Average Error: 26.6 → 14.8

Time: 11.4s

Precision: binary64

Cost: 15556

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

↓

\[\begin{array}{l} \mathbf{if}\;d \leq -2.623594142603243 \cdot 10^{+102}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -2.178304667980649 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{elif}\;d \leq 4.0526762739374186 \cdot 10^{-107}:\\ \;\;\;\;\frac{b}{c} - \frac{d \cdot a}{c \cdot c}\\ \mathbf{elif}\;d \leq 5.943546987088161 \cdot 10^{+45}:\\ \;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot c}{d \cdot d} - \frac{a}{d}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;d \leq -2.623594142603243 \cdot 10^{+102}:\\
\;\;\;\;\frac{-a}{d}\\

\mathbf{elif}\;d \leq -2.178304667980649 \cdot 10^{-161}:\\
\;\;\;\;\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 4.0526762739374186 \cdot 10^{-107}:\\
\;\;\;\;\frac{b}{c} - \frac{d \cdot a}{c \cdot c}\\

\mathbf{elif}\;d \leq 5.943546987088161 \cdot 10^{+45}:\\
\;\;\;\;\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 \cdot c}{d \cdot d} - \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 -2.623594142603243e+102)
   (/ (- a) d)
   (if (<= d -2.178304667980649e-161)
     (/
      (/ (- (* b c) (* d a)) (sqrt (+ (* c c) (* d d))))
      (sqrt (+ (* c c) (* d d))))
     (if (<= d 4.0526762739374186e-107)
       (- (/ b c) (/ (* d a) (* c c)))
       (if (<= d 5.943546987088161e+45)
         (/
          (/ (- (* b c) (* d a)) (sqrt (+ (* c c) (* d d))))
          (sqrt (+ (* c c) (* d d))))
         (- (/ (* b c) (* d d)) (/ 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 <= -2.623594142603243e+102) {
		tmp = -a / d;
	} else if (d <= -2.178304667980649e-161) {
		tmp = (((b * c) - (d * a)) / sqrt((c * c) + (d * d))) / sqrt((c * c) + (d * d));
	} else if (d <= 4.0526762739374186e-107) {
		tmp = (b / c) - ((d * a) / (c * c));
	} else if (d <= 5.943546987088161e+45) {
		tmp = (((b * c) - (d * a)) / sqrt((c * c) + (d * d))) / sqrt((c * c) + (d * d));
	} else {
		tmp = ((b * c) / (d * d)) - (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	26.6
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	27.1
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	45.3
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	27.0
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	26.7
Cost	34112

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

Alternative 5
Error	26.6
Cost	33984

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

Alternative 6
Error	27.0
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	27.0
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	27.0
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	27.0
Cost	21184

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

Alternative 10
Error	54.9
Cost	15168

\[\frac{b \cdot c - a \cdot d}{{c}^{6} + {d}^{6}} \cdot \left(\left(c \cdot c\right) \cdot \left(c \cdot c\right) + \left(\left(d \cdot d\right) \cdot \left(d \cdot d\right) - \left(c \cdot c\right) \cdot \left(d \cdot d\right)\right)\right)\]

Alternative 11
Error	40.3
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	26.6
Cost	14400

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

Alternative 13
Error	45.2
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 14
Error	26.5
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 15
Error	49.2
Cost	14144

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

Alternative 16
Error	41.7
Cost	13824

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

Alternative 17
Error	47.2
Cost	7552

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

Alternative 18
Error	47.1
Cost	7552

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

Alternative 19
Error	47.3
Cost	7488

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

Alternative 20
Error	46.9
Cost	7360

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

Alternative 21
Error	46.9
Cost	7360

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

Alternative 22
Error	47.1
Cost	7360

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

Alternative 23
Error	47.5
Cost	6976

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

Alternative 24
Error	41.8
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 25
Error	26.6
Cost	1472

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

Alternative 26
Error	26.7
Cost	1088

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

Alternative 27
Error	26.7
Cost	1088

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

Alternative 28
Error	26.6
Cost	960

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

Alternative 29
Error	40.0
Cost	768

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

Alternative 30
Error	34.9
Cost	704

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

Alternative 31
Error	34.8
Cost	704

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

Alternative 32
Error	39.3
Cost	704

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

Alternative 33
Error	37.5
Cost	256

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

Alternative 34
Error	37.3
Cost	192

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

Alternative 35
Error	61.6
Cost	64

\[1\]

Alternative 36
Error	52.3
Cost	64

\[0\]

Alternative 37
Error	61.6
Cost	64

\[-1\]

Derivation

Split input into 4 regimes
if d < -2.6235941426032431e102
1. Initial program 41.0
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Taylor expanded around 0 17.6
  \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}}\]
3. Simplified17.6
  \[\leadsto \color{blue}{\frac{-a}{d}}\]
4. Simplified17.6
  \[\leadsto \color{blue}{\frac{-a}{d}}\]
if -2.6235941426032431e102 < d < -2.178304667980649e-161 or 4.0526762739374186e-107 < d < 5.943546987088161e45
1. Initial program 15.6
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt_binary64_214615.6
  \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied associate-/r*_binary64_206815.6
  \[\leadsto \color{blue}{\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
5. Simplified15.6
  \[\leadsto \color{blue}{\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
if -2.178304667980649e-161 < d < 4.0526762739374186e-107
1. Initial program 22.6
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Taylor expanded around inf 10.2
  \[\leadsto \color{blue}{\frac{b}{c} - \frac{d \cdot a}{{c}^{2}}}\]
3. Simplified10.2
  \[\leadsto \color{blue}{\frac{b}{c} - \frac{a \cdot d}{c \cdot c}}\]
4. Simplified10.2
  \[\leadsto \color{blue}{\frac{b}{c} - \frac{a \cdot d}{c \cdot c}}\]
if 5.943546987088161e45 < d
1. Initial program 36.1
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Taylor expanded around 0 16.8
  \[\leadsto \color{blue}{\frac{b \cdot c}{{d}^{2}} - \frac{a}{d}}\]
3. Simplified16.8
  \[\leadsto \color{blue}{\frac{b \cdot c}{d \cdot d} - \frac{a}{d}}\]
4. Simplified16.8
  \[\leadsto \color{blue}{\frac{b \cdot c}{d \cdot d} - \frac{a}{d}}\]
Recombined 4 regimes into one program.
Final simplification14.8
\[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.623594142603243 \cdot 10^{+102}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -2.178304667980649 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{elif}\;d \leq 4.0526762739374186 \cdot 10^{-107}:\\ \;\;\;\;\frac{b}{c} - \frac{d \cdot a}{c \cdot c}\\ \mathbf{elif}\;d \leq 5.943546987088161 \cdot 10^{+45}:\\ \;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot c}{d \cdot d} - \frac{a}{d}\\ \end{array}\]

Reproduce

herbie shell --seed 2021022 
(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 < -2.6235941426032431e102`

`if -2.6235941426032431e102 < d < -2.178304667980649e-161 or 4.0526762739374186e-107 < d < 5.943546987088161e45`

`if -2.178304667980649e-161 < d < 4.0526762739374186e-107`

`if 5.943546987088161e45 < d`

Reproduce