Original	26.1
Target	0.3
Herbie	7.1

Derivation

Split input into 3 regimes
if c < -1.8988337831022309e+177
1. Initial program 43.4
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Simplified43.4
  \[\leadsto \color{blue}{\frac{b \cdot c - a \cdot d}{(d \cdot d + \left(c \cdot c\right))_*}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt43.4
  \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied *-un-lft-identity43.4
  \[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
6. Applied times-frac43.4
  \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
7. Simplified43.4
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Simplified29.8
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied associate-*l/29.8
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]
11. Simplified29.8
  \[\leadsto \frac{\color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
12. Taylor expanded around -inf 11.7
  \[\leadsto \frac{\color{blue}{-1 \cdot b}}{\sqrt{d^2 + c^2}^*}\]
13. Simplified11.7
  \[\leadsto \frac{\color{blue}{-b}}{\sqrt{d^2 + c^2}^*}\]
if -1.8988337831022309e+177 < c < 2.599647374459235e+114
1. Initial program 19.6
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Simplified19.6
  \[\leadsto \color{blue}{\frac{b \cdot c - a \cdot d}{(d \cdot d + \left(c \cdot c\right))_*}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt19.6
  \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied *-un-lft-identity19.6
  \[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
6. Applied times-frac19.6
  \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
7. Simplified19.6
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Simplified12.1
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied associate-*r/12.1
  \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{d^2 + c^2}^*} \cdot \left(b \cdot c - a \cdot d\right)}{\sqrt{d^2 + c^2}^*}}\]
11. Using strategy rm
12. Applied sub-neg12.1
  \[\leadsto \frac{\frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\left(b \cdot c + \left(-a \cdot d\right)\right)}}{\sqrt{d^2 + c^2}^*}\]
13. Applied distribute-lft-in12.1
  \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*} \cdot \left(b \cdot c\right) + \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \left(-a \cdot d\right)}}{\sqrt{d^2 + c^2}^*}\]
14. Simplified4.2
  \[\leadsto \frac{\frac{1}{\sqrt{d^2 + c^2}^*} \cdot \left(b \cdot c\right) + \color{blue}{\frac{-a}{\sqrt{d^2 + c^2}^*} \cdot d}}{\sqrt{d^2 + c^2}^*}\]
if 2.599647374459235e+114 < c
1. Initial program 41.9
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Simplified41.9
  \[\leadsto \color{blue}{\frac{b \cdot c - a \cdot d}{(d \cdot d + \left(c \cdot c\right))_*}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt41.9
  \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied *-un-lft-identity41.9
  \[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
6. Applied times-frac41.9
  \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
7. Simplified41.9
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Simplified27.0
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied associate-*r/27.0
  \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{d^2 + c^2}^*} \cdot \left(b \cdot c - a \cdot d\right)}{\sqrt{d^2 + c^2}^*}}\]
11. Taylor expanded around 0 16.4
  \[\leadsto \frac{\color{blue}{b}}{\sqrt{d^2 + c^2}^*}\]
Recombined 3 regimes into one program.
Final simplification7.1
\[\leadsto \begin{array}{l} \mathbf{if}\;c \le -1.8988337831022309 \cdot 10^{+177}:\\ \;\;\;\;\frac{-b}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;c \le 2.599647374459235 \cdot 10^{+114}:\\ \;\;\;\;\frac{\frac{-a}{\sqrt{d^2 + c^2}^*} \cdot d + \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \left(b \cdot c\right)}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Details

sample37.0ms

Algorithm

intervals

Results

25.0ms

267×

body

valid

simplify60.0ms

Counts

1 → 1

Calls

1 calls:

Slowest

60.0ms

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

prune23.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 28.1b

localize27.0ms

Local error

Found 3 expressions with local error:

28.1b	(/ (- (* b c) (* a d)) (fma d d (* c c)))
0.2b	(- (* b c) (* a d))
0.0b	(fma d d (* c c))

rewrite10.0ms

Algorithm

rewrite-expression-head

Rules

11×	`add-cube-cbrt`
11×	`*-un-lft-identity`
11×	`add-sqr-sqrt`
9×	`times-frac`
5×	`add-log-exp`
5×	`add-exp-log`
5×	`add-cbrt-cube`
3×	`associate-/l*`
3×	`log1p-expm1-u`
3×	`associate-/r*`
3×	`pow1`
3×	`expm1-log1p-u`
2×	`associate-/l/`
2×	`flip--`
2×	`flip3--`
1×	`div-inv`
1×	`fma-udef`
1×	`prod-diff`
1×	`div-exp`
1×	`div-sub`
1×	`diff-log`
1×	`fma-neg`
1×	`frac-2neg`
1×	`sub-neg`
1×	`clear-num`
1×	`cbrt-undiv`

Counts

3 → 58

Calls

3 calls:

Slowest

7.0ms	(/ (- (* b c) (* a d)) (fma d d (* c c)))
2.0ms	(- (* b c) (* a d))
0.0ms	(fma d d (* c c))

series108.0ms

Counts

3 → 9

Calls

3 calls:

Slowest

63.0ms	(/ (- (* b c) (* a d)) (fma d d (* c c)))
29.0ms	(- (* b c) (* a d))
17.0ms	(fma d d (* c c))

simplify1.4s

Counts

40 → 67

Calls

40 calls:

Slowest

478.0ms	(- (* (* b c) (* b c)) (* (* a d) (* a d)))
250.0ms	(/ (* (* (- (* b c) (* a d)) (- (* b c) (* a d))) (- (* b c) (* a d))) (* (* (fma d d (* c c)) (fma d d (* c c))) (fma d d (* c c))))
194.0ms	(- (pow (* b c) 3) (pow (* a d) 3))
44.0ms	(/ (* (cbrt (- (* b c) (* a d))) (cbrt (- (* b c) (* a d)))) (* (cbrt (fma d d (* c c))) (cbrt (fma d d (* c c)))))
38.0ms	(/ (- (* b c) (* a d)) (fma d d (* c c)))

prune620.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 16.8b

localize11.0ms

Local error

Found 4 expressions with local error:

16.5b	(/ (- (* b c) (* a d)) (hypot d c))
0.3b	(* (/ 1 (hypot d c)) (/ (- (* b c) (* a d)) (hypot d c)))
0.2b	(- (* b c) (* a d))
0.0b	(hypot d c)

rewrite19.0ms

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×	`add-log-exp`
6×	`pow1`
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×	`associate-/l/`
2×	`flip--`
2×	`cbrt-unprod`
2×	`flip3--`
2×	`rec-exp`
2×	`cbrt-undiv`
1×	`*-commutative`
1×	`associate-*r/`
1×	`associate-*l/`
1×	`pow-prod-down`
1×	`prod-diff`
1×	`div-sub`
1×	`diff-log`
1×	`fma-neg`
1×	`frac-2neg`
1×	`sub-neg`
1×	`clear-num`
1×	`hypot-udef`
1×	`frac-times`

Counts

4 → 95

Calls

4 calls:

Slowest

10.0ms	(* (/ 1 (hypot d c)) (/ (- (* b c) (* a d)) (hypot d c)))
6.0ms	(/ (- (* b c) (* a d)) (hypot d c))
2.0ms	(- (* b c) (* a d))
0.0ms	(hypot d c)

series238.0ms

Counts

4 → 12

Calls

4 calls:

Slowest

98.0ms	(* (/ 1 (hypot d c)) (/ (- (* b c) (* a d)) (hypot d c)))
89.0ms	(/ (- (* b c) (* a d)) (hypot d c))
28.0ms	(- (* b c) (* a d))
23.0ms	(hypot d c)

simplify2.3s

Counts

69 → 107

Calls

69 calls:

Slowest

430.0ms	(- (* (* b c) (* b c)) (* (* a d) (* a d)))
264.0ms	(- (pow (* b c) 3) (pow (* a d) 3))
219.0ms	(* 1 (/ (- (* b c) (* a d)) (hypot d c)))
186.0ms	(/ (* (* (- (* b c) (* a d)) (- (* b c) (* a d))) (- (* b c) (* a d))) (* (* (hypot d c) (hypot d c)) (hypot d c)))
98.0ms	(* (* (* (/ 1 (hypot d c)) (/ 1 (hypot d c))) (/ 1 (hypot d c))) (/ (* (* (- (* b c) (* a d)) (- (* b c) (* a d))) (- (* b c) (* a d))) (* (* (hypot d c) (hypot d c)) (hypot d c))))

prune812.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 4.5b

localize10.0ms

Local error

Found 4 expressions with local error:

16.5b	(/ (- (* c b) (* a d)) (hypot d c))
0.2b	(- (* c b) (* a d))
0.1b	(/ (/ (- (* c b) (* a d)) (hypot d c)) (hypot d c))
0.0b	(hypot d c)

rewrite17.0ms

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×	`add-log-exp`
6×	`div-inv`
6×	`associate-/r*`
4×	`log1p-expm1-u`
4×	`div-exp`
4×	`cbrt-undiv`
4×	`pow1`
4×	`expm1-log1p-u`
3×	`associate-/l/`
3×	`div-sub`
2×	`flip--`
2×	`flip3--`
2×	`frac-2neg`
2×	`clear-num`
1×	`prod-diff`
1×	`diff-log`
1×	`fma-neg`
1×	`sub-neg`
1×	`hypot-udef`

Counts

4 → 131

Calls

4 calls:

Slowest

7.0ms	(/ (/ (- (* c b) (* a d)) (hypot d c)) (hypot d c))
6.0ms	(/ (- (* c b) (* a d)) (hypot d c))
2.0ms	(- (* c b) (* a d))
0.0ms	(hypot d c)

series174.0ms

Counts

4 → 12

Calls

4 calls:

Slowest

63.0ms	(/ (/ (- (* c b) (* a d)) (hypot d c)) (hypot d c))
62.0ms	(/ (- (* c b) (* a d)) (hypot d c))
27.0ms	(- (* c b) (* a d))
22.0ms	(hypot d c)

simplify4.9s

Counts

142 → 143

Calls

142 calls:

Slowest

351.0ms	(- (* (* c b) (* c b)) (* (* a d) (* a d)))
337.0ms	(/ (/ (- (* c b) (* a d)) (sqrt (hypot d c))) (hypot d c))
332.0ms	(/ (* (* (/ (- (* c b) (* a d)) (hypot d c)) (/ (- (* c b) (* a d)) (hypot d c))) (/ (- (* c b) (* a d)) (hypot d c))) (* (* (hypot d c) (hypot d c)) (hypot d c)))
263.0ms	(/ (/ (* (* (- (* c b) (* a d)) (- (* c b) (* a d))) (- (* c b) (* a d))) (* (* (hypot d c) (hypot d c)) (hypot d c))) (* (* (hypot d c) (hypot d c)) (hypot d c)))
176.0ms	(- (pow (* c b) 3) (pow (* a d) 3))

prune999.0ms

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 4.5b

localize6.0ms

Local error

Found 4 expressions with local error:

16.6b	(* (/ 1 (hypot d c)) (- (* b c) (* a d)))
0.2b	(- (* b c) (* a d))
0.1b	(/ (* (/ 1 (hypot d c)) (- (* b c) (* a d))) (hypot d c))
0.0b	(hypot d c)

rewrite21.0ms

Algorithm

rewrite-expression-head

Rules

16×	`add-exp-log`
11×	`add-cbrt-cube`
8×	`add-cube-cbrt`
8×	`*-un-lft-identity`
8×	`add-sqr-sqrt`
6×	`add-log-exp`
6×	`pow1`
5×	`associate-/l/`
5×	`flip--`
5×	`flip3--`
4×	`log1p-expm1-u`
4×	`associate-*r/`
4×	`prod-exp`
4×	`associate-l`
4×	`frac-times`
4×	`expm1-log1p-u`
3×	`prod-diff`
3×	`associate-r`
3×	`div-exp`
3×	`times-frac`
3×	`distribute-lft-in`
3×	`sub-neg`
3×	`associate-/r*`
3×	`distribute-rgt-in`
2×	`div-inv`
2×	`cbrt-unprod`
2×	`associate-*l/`
2×	`rec-exp`
2×	`cbrt-undiv`
1×	`associate-/l*`
1×	`*-commutative`
1×	`pow-prod-down`
1×	`diff-log`
1×	`fma-neg`
1×	`frac-2neg`
1×	`clear-num`
1×	`hypot-udef`

Counts

4 → 87

Calls

4 calls:

Slowest

10.0ms	(/ (* (/ 1 (hypot d c)) (- (* b c) (* a d))) (hypot d c))
7.0ms	(* (/ 1 (hypot d c)) (- (* b c) (* a d)))
2.0ms	(- (* b c) (* a d))
0.0ms	(hypot d c)

series229.0ms

Counts

4 → 12

Calls

4 calls:

Slowest

90.0ms	(/ (* (/ 1 (hypot d c)) (- (* b c) (* a d))) (hypot d c))
83.0ms	(* (/ 1 (hypot d c)) (- (* b c) (* a d)))
31.0ms	(hypot d c)
26.0ms	(- (* b c) (* a d))

simplify3.9s

Counts

57 → 99

Calls

57 calls:

Slowest

312.0ms	(* (- (* a d)) (/ 1 (hypot d c)))
287.0ms	(* 1 (- (* (* b c) (* b c)) (* (* a d) (* a d))))
263.0ms	(* (/ 1 (hypot d c)) (- (pow (* b c) 3) (pow (* a d) 3)))
236.0ms	(* (- (* a d)) (/ 1 (hypot d c)))
207.0ms	(- (log (* (/ 1 (hypot d c)) (- (* b c) (* a d)))) (log (hypot d c)))

prune939.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.9b

regimes230.0ms

Accuracy

32.1% (5.4b remaining)

Error of 7.1b against oracle of 1.7b and baseline of 9.6b

bsearch88.0ms

end0.0ms

sample1.3s

Algorithm

intervals

Results

1.1s

8441×

body

valid

Error

Try it out

Target

Derivation

`if c < -1.8988337831022309e+177`

`if -1.8988337831022309e+177 < c < 2.599647374459235e+114`

`if 2.599647374459235e+114 < c`

Reproduce

Details

sample37.0ms

simplify60.0ms

prune23.0ms

localize27.0ms

rewrite10.0ms

series108.0ms

simplify1.4s

prune620.0ms

localize11.0ms

rewrite19.0ms

series238.0ms

simplify2.3s

prune812.0ms

localize10.0ms

rewrite17.0ms

series174.0ms

simplify4.9s

prune999.0ms

localize6.0ms

rewrite21.0ms

series229.0ms

simplify3.9s

prune939.0ms

regimes230.0ms

bsearch88.0ms

end0.0ms

sample1.3s

In
Out