Complex division, real part

?

Percentage Accurate: 61.1% → 84.9%
Time: 12.0s
Precision: binary64
Cost: 20932

?

\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 10^{+297}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \]
(FPCore (a b c d)
 :precision binary64
 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))
↓
(FPCore (a b c d)
 :precision binary64
 (if (<= (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) 1e+297)
   (/ (/ (fma a c (* b d)) (hypot c d)) (hypot c d))
   (+ (/ a c) (* (/ d c) (/ b 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) {
	double tmp;
	if ((((a * c) + (b * d)) / ((c * c) + (d * d))) <= 1e+297) {
		tmp = (fma(a, c, (b * d)) / hypot(c, d)) / hypot(c, d);
	} else {
		tmp = (a / c) + ((d / c) * (b / c));
	}
	return tmp;
}

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)
	tmp = 0.0
	if (Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d))) <= 1e+297)
		tmp = Float64(Float64(fma(a, c, Float64(b * d)) / hypot(c, d)) / hypot(c, d));
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	end
	return tmp
end

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_] := If[LessEqual[N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+297], N[(N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\begin{array}{l}
\mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 10^{+297}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\


\end{array}

Local Percentage Accuracy?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 12 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each of Herbie's proposed alternatives. Up and to the right is better. Each dot represents an alternative program; the red square represents the initial program.

Bogosity?

Bogosity

Target

Original61.1%
Target99.4%
Herbie84.9%
\[\begin{array}{l} \mathbf{if}\;\left|d\right| < \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array} \]

Derivation?

  1. Split input into 2 regimes

  2. if (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 1e297

    1. Initial program 86.8%

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

    2. Applied egg-rr97.9%

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

      [Start]86.8

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

      *-un-lft-identity [=>]86.8

      \[ \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{c \cdot c + d \cdot d} \]

      add-sqr-sqrt [=>]86.8

      \[ \frac{1 \cdot \left(a \cdot c + b \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]

      times-frac [=>]86.9

      \[ \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]

      hypot-def [=>]86.9

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

      fma-def [=>]86.9

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

      hypot-def [=>]97.9

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

    3. Applied egg-rr98.1%

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

      [Start]97.9

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

      associate-*l/ [=>]98.1

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

      *-un-lft-identity [<=]98.1

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

    if 1e297 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

    1. Initial program 10.6%

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

    2. Taylor expanded in c around inf 45.6%

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{d \cdot b}{{c}^{2}}} \]

    3. Simplified62.6%

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}} \]
      Step-by-step derivation

      [Start]45.6

      \[ \frac{a}{c} + \frac{d \cdot b}{{c}^{2}} \]

      unpow2 [=>]45.6

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

      times-frac [=>]62.6

      \[ \frac{a}{c} + \color{blue}{\frac{d}{c} \cdot \frac{b}{c}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification88.1%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 10^{+297}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy83.8%
Cost20304
\[\begin{array}{l} t_0 := \mathsf{fma}\left(d, b, a \cdot c\right) \cdot {\left(\mathsf{hypot}\left(c, d\right)\right)}^{-2}\\ t_1 := \frac{d}{\frac{c}{b}}\\ \mathbf{if}\;c \leq -9.5 \cdot 10^{+113}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, t_1, -a\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -8.8 \cdot 10^{-83}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 7.4 \cdot 10^{-134}:\\ \;\;\;\;\frac{b + \frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;c \leq 9.8 \cdot 10^{+60}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a + t_1}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
Alternative 2
Accuracy83.7%
Cost13636
\[\begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{d}{\frac{c}{b}}\\ \mathbf{if}\;c \leq -1.1 \cdot 10^{+113}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, t_1, -a\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -2.65 \cdot 10^{-83}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 3.1 \cdot 10^{-133}:\\ \;\;\;\;\frac{b + \frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;c \leq 9.5 \cdot 10^{+60}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a + t_1}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
Alternative 3
Accuracy83.5%
Cost7568
\[\begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -2 \cdot 10^{+114}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;c \leq -2.4 \cdot 10^{-83}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{-133}:\\ \;\;\;\;\frac{b + \frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{d}{\frac{c}{b}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
Alternative 4
Accuracy83.1%
Cost1488
\[\begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{if}\;c \leq -1.3 \cdot 10^{+113}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -2.4 \cdot 10^{-83}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{-133}:\\ \;\;\;\;\frac{b + \frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;c \leq 7.8 \cdot 10^{+60}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 5
Accuracy74.1%
Cost969
\[\begin{array}{l} \mathbf{if}\;d \leq -1.2 \cdot 10^{-34} \lor \neg \left(d \leq 3.6 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{b + \frac{c}{\frac{d}{a}}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + d \cdot \frac{b}{c \cdot c}\\ \end{array} \]
Alternative 6
Accuracy76.7%
Cost969
\[\begin{array}{l} \mathbf{if}\;d \leq -1.08 \cdot 10^{-39} \lor \neg \left(d \leq 6.2 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{b + \frac{c}{\frac{d}{a}}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \]
Alternative 7
Accuracy76.2%
Cost968
\[\begin{array}{l} \mathbf{if}\;d \leq -4.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d \cdot \frac{d}{a}}\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{c}{\frac{d}{a}}}{d}\\ \end{array} \]
Alternative 8
Accuracy69.4%
Cost841
\[\begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{-85} \lor \neg \left(d \leq 4 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{b + \frac{c}{\frac{d}{a}}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
Alternative 9
Accuracy63.6%
Cost456
\[\begin{array}{l} \mathbf{if}\;d \leq -1150:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 1.08 \cdot 10^{-16}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
Alternative 10
Accuracy43.9%
Cost324
\[\begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{+166}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
Alternative 11
Accuracy43.2%
Cost192
\[\frac{a}{c} \]

Reproduce?

herbie shell --seed 2023161 
(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))))