Average Error: 25.4 → 13.5
Time: 19.4s
Precision: 64
Internal Precision: 128
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le -9.234484326885406 \cdot 10^{+101}:\\ \;\;\;\;\frac{-1}{\sqrt{d^2 + c^2}^*} \cdot b\\ \mathbf{elif}\;d \le -1.1521206323075965 \cdot 10^{-237}:\\ \;\;\;\;\frac{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;d \le -7.3148665069542 \cdot 10^{-292}:\\ \;\;\;\;\frac{-a}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;d \le 1.0294661443825063 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original25.4
Target0.3
Herbie13.5
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if d < -9.234484326885406e+101

    1. Initial program 38.4

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]

    2. Simplified38.4

      \[\leadsto \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt38.4

      \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]

    5. Applied *-un-lft-identity38.4

      \[\leadsto \frac{\color{blue}{1 \cdot (a \cdot c + \left(b \cdot d\right))_*}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]

    6. Applied times-frac38.4

      \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]

    7. Simplified38.4

      \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]

    8. Simplified24.6

      \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]

    9. Taylor expanded around -inf 16.0

      \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\left(-1 \cdot b\right)}\]

    10. Simplified16.0

      \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\left(-b\right)}\]

    if -9.234484326885406e+101 < d < -1.1521206323075965e-237 or -7.3148665069542e-292 < d < 1.0294661443825063e+154

    1. Initial program 18.4

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]

    2. Simplified18.4

      \[\leadsto \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt18.4

      \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]

    5. Applied *-un-lft-identity18.4

      \[\leadsto \frac{\color{blue}{1 \cdot (a \cdot c + \left(b \cdot d\right))_*}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]

    6. Applied times-frac18.4

      \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]

    7. Simplified18.4

      \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]

    8. Simplified12.2

      \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
    9. Using strategy rm

    10. Applied associate-*l/12.0

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]

    11. Simplified12.0

      \[\leadsto \frac{\color{blue}{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]

    if -1.1521206323075965e-237 < d < -7.3148665069542e-292

    1. Initial program 23.5

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]

    2. Simplified23.6

      \[\leadsto \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt23.6

      \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]

    5. Applied *-un-lft-identity23.6

      \[\leadsto \frac{\color{blue}{1 \cdot (a \cdot c + \left(b \cdot d\right))_*}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]

    6. Applied times-frac23.5

      \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]

    7. Simplified23.5

      \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]

    8. Simplified13.9

      \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
    9. Using strategy rm

    10. Applied associate-*l/13.8

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]

    11. Simplified13.8

      \[\leadsto \frac{\color{blue}{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]

    12. Taylor expanded around -inf 30.6

      \[\leadsto \frac{\color{blue}{-1 \cdot a}}{\sqrt{d^2 + c^2}^*}\]

    13. Simplified30.6

      \[\leadsto \frac{\color{blue}{-a}}{\sqrt{d^2 + c^2}^*}\]

    if 1.0294661443825063e+154 < d

    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. Simplified28.1

      \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
    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 0 12.3

      \[\leadsto \frac{\color{blue}{b}}{\sqrt{d^2 + c^2}^*}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification13.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -9.234484326885406 \cdot 10^{+101}:\\ \;\;\;\;\frac{-1}{\sqrt{d^2 + c^2}^*} \cdot b\\ \mathbf{elif}\;d \le -1.1521206323075965 \cdot 10^{-237}:\\ \;\;\;\;\frac{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;d \le -7.3148665069542 \cdot 10^{-292}:\\ \;\;\;\;\frac{-a}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;d \le 1.0294661443825063 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(FPCore (a b c d)
  :name "Complex division, real part"

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d)))))

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

Details

Time bar (total: 18.4s)Debug log

sample50.0ms

Algorithm
intervals

simplify8.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

8.0ms
(/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 30.8b

localize17.0ms

Local error

Found 3 expressions with local error:

30.8b
(/ (fma a c (* b d)) (fma d d (* c c)))
0.2b
(fma a c (* b d))
0.0b
(fma d d (* c c))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
3 → 49
Calls

3 calls. Slowest were:

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))

series148.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

77.0ms
(/ (fma a c (* b d)) (fma d d (* c c)))
43.0ms
(fma a c (* b d))
27.0ms
(fma d d (* c c))

simplify662.0ms

Counts
34 → 58
Calls

34 calls. Slowest were:

216.0ms
(/ (* (* (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))))
46.0ms
(/ (* (cbrt (fma a c (* b d))) (cbrt (fma a c (* b d)))) (* (cbrt (fma d d (* c c))) (cbrt (fma d d (* c c)))))
24.0ms
(/ (sqrt (fma a c (* b d))) (* (cbrt (fma d d (* c c))) (cbrt (fma d d (* c c)))))

prune641.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 21.8b

localize18.0ms

Local error

Found 4 expressions with local error:

21.5b
(/ (fma d b (* a c)) (hypot d c))
1.0b
(fma d b (* a c))
0.3b
(* (/ 1 (hypot d c)) (/ (fma d b (* a c)) (hypot d c)))
0.0b
(/ 1 (hypot d c))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
4 → 94
Calls

4 calls. Slowest were:

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))

series250.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

127.0ms
(* (/ 1 (hypot d c)) (/ (fma d b (* a c)) (hypot d c)))
67.0ms
(/ (fma d b (* a c)) (hypot d c))
30.0ms
(fma d b (* a c))
25.0ms
(/ 1 (hypot d c))

simplify2.0s

Counts
67 → 106
Calls

67 calls. Slowest were:

604.0ms
(* (* (* (/ 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))))
247.0ms
(* (/ 1 (hypot d c)) (/ (fma d b (* a c)) (hypot d c)))
143.0ms
(* (* (* (/ 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))))

prune1.4s

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 8.1b

localize9.0ms

Local error

Found 3 expressions with local error:

21.5b
(/ (fma c a (* b d)) (hypot d c))
0.2b
(fma c a (* b d))
0.1b
(/ (/ (fma c a (* b d)) (hypot d c)) (hypot d c))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
3 → 111
Calls

3 calls. Slowest were:

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))

series229.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

108.0ms
(/ (/ (fma c a (* b d)) (hypot d c)) (hypot d c))
83.0ms
(/ (fma c a (* b d)) (hypot d c))
39.0ms
(fma c a (* b d))

simplify3.1s

Counts
132 → 120
Calls

132 calls. Slowest were:

327.0ms
(/ (/ (* (* (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)))
213.0ms
(/ (/ (* (cbrt (fma c a (* b d))) (cbrt (fma c a (* b d)))) (* (cbrt (hypot d c)) (cbrt (hypot d c)))) 1)

prune1.5s

Pruning

10 alts after pruning (8 fresh and 2 done)

Merged error: 3.3b

localize17.0ms

Local error

Found 4 expressions with local error:

22.1b
(cbrt (fma d b (* a c)))
22.1b
(cbrt (fma d b (* a c)))
22.1b
(cbrt (fma d b (* a c)))
4.7b
(* (/ 1 (hypot d c)) (/ (* (cbrt (fma d b (* a c))) (cbrt (fma d b (* a c)))) 1))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
4 → 64
Calls

4 calls. Slowest were:

7.0ms
(* (/ 1 (hypot d c)) (/ (* (cbrt (fma d b (* a c))) (cbrt (fma d b (* a c)))) 1))
3.0ms
(cbrt (fma d b (* a c)))
1.0ms
(cbrt (fma d b (* a c)))

series694.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

264.0ms
(* (/ 1 (hypot d c)) (/ (* (cbrt (fma d b (* a c))) (cbrt (fma d b (* a c)))) 1))
157.0ms
(cbrt (fma d b (* a c)))
143.0ms
(cbrt (fma d b (* a c)))
129.0ms
(cbrt (fma d b (* a c)))

simplify3.8s

Counts
34 → 76
Calls

34 calls. Slowest were:

396.0ms
(* (/ 1 (hypot d c)) (/ (* (cbrt (fma d b (* a c))) (cbrt (fma d b (* a c)))) 1))
375.0ms
(exp (* -1/3 (+ (log (/ 1 b)) (log (/ 1 d)))))
370.0ms
(exp (* -1/3 (+ (log (/ 1 b)) (log (/ 1 d)))))

prune1.1s

Pruning

10 alts after pruning (7 fresh and 3 done)

Merged error: 3.3b

regimes338.0ms

Accuracy

20.2% (11.4b remaining)

Error of 13.5b against oracle of 2.2b and baseline of 16.4b

bsearch272.0ms

end0.0ms

sample2.0s

Algorithm
intervals