Complex division, imag part

Average Error: 26.2 → 0.9

Time: 14.3s

Precision: binary64

Cost: 20224

Math

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

↓

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

FPCore

(FPCore (a b c d)
 :precision binary64
 (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))

↓

(FPCore (a b c d)
 :precision binary64
 (/ (- (* c (/ b (hypot c d))) (/ a (/ (hypot c d) d))) (hypot c d)))

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) {
	return ((c * (b / hypot(c, d))) - (a / (hypot(c, d) / d))) / hypot(c, d);
}

Java

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

↓

public static double code(double a, double b, double c, double d) {
	return ((c * (b / Math.hypot(c, d))) - (a / (Math.hypot(c, d) / d))) / Math.hypot(c, d);
}

Python

def code(a, b, c, d):
	return ((b * c) - (a * d)) / ((c * c) + (d * d))

↓

def code(a, b, c, d):
	return ((c * (b / math.hypot(c, d))) - (a / (math.hypot(c, d) / d))) / math.hypot(c, d)

Julia

function code(a, b, c, d)
	return Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
end

↓

function code(a, b, c, d)
	return Float64(Float64(Float64(c * Float64(b / hypot(c, d))) - Float64(a / Float64(hypot(c, d) / d))) / hypot(c, d))
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = ((b * c) - (a * d)) / ((c * c) + (d * d));
end

↓

function tmp = code(a, b, c, d)
	tmp = ((c * (b / hypot(c, d))) - (a / (hypot(c, d) / d))) / hypot(c, d);
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_, d_] := N[(N[(N[(c * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]

Target

Original	26.2
Target	0.4
Herbie	0.9

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

Derivation

1. Initial program 26.2

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

2. Applied egg-rr 16.9

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

3. Applied egg-rr 1.0

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

4. Applied egg-rr 0.8

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

5. Applied egg-rr 0.9

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

6. Final simplification 0.9

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

Alternatives

Alternative 1
Error	0.8
Cost	20224

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

Alternative 2
Error	6.7
Cost	15881

\[\begin{array}{l} t_0 := \frac{b \cdot c - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 5 \cdot 10^{+296}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(d \cdot a - b \cdot c\right)}{-\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 3
Error	6.7
Cost	15817

\[\begin{array}{l} t_0 := b \cdot c - d \cdot a\\ t_1 := \frac{t_0}{c \cdot c + d \cdot d}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 5 \cdot 10^{+296}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 4
Error	5.5
Cost	14672

\[\begin{array}{l} t_0 := \frac{d}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := \frac{b}{\frac{c \cdot c + d \cdot d}{c}} - t_0 \cdot \frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;c \leq -1.02 \cdot 10^{+129}:\\ \;\;\;\;\frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \mathbf{elif}\;c \leq -6.2 \cdot 10^{-182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 1.1 \cdot 10^{-165}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b \cdot \frac{c}{d} - a\right)\\ \mathbf{elif}\;c \leq 3.35 \cdot 10^{+69}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot t_0}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 5
Error	5.4
Cost	14672

\[\begin{array}{l} t_0 := \frac{b}{\frac{c \cdot c + d \cdot d}{c}}\\ t_1 := \frac{d}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;c \leq -1.02 \cdot 10^{+129}:\\ \;\;\;\;\frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \mathbf{elif}\;c \leq -8.2 \cdot 10^{-136}:\\ \;\;\;\;t_0 + \frac{a}{\frac{\mathsf{hypot}\left(c, d\right)}{d}} \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq 1.62 \cdot 10^{-165}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b \cdot \frac{c}{d} - a\right)\\ \mathbf{elif}\;c \leq 3.35 \cdot 10^{+69}:\\ \;\;\;\;t_0 - t_1 \cdot \frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot t_1}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 6
Error	10.3
Cost	13900

\[\begin{array}{l} t_0 := c \cdot c + d \cdot d\\ \mathbf{if}\;c \leq -7 \cdot 10^{+125}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{elif}\;c \leq -1.8 \cdot 10^{-114}:\\ \;\;\;\;\frac{b}{\frac{t_0}{c}} - \frac{d \cdot a}{t_0}\\ \mathbf{elif}\;c \leq 8.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b \cdot \frac{c}{d} - a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 7
Error	10.6
Cost	7568

\[\begin{array}{l} t_0 := c \cdot c + d \cdot d\\ t_1 := \frac{b}{\frac{t_0}{c}} - \frac{d \cdot a}{t_0}\\ \mathbf{if}\;d \leq -3.5 \cdot 10^{+90}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 5 \cdot 10^{-159}:\\ \;\;\;\;\frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \mathbf{elif}\;d \leq 3.1 \cdot 10^{+66}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 8
Error	10.8
Cost	2000

\[\begin{array}{l} t_0 := c \cdot c + d \cdot d\\ t_1 := \frac{b}{\frac{t_0}{c}} - \frac{d \cdot a}{t_0}\\ t_2 := \frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{if}\;d \leq -1.66 \cdot 10^{+96}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-159}:\\ \;\;\;\;\frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{+68}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	12.7
Cost	1488

\[\begin{array}{l} t_0 := \frac{b \cdot c - d \cdot a}{c \cdot c + d \cdot d}\\ t_1 := \frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{if}\;d \leq -1.5 \cdot 10^{+99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -3.4 \cdot 10^{-77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 4.3 \cdot 10^{-74}:\\ \;\;\;\;\frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \mathbf{elif}\;d \leq 1.06 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 10^{+20}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	15.0
Cost	1032

\[\begin{array}{l} t_0 := \frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{if}\;d \leq -5.4 \cdot 10^{+87}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.7 \cdot 10^{-35}:\\ \;\;\;\;\frac{-d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+20}:\\ \;\;\;\;\frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	19.1
Cost	841

\[\begin{array}{l} \mathbf{if}\;d \leq -1.85 \cdot 10^{-35} \lor \neg \left(d \leq 1.06 \cdot 10^{+20}\right):\\ \;\;\;\;-\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \end{array} \]

Alternative 12
Error	14.7
Cost	841

\[\begin{array}{l} \mathbf{if}\;d \leq -1.4 \cdot 10^{-23} \lor \neg \left(d \leq 3.6 \cdot 10^{+19}\right):\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \end{array} \]

Alternative 13
Error	24.3
Cost	520

\[\begin{array}{l} \mathbf{if}\;c \leq -9 \cdot 10^{-52}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.1 \cdot 10^{+113}:\\ \;\;\;\;-\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 14
Error	37.0
Cost	324

\[\begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{+136}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 15
Error	56.9
Cost	192

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

Reproduce

herbie shell --seed 2023031 
(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))))