Average Error: 25.9 → 26.0
Time: 1.1m
Precision: 64
Internal Precision: 320
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le 2.287685333116204 \cdot 10^{+71}:\\ \;\;\;\;\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{if}\;d \le 2.6646868450529295 \cdot 10^{+148}:\\ \;\;\;\;\frac{b}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{a \cdot c + b \cdot d} \cdot \sqrt[3]{a \cdot c + b \cdot d}}{\sqrt[3]{c \cdot c + d \cdot d} \cdot \sqrt[3]{c \cdot c + d \cdot d}} \cdot \frac{\sqrt[3]{a \cdot c + b \cdot d}}{\sqrt[3]{c \cdot c + d \cdot d}}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original25.9
Target0.5
Herbie26.0
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if d < 2.287685333116204e+71

    1. Initial program 22.6

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt22.6

      \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

    4. Applied associate-/r*22.5

      \[\leadsto \color{blue}{\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]

    if 2.287685333116204e+71 < d < 2.6646868450529295e+148

    1. Initial program 22.8

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt22.8

      \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

    4. Applied associate-/r*22.6

      \[\leadsto \color{blue}{\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]

    5. Taylor expanded around 0 25.2

      \[\leadsto \frac{\color{blue}{b}}{\sqrt{c \cdot c + d \cdot d}}\]

    if 2.6646868450529295e+148 < d

    1. Initial program 46.4

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt46.4

      \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\left(\sqrt[3]{c \cdot c + d \cdot d} \cdot \sqrt[3]{c \cdot c + d \cdot d}\right) \cdot \sqrt[3]{c \cdot c + d \cdot d}}}\]

    4. Applied add-cube-cbrt46.4

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{a \cdot c + b \cdot d} \cdot \sqrt[3]{a \cdot c + b \cdot d}\right) \cdot \sqrt[3]{a \cdot c + b \cdot d}}}{\left(\sqrt[3]{c \cdot c + d \cdot d} \cdot \sqrt[3]{c \cdot c + d \cdot d}\right) \cdot \sqrt[3]{c \cdot c + d \cdot d}}\]

    5. Applied times-frac46.4

      \[\leadsto \color{blue}{\frac{\sqrt[3]{a \cdot c + b \cdot d} \cdot \sqrt[3]{a \cdot c + b \cdot d}}{\sqrt[3]{c \cdot c + d \cdot d} \cdot \sqrt[3]{c \cdot c + d \cdot d}} \cdot \frac{\sqrt[3]{a \cdot c + b \cdot d}}{\sqrt[3]{c \cdot c + d \cdot d}}}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2019053 
(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))))