Complex division, imag part

Average Error: 26.4 → 13.0

Time: 3.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -1.5195216293899823 \cdot 10^{103}:\\ \;\;\;\;{\left(\frac{-1 \cdot b}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \mathbf{elif}\;c \le 9.14594638519536889 \cdot 10^{130}:\\ \;\;\;\;{\left(\frac{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{b}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \end{array}\]

C

double code(double a, double b, double c, double d) {
	return (((b * c) - (a * d)) / ((c * c) + (d * d)));
}

↓

double code(double a, double b, double c, double d) {
	double temp;
	if ((c <= -1.5195216293899823e+103)) {
		temp = pow(((-1.0 * b) / hypot(c, d)), 1.0);
	} else {
		double temp_1;
		if ((c <= 9.145946385195369e+130)) {
			temp_1 = pow(((((b * c) - (a * d)) / hypot(c, d)) / hypot(c, d)), 1.0);
		} else {
			temp_1 = pow((b / hypot(c, d)), 1.0);
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.4
Target	0.4
Herbie	13.0

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

Derivation

1. Split input into 3 regimes

2. if c < -1.5195216293899823e+103

   1. Initial program 40.0

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

   3. Applied add-sqr-sqrt 40.0

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

   4. Applied *-un-lft-identity 40.0

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

   5. Applied times-frac 40.0

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

   6. Simplified 40.0

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]

   7. Simplified 26.4

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}\]
   8. Using strategy rm

   9. Applied pow1 26.4

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1} \cdot \color{blue}{{\left(\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}}\]

   10. Applied pow1 26.4

       \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1}\right)}^{1}} \cdot {\left(\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\]

   11. Applied pow-prod-down 26.4

       \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1} \cdot \frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}}\]

   12. Simplified 26.4

       \[\leadsto {\color{blue}{\left(\frac{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\right)}}^{1}\]

   13. Taylor expanded around -inf 15.7

       \[\leadsto {\left(\frac{\color{blue}{-1 \cdot b}}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\]

   if -1.5195216293899823e+103 < c < 9.145946385195369e+130

   1. Initial program 19.1

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

   3. Applied add-sqr-sqrt 19.1

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

   4. Applied *-un-lft-identity 19.1

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

   5. Applied times-frac 19.1

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

   6. Simplified 19.1

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]

   7. Simplified 12.1

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}\]
   8. Using strategy rm

   9. Applied pow1 12.1

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1} \cdot \color{blue}{{\left(\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}}\]

   10. Applied pow1 12.1

       \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1}\right)}^{1}} \cdot {\left(\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\]

   11. Applied pow-prod-down 12.1

       \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1} \cdot \frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}}\]

   12. Simplified 12.0

       \[\leadsto {\color{blue}{\left(\frac{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\right)}}^{1}\]

   if 9.145946385195369e+130 < c

   1. Initial program 42.0

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

   3. Applied add-sqr-sqrt 42.0

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

   4. Applied *-un-lft-identity 42.0

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

   5. Applied times-frac 42.0

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

   6. Simplified 42.0

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]

   7. Simplified 28.3

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}\]
   8. Using strategy rm

   9. Applied pow1 28.3

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1} \cdot \color{blue}{{\left(\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}}\]

   10. Applied pow1 28.3

       \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1}\right)}^{1}} \cdot {\left(\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\]

   11. Applied pow-prod-down 28.3

       \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1} \cdot \frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}}\]

   12. Simplified 28.3

       \[\leadsto {\color{blue}{\left(\frac{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\right)}}^{1}\]

   13. Taylor expanded around inf 14.0

       \[\leadsto {\left(\frac{\color{blue}{b}}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\]
3. Recombined 3 regimes into one program.

4. Final simplification 13.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -1.5195216293899823 \cdot 10^{103}:\\ \;\;\;\;{\left(\frac{-1 \cdot b}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \mathbf{elif}\;c \le 9.14594638519536889 \cdot 10^{130}:\\ \;\;\;\;{\left(\frac{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{b}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 +o rules:numerics
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :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))))