Complex division, imag part

Average Accuracy: 58.2% → 98.5%

Time: 17.9s

Precision: binary64

Cost: 20352

Math

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

↓

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

FPCore

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

↓

(FPCore (a b c d)
 :precision binary64
 (* (/ 1.0 (hypot c d)) (- (* c (/ b (hypot c d))) (* (/ d (hypot c d)) a))))

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

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 (1.0 / Math.hypot(c, d)) * ((c * (b / Math.hypot(c, d))) - ((d / Math.hypot(c, d)) * a));
}

Python

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

↓

def code(a, b, c, d):
	return (1.0 / math.hypot(c, d)) * ((c * (b / math.hypot(c, d))) - ((d / math.hypot(c, d)) * a))

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(1.0 / hypot(c, d)) * Float64(Float64(c * Float64(b / hypot(c, d))) - Float64(Float64(d / hypot(c, d)) * a)))
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 = (1.0 / hypot(c, d)) * ((c * (b / hypot(c, d))) - ((d / hypot(c, d)) * a));
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[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(c * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(d / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	58.2%
Target	99.3%
Herbie	98.5%

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

Derivation

1. Initial program 58.2%

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

2. Applied egg-rr 72.6%

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

   [Start] 58.2	\[ \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   *-un-lft-identity [=>] 58.2	\[ \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{c \cdot c + d \cdot d} \]
   add-sqr-sqrt [=>] 58.2	\[ \frac{1 \cdot \left(b \cdot c - a \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]
   times-frac [=>] 58.2	\[ \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]
   hypot-def [=>] 58.2	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]
   hypot-def [=>] 72.6	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - a \cdot d}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

3. Applied egg-rr 98.5%

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

   [Start] 72.6	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)} \]
   div-sub [=>] 72.6	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(\frac{b \cdot c}{\mathsf{hypot}\left(c, d\right)} - \frac{a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right)} \]
   associate-/l* [=>] 84.7	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\color{blue}{\frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{c}}} - \frac{a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right) \]
   associate-/r/ [=>] 84.2	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\color{blue}{\frac{b}{\mathsf{hypot}\left(c, d\right)} \cdot c} - \frac{a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right) \]
   *-commutative [=>] 84.2	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\mathsf{hypot}\left(c, d\right)} \cdot c - \frac{\color{blue}{d \cdot a}}{\mathsf{hypot}\left(c, d\right)}\right) \]
   associate-/l* [=>] 97.8	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\mathsf{hypot}\left(c, d\right)} \cdot c - \color{blue}{\frac{d}{\frac{\mathsf{hypot}\left(c, d\right)}{a}}}\right) \]
   associate-/r/ [=>] 98.5	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\mathsf{hypot}\left(c, d\right)} \cdot c - \color{blue}{\frac{d}{\mathsf{hypot}\left(c, d\right)} \cdot a}\right) \]

4. Final simplification 98.5%

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

Alternatives

Alternative 1
Accuracy	87.3%
Cost	15817

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

Alternative 2
Accuracy	81.9%
Cost	14032

\[\begin{array}{l} \mathbf{if}\;c \leq -1.6 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{a}{\frac{c}{d}} - b\right)\\ \mathbf{elif}\;c \leq -9.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq -1.8 \cdot 10^{-233}:\\ \;\;\;\;b \cdot \frac{c}{d \cdot d} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 2.8 \cdot 10^{-153}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{\mathsf{hypot}\left(c, d\right)}{d}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 3
Accuracy	80.0%
Cost	7300

\[\begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -1.32 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{a}{\frac{c}{d}} - b\right)\\ \mathbf{elif}\;c \leq -6.8 \cdot 10^{-125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -1 \cdot 10^{-233}:\\ \;\;\;\;b \cdot \frac{c}{d \cdot d} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 6.4 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \end{array} \]

Alternative 4
Accuracy	80.1%
Cost	1620

\[\begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -2.05 \cdot 10^{+54}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{elif}\;c \leq -2.45 \cdot 10^{-124}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -1.2 \cdot 10^{-233}:\\ \;\;\;\;b \cdot \frac{c}{d \cdot d} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 2.45 \cdot 10^{-61}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 6.4 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \end{array} \]

Alternative 5
Accuracy	61.8%
Cost	1304

\[\begin{array}{l} t_0 := d \cdot \frac{\frac{-a}{c}}{c}\\ \mathbf{if}\;c \leq -3.35 \cdot 10^{+134}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -3.1 \cdot 10^{+105}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -0.46:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 7 \cdot 10^{-57}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{+141}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 9.8 \cdot 10^{+179}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 6
Accuracy	61.7%
Cost	1304

\[\begin{array}{l} t_0 := \frac{-a}{c}\\ \mathbf{if}\;c \leq -3.35 \cdot 10^{+134}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -3.1 \cdot 10^{+105}:\\ \;\;\;\;\frac{d \cdot t_0}{c}\\ \mathbf{elif}\;c \leq -2800:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.15 \cdot 10^{-54}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{+141}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 9.8 \cdot 10^{+179}:\\ \;\;\;\;d \cdot \frac{t_0}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 7
Accuracy	61.6%
Cost	1304

\[\begin{array}{l} t_0 := \frac{a \cdot \frac{-d}{c}}{c}\\ \mathbf{if}\;c \leq -6.8 \cdot 10^{+133}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -3 \cdot 10^{+105}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -3.4:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 3 \cdot 10^{-54}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 4 \cdot 10^{+141}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 9.8 \cdot 10^{+179}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 8
Accuracy	75.4%
Cost	969

\[\begin{array}{l} \mathbf{if}\;c \leq -7.6 \cdot 10^{+27} \lor \neg \left(c \leq 2.8 \cdot 10^{-54}\right):\\ \;\;\;\;\frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \end{array} \]

Alternative 9
Accuracy	75.3%
Cost	968

\[\begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{+27}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{elif}\;c \leq 2.9 \cdot 10^{-54}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \end{array} \]

Alternative 10
Accuracy	69.8%
Cost	841

\[\begin{array}{l} \mathbf{if}\;c \leq -51000 \lor \neg \left(c \leq 9.5 \cdot 10^{-59}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]

Alternative 11
Accuracy	70.3%
Cost	841

\[\begin{array}{l} \mathbf{if}\;c \leq -11200 \lor \neg \left(c \leq 10^{-54}\right):\\ \;\;\;\;\frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]

Alternative 12
Accuracy	63.6%
Cost	520

\[\begin{array}{l} \mathbf{if}\;c \leq -200000:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.95 \cdot 10^{-54}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 13
Accuracy	13.9%
Cost	456

\[\begin{array}{l} \mathbf{if}\;d \leq -2.7 \cdot 10^{+97}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{+123}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \]

Alternative 14
Accuracy	45.6%
Cost	456

\[\begin{array}{l} \mathbf{if}\;d \leq -5.1 \cdot 10^{+134}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{+184}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \]

Alternative 15
Accuracy	8.2%
Cost	192

\[\frac{a}{c} \]

Reproduce

herbie shell --seed 2023126 
(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))))