Average Error: 25.2 → 25.0

Time: 1.1m

Precision: 64

Internal Precision: 576

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -1.019095860764281 \cdot 10^{+69}:\\ \;\;\;\;\frac{-b}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{if}\;c \le 1.7442473790874188 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d}{\frac{c}{a}}}{\sqrt{d \cdot d + c \cdot c}}\\ \end{array}\]

Error

The X axis uses an exponential scale — Bits error versus `a`

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	25.2
Target	0.5
Herbie	25.0

\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

Split input into 3 regimes
if c < -1.019095860764281e+69
1. Initial program 36.9
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt36.9
  \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied associate-/r*36.9
  \[\leadsto \color{blue}{\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
5. Taylor expanded around -inf 37.6
  \[\leadsto \frac{\color{blue}{-1 \cdot b}}{\sqrt{c \cdot c + d \cdot d}}\]
6. Applied simplify37.6
  \[\leadsto \color{blue}{\frac{-b}{\sqrt{c \cdot c + d \cdot d}}}\]
if -1.019095860764281e+69 < c < 1.7442473790874188e+37
1. Initial program 17.5
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt17.5
  \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied associate-/r*17.4
  \[\leadsto \color{blue}{\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
if 1.7442473790874188e+37 < c
1. Initial program 33.6
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt33.6
  \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied associate-/r*33.5
  \[\leadsto \color{blue}{\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
5. Taylor expanded around inf 35.4
  \[\leadsto \frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\color{blue}{c}}\]
6. Applied simplify32.2
  \[\leadsto \color{blue}{\frac{b - \frac{d}{\frac{c}{a}}}{\sqrt{d \cdot d + c \cdot c}}}\]
Recombined 3 regimes into one program.

Runtime

herbie shell --seed '#(1072967564 1937075727 894099792 790700740 1036514779 1027793188)' 
(FPCore (a b c d)
  :name "Complex division, imag part"

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

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

Error

Try it out

Target

Derivation

`if c < -1.019095860764281e+69`

`if -1.019095860764281e+69 < c < 1.7442473790874188e+37`

`if 1.7442473790874188e+37 < c`

Runtime