Complex division, imag part

Average Accuracy: 59.2% → 79.9%

Time: 14.9s

Precision: binary64

Cost: 14024

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{+131}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \mathbf{elif}\;d \leq -1.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq 20000000000:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{\frac{d \cdot a}{c}}{c}, \frac{b}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} \cdot \frac{c}{d} - \frac{a}{d}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c d)
 :precision binary64
 (if (<= d -2.3e+131)
   (/ (- (/ c (/ d b)) a) d)
   (if (<= d -1.4e-48)
     (* (/ 1.0 (hypot c d)) (/ (- (* c b) (* d a)) (hypot c d)))
     (if (<= d 20000000000.0)
       (fma -1.0 (/ (/ (* d a) c) c) (/ b c))
       (- (* (/ b d) (/ c d)) (/ a 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 tmp;
	if (d <= -2.3e+131) {
		tmp = ((c / (d / b)) - a) / d;
	} else if (d <= -1.4e-48) {
		tmp = (1.0 / hypot(c, d)) * (((c * b) - (d * a)) / hypot(c, d));
	} else if (d <= 20000000000.0) {
		tmp = fma(-1.0, (((d * a) / c) / c), (b / c));
	} else {
		tmp = ((b / d) * (c / d)) - (a / 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)
	tmp = 0.0
	if (d <= -2.3e+131)
		tmp = Float64(Float64(Float64(c / Float64(d / b)) - a) / d);
	elseif (d <= -1.4e-48)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(Float64(Float64(c * b) - Float64(d * a)) / hypot(c, d)));
	elseif (d <= 20000000000.0)
		tmp = fma(-1.0, Float64(Float64(Float64(d * a) / c) / c), Float64(b / c));
	else
		tmp = Float64(Float64(Float64(b / d) * Float64(c / d)) - Float64(a / 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_] := If[LessEqual[d, -2.3e+131], N[(N[(N[(c / N[(d / b), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -1.4e-48], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 20000000000.0], N[(-1.0 * N[(N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b / d), $MachinePrecision] * N[(c / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	59.2%
Target	99.3%
Herbie	79.9%

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

Derivation

1. Split input into 4 regimes

2. if d < -2.29999999999999992e131

   1. Initial program 33.2%

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

   2. Taylor expanded in c around 0 76.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{c \cdot b}{{d}^{2}}} \]

   3. Simplified 87.3%

      \[\leadsto \color{blue}{\frac{b}{d} \cdot \frac{c}{d} - \frac{a}{d}} \]
      Proof

      [Start] 76.3	\[ -1 \cdot \frac{a}{d} + \frac{c \cdot b}{{d}^{2}} \]
      +-commutative [=>] 76.3	\[ \color{blue}{\frac{c \cdot b}{{d}^{2}} + -1 \cdot \frac{a}{d}} \]
      mul-1-neg [=>] 76.3	\[ \frac{c \cdot b}{{d}^{2}} + \color{blue}{\left(-\frac{a}{d}\right)} \]
      unsub-neg [=>] 76.3	\[ \color{blue}{\frac{c \cdot b}{{d}^{2}} - \frac{a}{d}} \]
      *-commutative [<=] 76.3	\[ \frac{\color{blue}{b \cdot c}}{{d}^{2}} - \frac{a}{d} \]
      unpow2 [=>] 76.3	\[ \frac{b \cdot c}{\color{blue}{d \cdot d}} - \frac{a}{d} \]
      times-frac [=>] 87.3	\[ \color{blue}{\frac{b}{d} \cdot \frac{c}{d}} - \frac{a}{d} \]

   4. Taylor expanded in b around 0 76.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{c \cdot b}{{d}^{2}}} \]

   5. Simplified 87.5%

      \[\leadsto \color{blue}{\frac{\frac{c}{\frac{d}{b}} - a}{d}} \]
      Proof

      [Start] 76.3	\[ -1 \cdot \frac{a}{d} + \frac{c \cdot b}{{d}^{2}} \]
      +-commutative [=>] 76.3	\[ \color{blue}{\frac{c \cdot b}{{d}^{2}} + -1 \cdot \frac{a}{d}} \]
      *-commutative [=>] 76.3	\[ \frac{\color{blue}{b \cdot c}}{{d}^{2}} + -1 \cdot \frac{a}{d} \]
      unpow2 [=>] 76.3	\[ \frac{b \cdot c}{\color{blue}{d \cdot d}} + -1 \cdot \frac{a}{d} \]
      associate-/l* [=>] 77.9	\[ \color{blue}{\frac{b}{\frac{d \cdot d}{c}}} + -1 \cdot \frac{a}{d} \]
      mul-1-neg [=>] 77.9	\[ \frac{b}{\frac{d \cdot d}{c}} + \color{blue}{\left(-\frac{a}{d}\right)} \]
      sub-neg [<=] 77.9	\[ \color{blue}{\frac{b}{\frac{d \cdot d}{c}} - \frac{a}{d}} \]
      associate-/l* [<=] 76.3	\[ \color{blue}{\frac{b \cdot c}{d \cdot d}} - \frac{a}{d} \]
      associate-/r* [=>] 80.4	\[ \color{blue}{\frac{\frac{b \cdot c}{d}}{d}} - \frac{a}{d} \]
      div-sub [<=] 80.4	\[ \color{blue}{\frac{\frac{b \cdot c}{d} - a}{d}} \]
      *-commutative [<=] 80.4	\[ \frac{\frac{\color{blue}{c \cdot b}}{d} - a}{d} \]
      associate-/l* [=>] 87.5	\[ \frac{\color{blue}{\frac{c}{\frac{d}{b}}} - a}{d} \]

   if -2.29999999999999992e131 < d < -1.40000000000000002e-48

   1. Initial program 73.0%

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

   2. Applied egg-rr 80.3%

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

      [Start] 73.0	\[ \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
      *-un-lft-identity [=>] 73.0	\[ \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{c \cdot c + d \cdot d} \]
      add-sqr-sqrt [=>] 73.0	\[ \frac{1 \cdot \left(b \cdot c - a \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]
      times-frac [=>] 73.0	\[ \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]
      hypot-def [=>] 73.0	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]
      hypot-def [=>] 80.3	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - a \cdot d}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   if -1.40000000000000002e-48 < d < 2e10

   1. Initial program 69.5%

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

   2. Applied egg-rr 82.4%

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

      [Start] 69.5	\[ \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
      *-un-lft-identity [=>] 69.5	\[ \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{c \cdot c + d \cdot d} \]
      add-sqr-sqrt [=>] 69.5	\[ \frac{1 \cdot \left(b \cdot c - a \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]
      times-frac [=>] 69.5	\[ \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]
      hypot-def [=>] 69.5	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]
      hypot-def [=>] 82.4	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - a \cdot d}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   3. Taylor expanded in c around inf 74.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c}} \]

   4. Simplified 78.5%

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

      [Start] 74.3	\[ -1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c} \]
      fma-def [=>] 74.3	\[ \color{blue}{\mathsf{fma}\left(-1, \frac{a \cdot d}{{c}^{2}}, \frac{b}{c}\right)} \]
      unpow2 [=>] 74.3	\[ \mathsf{fma}\left(-1, \frac{a \cdot d}{\color{blue}{c \cdot c}}, \frac{b}{c}\right) \]
      associate-/r* [=>] 78.7	\[ \mathsf{fma}\left(-1, \color{blue}{\frac{\frac{a \cdot d}{c}}{c}}, \frac{b}{c}\right) \]
      associate-/l* [=>] 78.5	\[ \mathsf{fma}\left(-1, \frac{\color{blue}{\frac{a}{\frac{c}{d}}}}{c}, \frac{b}{c}\right) \]

   5. Taylor expanded in a around 0 78.7%

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

   if 2e10 < d

   1. Initial program 47.9%

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

   2. Taylor expanded in c around 0 71.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{c \cdot b}{{d}^{2}}} \]

   3. Simplified 77.2%

      \[\leadsto \color{blue}{\frac{b}{d} \cdot \frac{c}{d} - \frac{a}{d}} \]
      Proof

      [Start] 71.0	\[ -1 \cdot \frac{a}{d} + \frac{c \cdot b}{{d}^{2}} \]
      +-commutative [=>] 71.0	\[ \color{blue}{\frac{c \cdot b}{{d}^{2}} + -1 \cdot \frac{a}{d}} \]
      mul-1-neg [=>] 71.0	\[ \frac{c \cdot b}{{d}^{2}} + \color{blue}{\left(-\frac{a}{d}\right)} \]
      unsub-neg [=>] 71.0	\[ \color{blue}{\frac{c \cdot b}{{d}^{2}} - \frac{a}{d}} \]
      *-commutative [<=] 71.0	\[ \frac{\color{blue}{b \cdot c}}{{d}^{2}} - \frac{a}{d} \]
      unpow2 [=>] 71.0	\[ \frac{b \cdot c}{\color{blue}{d \cdot d}} - \frac{a}{d} \]
      times-frac [=>] 77.2	\[ \color{blue}{\frac{b}{d} \cdot \frac{c}{d}} - \frac{a}{d} \]

3. Recombined 4 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{+131}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \mathbf{elif}\;d \leq -1.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq 20000000000:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{\frac{d \cdot a}{c}}{c}, \frac{b}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} \cdot \frac{c}{d} - \frac{a}{d}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	79.3%
Cost	7500

\[\begin{array}{l} t_0 := d \cdot d + c \cdot c\\ \mathbf{if}\;d \leq -1.7 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \mathbf{elif}\;d \leq -1.05 \cdot 10^{-154}:\\ \;\;\;\;\frac{c}{\frac{t_0}{b}} - \frac{a}{\frac{t_0}{d}}\\ \mathbf{elif}\;d \leq 78000:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{\frac{a}{\frac{c}{d}}}{c}, \frac{b}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} \cdot \frac{c}{d} - \frac{a}{d}\\ \end{array} \]

Alternative 2
Accuracy	79.4%
Cost	7500

\[\begin{array}{l} t_0 := d \cdot d + c \cdot c\\ \mathbf{if}\;d \leq -1.7 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \mathbf{elif}\;d \leq -2.05 \cdot 10^{-155}:\\ \;\;\;\;\frac{c}{\frac{t_0}{b}} - \frac{a}{\frac{t_0}{d}}\\ \mathbf{elif}\;d \leq 185000000000:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{\frac{d \cdot a}{c}}{c}, \frac{b}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} \cdot \frac{c}{d} - \frac{a}{d}\\ \end{array} \]

Alternative 3
Accuracy	78.6%
Cost	1736

\[\begin{array}{l} t_0 := d \cdot d + c \cdot c\\ \mathbf{if}\;d \leq -1.7 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \mathbf{elif}\;d \leq -2.15 \cdot 10^{-156}:\\ \;\;\;\;\frac{c}{\frac{t_0}{b}} - \frac{a}{\frac{t_0}{d}}\\ \mathbf{elif}\;d \leq 38000000000:\\ \;\;\;\;\frac{1}{c} \cdot \left(b - d \cdot \frac{a}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} \cdot \frac{c}{d} - \frac{a}{d}\\ \end{array} \]

Alternative 4
Accuracy	78.0%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;d \leq -2.25 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \mathbf{elif}\;d \leq -1.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;d \leq 950000:\\ \;\;\;\;\frac{1}{c} \cdot \left(b - d \cdot \frac{a}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} \cdot \frac{c}{d} - \frac{a}{d}\\ \end{array} \]

Alternative 5
Accuracy	70.7%
Cost	968

\[\begin{array}{l} \mathbf{if}\;d \leq -3.5 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \mathbf{elif}\;d \leq 13200000:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} \cdot \frac{c}{d} - \frac{a}{d}\\ \end{array} \]

Alternative 6
Accuracy	75.7%
Cost	968

\[\begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \mathbf{elif}\;d \leq 550:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} \cdot \frac{c}{d} - \frac{a}{d}\\ \end{array} \]

Alternative 7
Accuracy	75.9%
Cost	968

\[\begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \mathbf{elif}\;d \leq 2000000:\\ \;\;\;\;\frac{1}{c} \cdot \left(b - d \cdot \frac{a}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} \cdot \frac{c}{d} - \frac{a}{d}\\ \end{array} \]

Alternative 8
Accuracy	61.3%
Cost	844

\[\begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;c \leq -3.4 \cdot 10^{-40}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -2 \cdot 10^{-82}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -6.5 \cdot 10^{-188}:\\ \;\;\;\;\frac{b}{d} \cdot \frac{c}{d}\\ \mathbf{elif}\;c \leq 9.2 \cdot 10^{+32}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 9
Accuracy	70.8%
Cost	841

\[\begin{array}{l} \mathbf{if}\;d \leq -4.3 \cdot 10^{-27} \lor \neg \left(d \leq 92000\right):\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 10
Accuracy	63.9%
Cost	521

\[\begin{array}{l} \mathbf{if}\;d \leq -1.52 \cdot 10^{-46} \lor \neg \left(d \leq 1700\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 11
Accuracy	41.2%
Cost	192

\[\frac{b}{c} \]

Reproduce

herbie shell --seed 2023135 
(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))))