Complex division, imag part

Average Accuracy: 59.8% → 97.5%

Time: 16.5s

Precision: binary64

Cost: 20352

Math

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

↓

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

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(d / Float64(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[(d / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	59.8%
Target	99.3%
Herbie	97.5%

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

Derivation

1. Initial program 59.8%

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

2. Applied egg-rr 73.8%

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

   [Start] 59.8	\[ \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   *-un-lft-identity [=>] 59.8	\[ \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{c \cdot c + d \cdot d} \]
   add-sqr-sqrt [=>] 59.8	\[ \frac{1 \cdot \left(b \cdot c - a \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]
   times-frac [=>] 59.8	\[ \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]
   hypot-def [=>] 59.8	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]
   hypot-def [=>] 73.8	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - a \cdot d}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

3. Applied egg-rr 98.2%

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

   [Start] 73.8	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)} \]
   div-sub [=>] 73.8	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(\frac{b \cdot c}{\mathsf{hypot}\left(c, d\right)} - \frac{a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right)} \]
   associate-/l* [=>] 85.4	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\color{blue}{\frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{c}}} - \frac{a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right) \]
   associate-/r/ [=>] 84.7	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\color{blue}{\frac{b}{\mathsf{hypot}\left(c, d\right)} \cdot c} - \frac{a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right) \]
   *-commutative [=>] 84.7	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\mathsf{hypot}\left(c, d\right)} \cdot c - \frac{\color{blue}{d \cdot a}}{\mathsf{hypot}\left(c, d\right)}\right) \]
   associate-/l* [=>] 97.5	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\mathsf{hypot}\left(c, d\right)} \cdot c - \color{blue}{\frac{d}{\frac{\mathsf{hypot}\left(c, d\right)}{a}}}\right) \]
   associate-/r/ [=>] 98.2	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\mathsf{hypot}\left(c, d\right)} \cdot c - \color{blue}{\frac{d}{\mathsf{hypot}\left(c, d\right)} \cdot a}\right) \]

4. Applied egg-rr 68.4%

   \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\mathsf{hypot}\left(c, d\right)} \cdot c - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{d}{\frac{\mathsf{hypot}\left(c, d\right)}{a}}\right)} - 1\right)}\right) \]
   Proof

   [Start] 98.2	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\mathsf{hypot}\left(c, d\right)} \cdot c - \frac{d}{\mathsf{hypot}\left(c, d\right)} \cdot a\right) \]
   expm1-log1p-u [=>] 80.6	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\mathsf{hypot}\left(c, d\right)} \cdot c - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\mathsf{hypot}\left(c, d\right)} \cdot a\right)\right)}\right) \]
   expm1-udef [=>] 68.7	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\mathsf{hypot}\left(c, d\right)} \cdot c - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{d}{\mathsf{hypot}\left(c, d\right)} \cdot a\right)} - 1\right)}\right) \]
   associate-/r/ [<=] 68.4	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\mathsf{hypot}\left(c, d\right)} \cdot c - \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{d}{\frac{\mathsf{hypot}\left(c, d\right)}{a}}}\right)} - 1\right)\right) \]

5. Simplified 97.5%

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

   [Start] 68.4	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\mathsf{hypot}\left(c, d\right)} \cdot c - \left(e^{\mathsf{log1p}\left(\frac{d}{\frac{\mathsf{hypot}\left(c, d\right)}{a}}\right)} - 1\right)\right) \]
   expm1-def [=>] 80.3	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\mathsf{hypot}\left(c, d\right)} \cdot c - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\frac{\mathsf{hypot}\left(c, d\right)}{a}}\right)\right)}\right) \]
   expm1-log1p [=>] 97.5	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\mathsf{hypot}\left(c, d\right)} \cdot c - \color{blue}{\frac{d}{\frac{\mathsf{hypot}\left(c, d\right)}{a}}}\right) \]

6. Final simplification 97.5%

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

Alternatives

Alternative 1
Accuracy	98.2%
Cost	20352

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

Alternative 2
Accuracy	85.9%
Cost	14160

\[\begin{array}{l} t_0 := \frac{\frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;c \leq -2.8 \cdot 10^{+63}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq -9 \cdot 10^{-169}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 2.15 \cdot 10^{-110}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b \cdot \frac{c}{d} - a\right)\\ \mathbf{elif}\;c \leq 1.32 \cdot 10^{+123}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b - a \cdot \frac{d}{\mathsf{hypot}\left(c, d\right)}\right)\\ \end{array} \]

Alternative 3
Accuracy	82.2%
Cost	14032

\[\begin{array}{l} t_0 := \left(c \cdot b - d \cdot a\right) \cdot {\left(\mathsf{hypot}\left(c, d\right)\right)}^{-2}\\ \mathbf{if}\;c \leq -3.05 \cdot 10^{+62}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq -3.5 \cdot 10^{-123}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 3.6 \cdot 10^{-115}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b \cdot \frac{c}{d} - a\right)\\ \mathbf{elif}\;c \leq 8.5 \cdot 10^{+127}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \end{array} \]

Alternative 4
Accuracy	83.7%
Cost	14028

\[\begin{array}{l} \mathbf{if}\;c \leq -3.05 \cdot 10^{+63}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq -1.05 \cdot 10^{-124}:\\ \;\;\;\;\left(c \cdot b - d \cdot a\right) \cdot {\left(\mathsf{hypot}\left(c, d\right)\right)}^{-2}\\ \mathbf{elif}\;c \leq 3 \cdot 10^{-99}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b \cdot \frac{c}{d} - a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b - a \cdot \frac{d}{\mathsf{hypot}\left(c, d\right)}\right)\\ \end{array} \]

Alternative 5
Accuracy	83.6%
Cost	14028

\[\begin{array}{l} \mathbf{if}\;c \leq -3.05 \cdot 10^{+63}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq -2 \cdot 10^{-120}:\\ \;\;\;\;\left(c \cdot b - d \cdot a\right) \cdot {\left(\mathsf{hypot}\left(c, d\right)\right)}^{-2}\\ \mathbf{elif}\;c \leq 2.8 \cdot 10^{-99}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b \cdot \frac{c}{d} - a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b - \frac{d}{\frac{\mathsf{hypot}\left(c, d\right)}{a}}\right)\\ \end{array} \]

Alternative 6
Accuracy	82.0%
Cost	2000

\[\begin{array}{l} t_0 := c \cdot b - d \cdot a\\ t_1 := c \cdot c + d \cdot d\\ \mathbf{if}\;c \leq -1.9 \cdot 10^{+63}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq -2.1 \cdot 10^{-129}:\\ \;\;\;\;\frac{t_0}{t_1}\\ \mathbf{elif}\;c \leq 2.1 \cdot 10^{-104}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b \cdot \frac{c}{d} - a\right)\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{+123}:\\ \;\;\;\;\frac{d \cdot a + \left(t_0 - d \cdot a\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \end{array} \]

Alternative 7
Accuracy	73.6%
Cost	1497

\[\begin{array}{l} t_0 := \frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{if}\;d \leq -1.18 \cdot 10^{+173}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{+130}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;d \leq -7.5 \cdot 10^{+14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 2700000000000:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{+97} \lor \neg \left(d \leq 1.15 \cdot 10^{+114}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 8
Accuracy	73.3%
Cost	1496

\[\begin{array}{l} t_0 := \frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{if}\;d \leq -1.18 \cdot 10^{+173}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{+130}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;d \leq -9.5 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 2000000000000:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 9.8 \cdot 10^{+88}:\\ \;\;\;\;\frac{-a}{\frac{c \cdot c + d \cdot d}{d}}\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+114}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Accuracy	82.0%
Cost	1488

\[\begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -2.7 \cdot 10^{+63}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq -5.3 \cdot 10^{-129}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 9.8 \cdot 10^{-104}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b \cdot \frac{c}{d} - a\right)\\ \mathbf{elif}\;c \leq 5.7 \cdot 10^{+128}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \end{array} \]

Alternative 10
Accuracy	68.2%
Cost	1106

\[\begin{array}{l} \mathbf{if}\;d \leq -1.18 \cdot 10^{+173} \lor \neg \left(d \leq -1.7 \cdot 10^{+130}\right) \land \left(d \leq -135000000000 \lor \neg \left(d \leq 102000000000\right)\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]

Alternative 11
Accuracy	68.9%
Cost	1105

\[\begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -1.55 \cdot 10^{+173}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.7 \cdot 10^{+130}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;d \leq -1.25 \cdot 10^{+17} \lor \neg \left(d \leq 2.5 \cdot 10^{+15}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \]

Alternative 12
Accuracy	62.8%
Cost	520

\[\begin{array}{l} \mathbf{if}\;c \leq -1.9 \cdot 10^{-7}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.12 \cdot 10^{-61}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 13
Accuracy	44.9%
Cost	456

\[\begin{array}{l} \mathbf{if}\;d \leq -3.25 \cdot 10^{+184}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{+219}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \]

Alternative 14
Accuracy	10.8%
Cost	192

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

Reproduce

herbie shell --seed 2023131 
(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))))