Average Error: 26.1 → 10.4
Time: 7.8s
Precision: binary64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
↓
\[\begin{array}{l} t_0 := b \cdot c - a \cdot d\\ t_1 := \frac{t_0}{c \cdot c + d \cdot d}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;t_1 \leq 7.916655966912562 \cdot 10^{+295}:\\ \;\;\;\;\frac{\frac{t_0}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{a}{d}\\ \end{array} \]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\begin{array}{l}
t_0 := b \cdot c - a \cdot d\\
t_1 := \frac{t_0}{c \cdot c + d \cdot d}\\
\mathbf{if}\;t_1 \leq -\infty:\\
\;\;\;\;\frac{b}{c}\\

\mathbf{elif}\;t_1 \leq 7.916655966912562 \cdot 10^{+295}:\\
\;\;\;\;\frac{\frac{t_0}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\

\mathbf{else}:\\
\;\;\;\;-\frac{a}{d}\\


\end{array}

(FPCore (a b c d)
 :precision binary64
 (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))
↓
(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* b c) (* a d))) (t_1 (/ t_0 (+ (* c c) (* d d)))))
   (if (<= t_1 (- INFINITY))
     (/ b c)
     (if (<= t_1 7.916655966912562e+295)
       (/ (/ t_0 (hypot d c)) (hypot d c))
       (- (/ a d))))))
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 t_0 = (b * c) - (a * d);
	double t_1 = t_0 / ((c * c) + (d * d));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = b / c;
	} else if (t_1 <= 7.916655966912562e+295) {
		tmp = (t_0 / hypot(d, c)) / hypot(d, c);
	} else {
		tmp = -(a / d);
	}
	return tmp;
}

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

Original26.1
Target0.5
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;\left|d\right| < \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < -inf.0

    1. Initial program 64.0

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

    2. Simplified64.0

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

    3. Taylor expanded in c around inf 36.8

      \[\leadsto \color{blue}{\frac{b}{c}} \]

    if -inf.0 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 7.9166559669125623e295

    1. Initial program 11.7

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

    2. Simplified11.7

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

    3. Applied add-sqr-sqrt_binary6411.7

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

    4. Applied *-un-lft-identity_binary6411.7

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

    5. Applied times-frac_binary6411.7

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

    6. Simplified11.7

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

    7. Simplified0.7

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

    8. Applied associate-*l/_binary640.6

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

    9. Simplified0.6

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

    if 7.9166559669125623e295 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

    1. Initial program 63.1

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

    2. Simplified63.1

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

    3. Taylor expanded in c around 0 35.6

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]

    4. Simplified35.6

      \[\leadsto \color{blue}{-\frac{a}{d}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq -\infty:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq 7.916655966912562 \cdot 10^{+295}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{a}{d}\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(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))))