Average Error: 25.7 → 14.8

Time: 8.7s

Precision: binary64

Cost: 15877

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

↓

\[\begin{array}{l} \mathbf{if}\;d \leq -1.1437132202364462 \cdot 10^{+93}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -7.2230969502839745 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{elif}\;d \leq -1.3434382879141502 \cdot 10^{-278}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{\frac{c \cdot c}{d}}\\ \mathbf{elif}\;d \leq 1.0231125141194154 \cdot 10^{-128}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 2.405852373616342 \cdot 10^{+98}:\\ \;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\frac{d \cdot d}{c}} - \frac{a}{d}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;d \leq -1.1437132202364462 \cdot 10^{+93}:\\
\;\;\;\;\frac{-a}{d}\\

\mathbf{elif}\;d \leq -7.2230969502839745 \cdot 10^{-140}:\\
\;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\

\mathbf{elif}\;d \leq -1.3434382879141502 \cdot 10^{-278}:\\
\;\;\;\;\frac{b}{c} - \frac{a}{\frac{c \cdot c}{d}}\\

\mathbf{elif}\;d \leq 1.0231125141194154 \cdot 10^{-128}:\\
\;\;\;\;\frac{b}{c}\\

\mathbf{elif}\;d \leq 2.405852373616342 \cdot 10^{+98}:\\
\;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\

\mathbf{else}:\\
\;\;\;\;\frac{b}{\frac{d \cdot d}{c}} - \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
 (if (<= d -1.1437132202364462e+93)
   (/ (- a) d)
   (if (<= d -7.2230969502839745e-140)
     (/
      (/ (- (* b c) (* d a)) (sqrt (+ (* c c) (* d d))))
      (sqrt (+ (* c c) (* d d))))
     (if (<= d -1.3434382879141502e-278)
       (- (/ b c) (/ a (/ (* c c) d)))
       (if (<= d 1.0231125141194154e-128)
         (/ b c)
         (if (<= d 2.405852373616342e+98)
           (/
            (/ (- (* b c) (* d a)) (sqrt (+ (* c c) (* d d))))
            (sqrt (+ (* c c) (* d d))))
           (- (/ b (/ (* d 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 tmp;
	if (d <= -1.1437132202364462e+93) {
		tmp = -a / d;
	} else if (d <= -7.2230969502839745e-140) {
		tmp = (((b * c) - (d * a)) / sqrt((c * c) + (d * d))) / sqrt((c * c) + (d * d));
	} else if (d <= -1.3434382879141502e-278) {
		tmp = (b / c) - (a / ((c * c) / d));
	} else if (d <= 1.0231125141194154e-128) {
		tmp = b / c;
	} else if (d <= 2.405852373616342e+98) {
		tmp = (((b * c) - (d * a)) / sqrt((c * c) + (d * d))) / sqrt((c * c) + (d * d));
	} else {
		tmp = (b / ((d * d) / c)) - (a / d);
	}
	return tmp;
}

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	25.7
Target	0.5
Herbie	14.8

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

Alternatives

Alternative 1
Error	26.3
Cost	41408

\[\frac{\sqrt[3]{b \cdot c - a \cdot d} \cdot \sqrt[3]{b \cdot c - a \cdot d}}{\sqrt[3]{c \cdot c + d \cdot d} \cdot \sqrt[3]{c \cdot c + d \cdot d}} \cdot \frac{\sqrt[3]{b \cdot c - a \cdot d}}{\sqrt[3]{c \cdot c + d \cdot d}}\]

Alternative 2
Error	45.0
Cost	34496

\[\frac{\sqrt{b \cdot c - a \cdot d}}{\sqrt[3]{c \cdot c + d \cdot d} \cdot \sqrt[3]{c \cdot c + d \cdot d}} \cdot \frac{\sqrt{b \cdot c - a \cdot d}}{\sqrt[3]{c \cdot c + d \cdot d}}\]

Alternative 3
Error	26.2
Cost	34496

\[\frac{\sqrt[3]{b \cdot c - a \cdot d} \cdot \sqrt[3]{b \cdot c - a \cdot d}}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{\sqrt[3]{b \cdot c - a \cdot d}}{\sqrt{c \cdot c + d \cdot d}}\]

Alternative 4
Error	44.8
Cost	27584

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

Alternative 5
Error	27.5
Cost	27072

\[\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{e^{\log \left(\sqrt{c \cdot c + d \cdot d}\right)}}\]

Alternative 6
Error	26.2
Cost	22208

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

Alternative 7
Error	26.2
Cost	21312

\[\frac{1}{\sqrt[3]{c \cdot c + d \cdot d} \cdot \sqrt[3]{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt[3]{c \cdot c + d \cdot d}}\]

Alternative 8
Error	26.2
Cost	21184

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

Alternative 9
Error	47.1
Cost	15424

\[\frac{\frac{{\left(b \cdot c\right)}^{3} - {\left(a \cdot d\right)}^{3}}{c \cdot c + d \cdot d}}{\left(b \cdot c\right) \cdot \left(b \cdot c\right) + \left(\left(a \cdot d\right) \cdot \left(a \cdot d\right) + \left(b \cdot c\right) \cdot \left(a \cdot d\right)\right)}\]

Alternative 10
Error	47.7
Cost	15168

\[\frac{{\left(b \cdot c\right)}^{3} - {\left(a \cdot d\right)}^{3}}{\left(c \cdot c + d \cdot d\right) \cdot \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) + a \cdot \left(d \cdot \left(b \cdot c + a \cdot d\right)\right)\right)}\]

Alternative 11
Error	39.6
Cost	14784

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

Alternative 12
Error	26.0
Cost	14528

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

Alternative 13
Error	26.0
Cost	14528

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

Alternative 14
Error	25.7
Cost	14400

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

Alternative 15
Error	25.9
Cost	14400

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

Alternative 16
Error	44.8
Cost	14272

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

Alternative 17
Error	25.7
Cost	14272

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

Alternative 18
Error	48.5
Cost	14144

\[\frac{b \cdot c - a \cdot d}{{c}^{4} - {d}^{4}} \cdot \left(c \cdot c - d \cdot d\right)\]

Alternative 19
Error	40.9
Cost	13824

\[\sqrt[3]{{\left(\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\right)}^{3}}\]

Alternative 20
Error	46.5
Cost	7680

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

Alternative 21
Error	46.3
Cost	7616

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

Alternative 22
Error	46.2
Cost	7488

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

Alternative 23
Error	46.1
Cost	7360

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

Alternative 24
Error	47.1
Cost	7040

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

Alternative 25
Error	46.7
Cost	7040

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

Alternative 26
Error	40.9
Cost	1984

\[\frac{\left(b \cdot c\right) \cdot \left(b \cdot c\right) - \left(a \cdot d\right) \cdot \left(a \cdot d\right)}{\left(c \cdot c + d \cdot d\right) \cdot \left(b \cdot c + a \cdot d\right)}\]

Alternative 27
Error	25.9
Cost	1088

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

Alternative 28
Error	25.9
Cost	1088

\[\frac{1}{\frac{c \cdot c + d \cdot d}{b \cdot c - a \cdot d}}\]

Alternative 29
Error	25.7
Cost	960

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

Alternative 30
Error	43.8
Cost	832

\[\left(b \cdot c - a \cdot d\right) \cdot \frac{1}{d \cdot d}\]

Alternative 31
Error	38.9
Cost	768

\[\frac{a \cdot \left(-d\right)}{c \cdot c + d \cdot d}\]

Alternative 32
Error	35.2
Cost	704

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

Alternative 33
Error	34.1
Cost	704

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

Alternative 34
Error	38.8
Cost	704

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

Alternative 35
Error	35.0
Cost	704

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

Alternative 36
Error	34.1
Cost	704

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

Alternative 37
Error	37.0
Cost	256

\[\frac{-a}{d}\]

Alternative 38
Error	37.7
Cost	192

\[\frac{b}{c}\]

Alternative 39
Error	61.6
Cost	64

\[1\]

Alternative 40
Error	51.6
Cost	64

\[0\]

Alternative 41
Error	61.7
Cost	64

\[-1\]

Derivation

Split input into 5 regimes
if d < -1.1437132202364462e93
1. Initial program 38.0
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Taylor expanded around 0 16.1
  \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}}\]
3. Simplified16.1
  \[\leadsto \color{blue}{\frac{-a}{d}}\]
4. Simplified16.1
  \[\leadsto \color{blue}{\frac{-a}{d}}\]
if -1.1437132202364462e93 < d < -7.22309695028397451e-140 or 1.0231125141194154e-128 < d < 2.40585237361634206e98
1. Initial program 15.2
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt_binary64_442215.2
  \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied associate-/r*_binary64_434415.1
  \[\leadsto \color{blue}{\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
5. Simplified15.1
  \[\leadsto \color{blue}{\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
if -7.22309695028397451e-140 < d < -1.3434382879141502e-278
1. Initial program 22.8
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied div-inv_binary64_439723.3
  \[\leadsto \color{blue}{\left(b \cdot c - a \cdot d\right) \cdot \frac{1}{c \cdot c + d \cdot d}}\]
4. Using strategy rm
5. Applied add-sqr-sqrt_binary64_442223.3
  \[\leadsto \left(b \cdot c - a \cdot d\right) \cdot \frac{1}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
6. Applied associate-/r*_binary64_434423.3
  \[\leadsto \left(b \cdot c - a \cdot d\right) \cdot \color{blue}{\frac{\frac{1}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
7. Taylor expanded around inf 10.3
  \[\leadsto \color{blue}{\frac{b}{c} - \frac{d \cdot a}{{c}^{2}}}\]
8. Simplified10.7
  \[\leadsto \color{blue}{\frac{b}{c} - \frac{a}{\frac{c \cdot c}{d}}}\]
9. Simplified10.7
  \[\leadsto \color{blue}{\frac{b}{c} - \frac{a}{\frac{c \cdot c}{d}}}\]
if -1.3434382879141502e-278 < d < 1.0231125141194154e-128
1. Initial program 23.5
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Taylor expanded around inf 15.0
  \[\leadsto \color{blue}{\frac{b}{c}}\]
3. Simplified15.0
  \[\leadsto \color{blue}{\frac{b}{c}}\]
if 2.40585237361634206e98 < d
1. Initial program 38.7
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied div-inv_binary64_439738.7
  \[\leadsto \color{blue}{\left(b \cdot c - a \cdot d\right) \cdot \frac{1}{c \cdot c + d \cdot d}}\]
4. Using strategy rm
5. Applied add-sqr-sqrt_binary64_442238.7
  \[\leadsto \left(b \cdot c - a \cdot d\right) \cdot \frac{1}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
6. Applied associate-/r*_binary64_434438.7
  \[\leadsto \left(b \cdot c - a \cdot d\right) \cdot \color{blue}{\frac{\frac{1}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
7. Taylor expanded around 0 15.7
  \[\leadsto \color{blue}{\frac{b \cdot c}{{d}^{2}} - \frac{a}{d}}\]
8. Simplified15.0
  \[\leadsto \color{blue}{\frac{b}{\frac{d \cdot d}{c}} - \frac{a}{d}}\]
9. Simplified15.0
  \[\leadsto \color{blue}{\frac{b}{\frac{d \cdot d}{c}} - \frac{a}{d}}\]
Recombined 5 regimes into one program.
Final simplification14.8
\[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.1437132202364462 \cdot 10^{+93}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -7.2230969502839745 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{elif}\;d \leq -1.3434382879141502 \cdot 10^{-278}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{\frac{c \cdot c}{d}}\\ \mathbf{elif}\;d \leq 1.0231125141194154 \cdot 10^{-128}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 2.405852373616342 \cdot 10^{+98}:\\ \;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\frac{d \cdot d}{c}} - \frac{a}{d}\\ \end{array}\]

Reproduce

herbie shell --seed 2021042 
(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))))

Error

Try it out

Target

Alternatives

Error

Derivation

`if d < -1.1437132202364462e93`

`if -1.1437132202364462e93 < d < -7.22309695028397451e-140 or 1.0231125141194154e-128 < d < 2.40585237361634206e98`

`if -7.22309695028397451e-140 < d < -1.3434382879141502e-278`

`if -1.3434382879141502e-278 < d < 1.0231125141194154e-128`

`if 2.40585237361634206e98 < d`

Reproduce