Average Error: 25.9 → 25.0

Time: 1.2m

Precision: 64

Internal Precision: 576

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le -3.607046218183898 \cdot 10^{+307}:\\ \;\;\;\;\frac{-b}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le 6.205643929172327 \cdot 10^{+293}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{\sqrt{d \cdot d + c \cdot c}}\\ \end{array}\]

Error

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

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In
Out

Enter valid numbers for all inputs

Target

Original	25.9
Target	0.4
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 (/ (- (* b c) (* a d)) (+ (* c c) (* d d))) < -3.607046218183898e+307
1. Initial program 60.6
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt60.6
  \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied associate-/r*60.6
  \[\leadsto \color{blue}{\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
5. Taylor expanded around -inf 52.7
  \[\leadsto \frac{\color{blue}{-1 \cdot b}}{\sqrt{c \cdot c + d \cdot d}}\]
6. Applied simplify52.7
  \[\leadsto \color{blue}{\frac{-b}{\sqrt{c \cdot c + d \cdot d}}}\]
if -3.607046218183898e+307 < (/ (- (* b c) (* a d)) (+ (* c c) (* d d))) < 6.205643929172327e+293
1. Initial program 11.4
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt11.4
  \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied associate-/r*11.3
  \[\leadsto \color{blue}{\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
if 6.205643929172327e+293 < (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))
1. Initial program 61.9
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt61.9
  \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied associate-/r*61.9
  \[\leadsto \color{blue}{\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
5. Taylor expanded around 0 62.5
  \[\leadsto \frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\color{blue}{d}}\]
6. Applied simplify60.1
  \[\leadsto \color{blue}{\frac{c \cdot \frac{b}{d} - a}{\sqrt{d \cdot d + c \cdot c}}}\]
Recombined 3 regimes into one program.

Runtime

herbie shell --seed 2018201 
(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 (/ (- (* b c) (* a d)) (+ (* c c) (* d d))) < -3.607046218183898e+307`

`if -3.607046218183898e+307 < (/ (- (* b c) (* a d)) (+ (* c c) (* d d))) < 6.205643929172327e+293`

`if 6.205643929172327e+293 < (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))`

Runtime