Complex division, imag part

Average Accuracy: 59.6% → 98.4%

Time: 13.5s

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	59.6%
Target	99.3%
Herbie	98.4%

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

Derivation

1. Initial program 59.6%

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

2. Applied egg-rr 74.2%

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

   [Start] 59.6	\[ \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   *-un-lft-identity [=>] 59.6	\[ \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{c \cdot c + d \cdot d} \]
   add-sqr-sqrt [=>] 59.6	\[ \frac{1 \cdot \left(b \cdot c - a \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]
   times-frac [=>] 59.5	\[ \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]
   hypot-def [=>] 59.5	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]
   hypot-def [=>] 74.2	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - a \cdot d}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

3. Applied egg-rr 98.4%

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

   [Start] 74.2	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)} \]
   div-sub [=>] 74.2	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(\frac{b \cdot c}{\mathsf{hypot}\left(c, d\right)} - \frac{a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right)} \]
   associate-/l* [=>] 85.7	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\color{blue}{\frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{c}}} - \frac{a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right) \]
   associate-/r/ [=>] 85.2	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\color{blue}{\frac{b}{\mathsf{hypot}\left(c, d\right)} \cdot c} - \frac{a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right) \]
   *-commutative [=>] 85.2	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\mathsf{hypot}\left(c, d\right)} \cdot c - \frac{\color{blue}{d \cdot a}}{\mathsf{hypot}\left(c, d\right)}\right) \]
   associate-/l* [=>] 97.6	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\mathsf{hypot}\left(c, d\right)} \cdot c - \color{blue}{\frac{d}{\frac{\mathsf{hypot}\left(c, d\right)}{a}}}\right) \]
   associate-/r/ [=>] 98.4	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\mathsf{hypot}\left(c, d\right)} \cdot c - \color{blue}{\frac{d}{\mathsf{hypot}\left(c, d\right)} \cdot a}\right) \]

4. Final simplification 98.4%

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

Alternatives

Alternative 1
Accuracy	88.9%
Cost	15816

\[\begin{array}{l} t_0 := c \cdot b - d \cdot a\\ t_1 := \frac{1}{\mathsf{hypot}\left(c, d\right)}\\ t_2 := c \cdot c + d \cdot d\\ t_3 := \frac{t_0}{t_2}\\ \mathbf{if}\;t_3 \leq -2 \cdot 10^{+240}:\\ \;\;\;\;\frac{c}{\frac{t_2}{b}} - \frac{a}{\frac{t_2}{d}}\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+274}:\\ \;\;\;\;t_1 \cdot \frac{t_0}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(b - \frac{d}{\mathsf{hypot}\left(c, d\right)} \cdot a\right)\\ \end{array} \]

Alternative 2
Accuracy	83.5%
Cost	14028

\[\begin{array}{l} t_0 := c \cdot c + d \cdot d\\ t_1 := \frac{1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;c \leq -1.16 \cdot 10^{+107}:\\ \;\;\;\;t_1 \cdot \left(\frac{a}{\frac{c}{d}} - b\right)\\ \mathbf{elif}\;c \leq -2.05 \cdot 10^{-153}:\\ \;\;\;\;\frac{c}{\frac{t_0}{b}} - \frac{a}{\frac{t_0}{d}}\\ \mathbf{elif}\;c \leq 1.42 \cdot 10^{-83}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(b - \frac{d}{\mathsf{hypot}\left(c, d\right)} \cdot a\right)\\ \end{array} \]

Alternative 3
Accuracy	79.9%
Cost	7500

\[\begin{array}{l} t_0 := c \cdot c + d \cdot d\\ \mathbf{if}\;d \leq -5 \cdot 10^{+156}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-103}:\\ \;\;\;\;\frac{c}{\frac{t_0}{b}} - \frac{a}{\frac{t_0}{d}}\\ \mathbf{elif}\;d \leq 56000000:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{\frac{a}{\frac{c}{d}}}{c}, \frac{b}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]

Alternative 4
Accuracy	79.6%
Cost	1736

\[\begin{array}{l} t_0 := c \cdot c + d \cdot d\\ \mathbf{if}\;d \leq -3 \cdot 10^{+151}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{elif}\;d \leq -1.6 \cdot 10^{-103}:\\ \;\;\;\;\frac{c}{\frac{t_0}{b}} - \frac{a}{\frac{t_0}{d}}\\ \mathbf{elif}\;d \leq 3100:\\ \;\;\;\;\frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]

Alternative 5
Accuracy	78.7%
Cost	1224

\[\begin{array}{l} t_0 := \frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{if}\;d \leq -1.55 \cdot 10^{+69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -2.9 \cdot 10^{-142}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 30500:\\ \;\;\;\;\frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	69.2%
Cost	841

\[\begin{array}{l} \mathbf{if}\;d \leq -1.32 \cdot 10^{-36} \lor \neg \left(d \leq 16000000000\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \end{array} \]

Alternative 7
Accuracy	75.6%
Cost	841

\[\begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-37} \lor \neg \left(d \leq 15000\right):\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \end{array} \]

Alternative 8
Accuracy	63.4%
Cost	521

\[\begin{array}{l} \mathbf{if}\;d \leq -1.95 \cdot 10^{-38} \lor \neg \left(d \leq 1.7 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 9
Accuracy	14.0%
Cost	456

\[\begin{array}{l} \mathbf{if}\;d \leq -8.5 \cdot 10^{+60}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{elif}\;d \leq 4.7 \cdot 10^{+37}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \]

Alternative 10
Accuracy	45.5%
Cost	456

\[\begin{array}{l} \mathbf{if}\;d \leq -9 \cdot 10^{+116}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{elif}\;d \leq 5.4 \cdot 10^{+164}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \]

Alternative 11
Accuracy	8.3%
Cost	192

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

Reproduce

herbie shell --seed 2023140 
(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))))