Complex division, imag part

Percentage Accurate: 62.1% → 89.3%

Time: 8.7s

Precision: binary64

Cost: 22025

Math

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

↓

\[\begin{array}{l} t_0 := b \cdot c - a \cdot d\\ t_1 := \frac{t_0}{c \cdot c + d \cdot d}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 10^{+281}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* b c) (* a d))) (t_1 (/ t_0 (+ (* c c) (* d d)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 1e+281)))
     (fma (/ c (hypot c d)) (/ b (hypot c d)) (/ (- a) d))
     (* (/ 1.0 (hypot c d)) (/ t_0 (hypot c d))))))

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) {
	double t_0 = (b * c) - (a * d);
	double t_1 = t_0 / ((c * c) + (d * d));
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 1e+281)) {
		tmp = fma((c / hypot(c, d)), (b / hypot(c, d)), (-a / d));
	} else {
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	}
	return tmp;
}

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)
	t_0 = Float64(Float64(b * c) - Float64(a * d))
	t_1 = Float64(t_0 / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 1e+281))
		tmp = fma(Float64(c / hypot(c, d)), Float64(b / hypot(c, d)), Float64(Float64(-a) / d));
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(t_0 / hypot(c, d)));
	end
	return tmp
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_] := Block[{t$95$0 = N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 1e+281]], $MachinePrecision]], N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] + N[((-a) / d), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	62.1%
Target	99.4%
Herbie	89.3%

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < -inf.0 or 1e281 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 17.4%

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

   2. Applied egg-rr 56.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{a}{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}\right)} \]
      Step-by-step derivation

      [Start] 17.4%	\[ \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
      div-sub [=>] 7.3%	\[ \color{blue}{\frac{b \cdot c}{c \cdot c + d \cdot d} - \frac{a \cdot d}{c \cdot c + d \cdot d}} \]
      *-commutative [=>] 7.3%	\[ \frac{\color{blue}{c \cdot b}}{c \cdot c + d \cdot d} - \frac{a \cdot d}{c \cdot c + d \cdot d} \]
      add-sqr-sqrt [=>] 7.3%	\[ \frac{c \cdot b}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} - \frac{a \cdot d}{c \cdot c + d \cdot d} \]
      times-frac [=>] 15.9%	\[ \color{blue}{\frac{c}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b}{\sqrt{c \cdot c + d \cdot d}}} - \frac{a \cdot d}{c \cdot c + d \cdot d} \]
      fma-neg [=>] 15.9%	\[ \color{blue}{\mathsf{fma}\left(\frac{c}{\sqrt{c \cdot c + d \cdot d}}, \frac{b}{\sqrt{c \cdot c + d \cdot d}}, -\frac{a \cdot d}{c \cdot c + d \cdot d}\right)} \]
      hypot-def [=>] 15.9%	\[ \mathsf{fma}\left(\frac{c}{\color{blue}{\mathsf{hypot}\left(c, d\right)}}, \frac{b}{\sqrt{c \cdot c + d \cdot d}}, -\frac{a \cdot d}{c \cdot c + d \cdot d}\right) \]
      hypot-def [=>] 45.1%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\color{blue}{\mathsf{hypot}\left(c, d\right)}}, -\frac{a \cdot d}{c \cdot c + d \cdot d}\right) \]
      associate-/l* [=>] 56.7%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\color{blue}{\frac{a}{\frac{c \cdot c + d \cdot d}{d}}}\right) \]
      add-sqr-sqrt [=>] 56.7%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{a}{\frac{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}{d}}\right) \]
      pow2 [=>] 56.7%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{a}{\frac{\color{blue}{{\left(\sqrt{c \cdot c + d \cdot d}\right)}^{2}}}{d}}\right) \]
      hypot-def [=>] 56.7%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{a}{\frac{{\color{blue}{\left(\mathsf{hypot}\left(c, d\right)\right)}}^{2}}{d}}\right) \]

   3. Taylor expanded in c around 0 80.0%

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

   if -inf.0 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 1e281

   1. Initial program 83.7%

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

   2. Applied egg-rr 98.1%

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

      [Start] 83.7%	\[ \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
      *-un-lft-identity [=>] 83.7%	\[ \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{c \cdot c + d \cdot d} \]
      add-sqr-sqrt [=>] 83.7%	\[ \frac{1 \cdot \left(b \cdot c - a \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]
      times-frac [=>] 83.7%	\[ \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]
      hypot-def [=>] 83.7%	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]
      hypot-def [=>] 98.1%	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - a \cdot d}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq -\infty \lor \neg \left(\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq 10^{+281}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	89.3%
Cost	22025

\[\begin{array}{l} t_0 := b \cdot c - a \cdot d\\ t_1 := \frac{t_0}{c \cdot c + d \cdot d}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 10^{+281}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 2
Accuracy	84.8%
Cost	14788

\[\begin{array}{l} t_0 := b \cdot c - a \cdot d\\ \mathbf{if}\;\frac{t_0}{c \cdot c + d \cdot d} \leq 10^{+281}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{c}{\frac{d}{b}} - a\right)\\ \end{array} \]

Alternative 3
Accuracy	82.5%
Cost	7824

\[\begin{array}{l} t_0 := c \cdot c + d \cdot d\\ \mathbf{if}\;c \leq -5.5 \cdot 10^{+42}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \mathbf{elif}\;c \leq -6.5 \cdot 10^{-97}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{t_0}\\ \mathbf{elif}\;c \leq 8.5 \cdot 10^{-137}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{c}{\frac{d}{b}} - a\right)\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{+66}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, a, b \cdot c\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]

Alternative 4
Accuracy	82.6%
Cost	1488

\[\begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -4.8 \cdot 10^{+42}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \mathbf{elif}\;c \leq -2.8 \cdot 10^{-98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.05 \cdot 10^{-136}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{c}{\frac{d}{b}} - a\right)\\ \mathbf{elif}\;c \leq 5.8 \cdot 10^{+69}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]

Alternative 5
Accuracy	76.4%
Cost	969

\[\begin{array}{l} \mathbf{if}\;c \leq -10500 \lor \neg \left(c \leq 9.7 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \end{array} \]

Alternative 6
Accuracy	77.1%
Cost	969

\[\begin{array}{l} \mathbf{if}\;c \leq -6200000000 \lor \neg \left(c \leq 2.75 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{c}{\frac{d}{b}} - a\right)\\ \end{array} \]

Alternative 7
Accuracy	69.8%
Cost	841

\[\begin{array}{l} \mathbf{if}\;c \leq -8.5 \cdot 10^{-14} \lor \neg \left(c \leq 3.1 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]

Alternative 8
Accuracy	62.6%
Cost	520

\[\begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{-15}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 5.4 \cdot 10^{-68}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 9
Accuracy	46.0%
Cost	456

\[\begin{array}{l} \mathbf{if}\;d \leq -3.6 \cdot 10^{+90}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{+191}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \]

Alternative 10
Accuracy	11.2%
Cost	192

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

Reproduce

herbie shell --seed 2023178 
(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))))