (hypot d c)))
125.0ms	(* (* (* (/ 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))))
103.0ms	(+ (log (/ 1 (hypot d c))) (- (log (fma d b (* a c))) (log (hypot d c))))

prune1.1s

Pruning

12 alts after pruning (10 fresh and 2 done)

Merged error: 7.6b

localize13.0ms

Local error

Found 4 expressions with local error:

16.2b	(/ (fma c a (* b d)) (hypot d c))
0.3b	(fma c a (* b d))
0.1b	(/ (/ (fma c a (* b d)) (hypot d c)) (hypot d c))
0.0b	(hypot d c)

rewrite11.0ms

Algorithm

rewrite-expression-head

Rules

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

Counts

4 → 121

Calls

4 calls:

Slowest

6.0ms	(/ (/ (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))
0.0ms	(hypot d c)

series236.0ms

Counts

4 → 12

Calls

4 calls:

Slowest

91.0ms	(/ (/ (fma c a (* b d)) (hypot d c)) (hypot d c))
80.0ms	(/ (fma c a (* b d)) (hypot d c))
38.0ms	(fma c a (* b d))
27.0ms	(hypot d c)

simplify2.5s

Counts

135 → 133

Calls

135 calls:

Slowest

287.0ms	(/ (/ (* (* (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)))
190.0ms	(/ (* (* (/ (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)))
150.0ms	(/ (/ (* (cbrt (fma c a (* b d))) (cbrt (fma c a (* b d)))) (* (cbrt (hypot d c)) (cbrt (hypot d c)))) 1)
148.0ms	(/ (* (* (fma c a (* b d)) (fma c a (* b d))) (fma c a (* b d))) (* (* (hypot d c) (hypot d c)) (hypot d c)))
126.0ms	(/ (/ (* (cbrt (fma c a (* b d))) (cbrt (fma c a (* b d)))) 1) (* (cbrt (hypot d c)) (cbrt (hypot d c))))

prune1.3s

Pruning

10 alts after pruning (8 fresh and 2 done)

Merged error: 2.5b

localize13.0ms

Local error

Found 4 expressions with local error:

16.3b	(/ (hypot d c) (fma c a (* b d)))
0.3b	(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))

rewrite18.0ms

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))
5.0ms	(/ (hypot d c) (fma c a (* b d)))
2.0ms	(/ 1 (/ (hypot d c) (fma c a (* b d))))
0.0ms	(fma c a (* b d))

series306.0ms

Counts

4 → 12

Calls

4 calls:

Slowest

108.0ms	(/ (/ 1 (/ (hypot d c) (fma c a (* b d)))) (hypot d c))
91.0ms	(/ 1 (/ (hypot d c) (fma c a (* b d))))
78.0ms	(/ (hypot d c) (fma c a (* b d)))
29.0ms	(fma c a (* b d))

simplify4.3s

Counts

92 → 121

Calls

92 calls:

Slowest

365.0ms	(/ (/ 1 (/ (hypot d c) (fma c a (* b d)))) (sqrt (hypot d c)))
347.0ms	(/ (/ 1 (/ (hypot d c) (fma c a (* b d)))) (cbrt (hypot d c)))
341.0ms	(/ (/ 1 (/ (hypot d c) (fma c a (* b d)))) (sqrt (hypot d c)))
327.0ms	(/ (/ 1 (/ (hypot d c) (fma c a (* b d)))) (cbrt (hypot d c)))
257.0ms	(/ (/ 1 (/ (hypot d c) (fma c a (* b d)))) (hypot d c))

prune887.0ms

Pruning

10 alts after pruning (7 fresh and 3 done)

Merged error: 2.5b

regimes258.0ms

Accuracy

28.9% (10.0b remaining)

Error of 12.5b against oracle of 2.5b and baseline of 16.5b

bsearch85.0ms

end0.0ms

sample949.0ms

Algorithm

intervals

Results

795.0ms

8447×

body

valid

Error

Target

Derivation

`if c < -1.8506614466369815e+150`

`if -1.8506614466369815e+150 < c < 6.560820932670249e+185`

`if 6.560820932670249e+185 < c`

Reproduce

Details

sample64.0ms

simplify10.0ms

prune10.0ms

localize16.0ms

rewrite4.0ms

series144.0ms

simplify471.0ms

prune616.0ms

localize30.0ms

rewrite10.0ms

series275.0ms

simplify1.7s

prune1.1s

localize13.0ms

rewrite11.0ms

series236.0ms

simplify2.5s

prune1.3s

localize13.0ms

rewrite18.0ms

series306.0ms

simplify4.3s

prune887.0ms

regimes258.0ms

bsearch85.0ms

end0.0ms

sample949.0ms