Complex division, real part

Average Accuracy: 59.2% → 98.9%

Time: 13.9s

Precision: binary64

Cost: 20352

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double a, double b, double c, double d) {
	return (1.0 / hypot(c, d)) * ((b / (hypot(c, d) / d)) + (a / (hypot(c, d) / c)));
}

Java

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

↓

public static double code(double a, double b, double c, double d) {
	return (1.0 / Math.hypot(c, d)) * ((b / (Math.hypot(c, d) / d)) + (a / (Math.hypot(c, d) / c)));
}

Python

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

↓

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

Julia

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

MATLAB

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

↓

function tmp = code(a, b, c, d)
	tmp = (1.0 / hypot(c, d)) * ((b / (hypot(c, d) / d)) + (a / (hypot(c, d) / c)));
end

Wolfram

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

Target

Original	59.2%
Target	99.4%
Herbie	98.9%

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

Derivation

1. Initial program 59.2%

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

2. Applied egg-rr 74.0%

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

   [Start] 59.2	\[ \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   *-un-lft-identity [=>] 59.2	\[ \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{c \cdot c + d \cdot d} \]
   add-sqr-sqrt [=>] 59.2	\[ \frac{1 \cdot \left(a \cdot c + b \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]
   times-frac [=>] 59.2	\[ \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]
   hypot-def [=>] 59.2	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]
   fma-def [=>] 59.2	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d}} \]
   hypot-def [=>] 74.0	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

3. Applied egg-rr 74.0%

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

   [Start] 74.0	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)} \]
   div-inv [=>] 73.8	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{\mathsf{hypot}\left(c, d\right)}\right)} \]
   *-commutative [<=] 73.8	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \mathsf{fma}\left(a, c, b \cdot d\right)\right)} \]
   fma-udef [=>] 73.8	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(a \cdot c + b \cdot d\right)}\right) \]
   distribute-lft-in [=>] 73.8	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(a \cdot c\right) + \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b \cdot d\right)\right)} \]
   associate-*l/ [=>] 73.9	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\color{blue}{\frac{1 \cdot \left(a \cdot c\right)}{\mathsf{hypot}\left(c, d\right)}} + \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b \cdot d\right)\right) \]
   *-un-lft-identity [<=] 73.9	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{\color{blue}{a \cdot c}}{\mathsf{hypot}\left(c, d\right)} + \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b \cdot d\right)\right) \]
   associate-*l/ [=>] 74.0	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{a \cdot c}{\mathsf{hypot}\left(c, d\right)} + \color{blue}{\frac{1 \cdot \left(b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}\right) \]
   *-un-lft-identity [<=] 74.0	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{a \cdot c}{\mathsf{hypot}\left(c, d\right)} + \frac{\color{blue}{b \cdot d}}{\mathsf{hypot}\left(c, d\right)}\right) \]

4. Simplified 98.9%

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

   [Start] 74.0	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{a \cdot c}{\mathsf{hypot}\left(c, d\right)} + \frac{b \cdot d}{\mathsf{hypot}\left(c, d\right)}\right) \]
   +-commutative [=>] 74.0	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(\frac{b \cdot d}{\mathsf{hypot}\left(c, d\right)} + \frac{a \cdot c}{\mathsf{hypot}\left(c, d\right)}\right)} \]
   associate-/l* [=>] 85.6	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\color{blue}{\frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{d}}} + \frac{a \cdot c}{\mathsf{hypot}\left(c, d\right)}\right) \]
   associate-/l* [=>] 98.9	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{d}} + \color{blue}{\frac{a}{\frac{\mathsf{hypot}\left(c, d\right)}{c}}}\right) \]

5. Final simplification 98.9%

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

Alternatives

Alternative 1
Accuracy	88.8%
Cost	22089

\[\begin{array}{l} t_0 := \frac{c \cdot a + d \cdot b}{c \cdot c + d \cdot d}\\ t_1 := \frac{1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 5 \cdot 10^{+259}\right):\\ \;\;\;\;t_1 \cdot \left(\frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{d}} + a\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 2
Accuracy	77.8%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{+116}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d} \cdot \frac{c}{d}\\ \mathbf{elif}\;d \leq -1.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{c \cdot a + d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 206000000:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{\frac{c}{b}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d \cdot \frac{d}{a}}\\ \end{array} \]

Alternative 3
Accuracy	70.3%
Cost	969

\[\begin{array}{l} \mathbf{if}\;d \leq -6 \cdot 10^{+33} \lor \neg \left(d \leq 64000000000\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b}{\frac{c}{d}}\right)\\ \end{array} \]

Alternative 4
Accuracy	70.0%
Cost	969

\[\begin{array}{l} \mathbf{if}\;d \leq -1.8 \cdot 10^{+32} \lor \neg \left(d \leq 18000000000\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{\frac{c}{b}}}{c}\\ \end{array} \]

Alternative 5
Accuracy	76.3%
Cost	969

\[\begin{array}{l} \mathbf{if}\;d \leq -2.2 \cdot 10^{+19} \lor \neg \left(d \leq 72\right):\\ \;\;\;\;\frac{b}{d} + \frac{a}{d} \cdot \frac{c}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{\frac{c}{b}}}{c}\\ \end{array} \]

Alternative 6
Accuracy	75.8%
Cost	968

\[\begin{array}{l} \mathbf{if}\;d \leq -2.15 \cdot 10^{+19}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d} \cdot \frac{c}{d}\\ \mathbf{elif}\;d \leq 185:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{\frac{c}{b}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d \cdot \frac{d}{a}}\\ \end{array} \]

Alternative 7
Accuracy	63.4%
Cost	844

\[\begin{array}{l} \mathbf{if}\;c \leq -8.6 \cdot 10^{+19}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 1.05 \cdot 10^{-6}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 2.3 \cdot 10^{+34}:\\ \;\;\;\;\frac{a}{d} \cdot \frac{c}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 8
Accuracy	64.0%
Cost	456

\[\begin{array}{l} \mathbf{if}\;c \leq -2.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 4.05 \cdot 10^{+34}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 9
Accuracy	41.8%
Cost	192

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

Reproduce

herbie shell --seed 2023135 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d)))))

  (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))