Complex division, imag part

Average Error: 26.9 → 11.9

Time: 13.8s

Precision: binary64

Cost: 14420

Math

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

↓

\[\begin{array}{l} t_0 := \frac{b}{d} \cdot \frac{c}{d} - \frac{a}{d}\\ t_1 := \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;d \leq -3.3 \cdot 10^{+33}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-59}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{elif}\;d \leq -1.8 \cdot 10^{-197}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 6.9 \cdot 10^{-152}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{+137}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 d) (/ c d)) (/ a d)))
        (t_1 (* (/ 1.0 (hypot c d)) (/ (- (* b c) (* d a)) (hypot c d)))))
   (if (<= d -3.3e+33)
     t_0
     (if (<= d -2e-59)
       (- (/ b c) (* (/ d c) (/ a c)))
       (if (<= d -1.8e-197)
         t_1
         (if (<= d 6.9e-152)
           (/ (- b (* d (/ a c))) c)
           (if (<= d 2.4e+137) t_1 t_0)))))))

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 / d) * (c / d)) - (a / d);
	double t_1 = (1.0 / hypot(c, d)) * (((b * c) - (d * a)) / hypot(c, d));
	double tmp;
	if (d <= -3.3e+33) {
		tmp = t_0;
	} else if (d <= -2e-59) {
		tmp = (b / c) - ((d / c) * (a / c));
	} else if (d <= -1.8e-197) {
		tmp = t_1;
	} else if (d <= 6.9e-152) {
		tmp = (b - (d * (a / c))) / c;
	} else if (d <= 2.4e+137) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

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) {
	double t_0 = ((b / d) * (c / d)) - (a / d);
	double t_1 = (1.0 / Math.hypot(c, d)) * (((b * c) - (d * a)) / Math.hypot(c, d));
	double tmp;
	if (d <= -3.3e+33) {
		tmp = t_0;
	} else if (d <= -2e-59) {
		tmp = (b / c) - ((d / c) * (a / c));
	} else if (d <= -1.8e-197) {
		tmp = t_1;
	} else if (d <= 6.9e-152) {
		tmp = (b - (d * (a / c))) / c;
	} else if (d <= 2.4e+137) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(a, b, c, d):
	t_0 = ((b / d) * (c / d)) - (a / d)
	t_1 = (1.0 / math.hypot(c, d)) * (((b * c) - (d * a)) / math.hypot(c, d))
	tmp = 0
	if d <= -3.3e+33:
		tmp = t_0
	elif d <= -2e-59:
		tmp = (b / c) - ((d / c) * (a / c))
	elif d <= -1.8e-197:
		tmp = t_1
	elif d <= 6.9e-152:
		tmp = (b - (d * (a / c))) / c
	elif d <= 2.4e+137:
		tmp = t_1
	else:
		tmp = t_0
	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(Float64(b / d) * Float64(c / d)) - Float64(a / d))
	t_1 = Float64(Float64(1.0 / hypot(c, d)) * Float64(Float64(Float64(b * c) - Float64(d * a)) / hypot(c, d)))
	tmp = 0.0
	if (d <= -3.3e+33)
		tmp = t_0;
	elseif (d <= -2e-59)
		tmp = Float64(Float64(b / c) - Float64(Float64(d / c) * Float64(a / c)));
	elseif (d <= -1.8e-197)
		tmp = t_1;
	elseif (d <= 6.9e-152)
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	elseif (d <= 2.4e+137)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(a, b, c, d)
	t_0 = ((b / d) * (c / d)) - (a / d);
	t_1 = (1.0 / hypot(c, d)) * (((b * c) - (d * a)) / hypot(c, d));
	tmp = 0.0;
	if (d <= -3.3e+33)
		tmp = t_0;
	elseif (d <= -2e-59)
		tmp = (b / c) - ((d / c) * (a / c));
	elseif (d <= -1.8e-197)
		tmp = t_1;
	elseif (d <= 6.9e-152)
		tmp = (b - (d * (a / c))) / c;
	elseif (d <= 2.4e+137)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = 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[(N[(b / d), $MachinePrecision] * N[(c / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.3e+33], t$95$0, If[LessEqual[d, -2e-59], N[(N[(b / c), $MachinePrecision] - N[(N[(d / c), $MachinePrecision] * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.8e-197], t$95$1, If[LessEqual[d, 6.9e-152], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2.4e+137], t$95$1, t$95$0]]]]]]]

Target

Original	26.9
Target	0.5
Herbie	11.9

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

Derivation

1. Split input into 4 regimes

2. if d < -3.29999999999999976e33 or 2.39999999999999983e137 < d

   1. Initial program 38.9

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

   2. Simplified 38.9

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

      [Start] 38.9	\[ \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
      fma-def [=>] 38.9	\[ \frac{b \cdot c - a \cdot d}{\color{blue}{\mathsf{fma}\left(c, c, d \cdot d\right)}} \]

   3. Taylor expanded in c around 0 16.8

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

   4. Simplified 11.1

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

      [Start] 16.8	\[ -1 \cdot \frac{a}{d} + \frac{c \cdot b}{{d}^{2}} \]
      +-commutative [=>] 16.8	\[ \color{blue}{\frac{c \cdot b}{{d}^{2}} + -1 \cdot \frac{a}{d}} \]
      mul-1-neg [=>] 16.8	\[ \frac{c \cdot b}{{d}^{2}} + \color{blue}{\left(-\frac{a}{d}\right)} \]
      unsub-neg [=>] 16.8	\[ \color{blue}{\frac{c \cdot b}{{d}^{2}} - \frac{a}{d}} \]
      *-commutative [<=] 16.8	\[ \frac{\color{blue}{b \cdot c}}{{d}^{2}} - \frac{a}{d} \]
      unpow2 [=>] 16.8	\[ \frac{b \cdot c}{\color{blue}{d \cdot d}} - \frac{a}{d} \]
      times-frac [=>] 11.1	\[ \color{blue}{\frac{b}{d} \cdot \frac{c}{d}} - \frac{a}{d} \]

   if -3.29999999999999976e33 < d < -2.0000000000000001e-59

   1. Initial program 13.4

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

   2. Taylor expanded in c around inf 32.1

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

   3. Simplified 30.1

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

      [Start] 32.1	\[ -1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c} \]
      +-commutative [=>] 32.1	\[ \color{blue}{\frac{b}{c} + -1 \cdot \frac{a \cdot d}{{c}^{2}}} \]
      mul-1-neg [=>] 32.1	\[ \frac{b}{c} + \color{blue}{\left(-\frac{a \cdot d}{{c}^{2}}\right)} \]
      unsub-neg [=>] 32.1	\[ \color{blue}{\frac{b}{c} - \frac{a \cdot d}{{c}^{2}}} \]
      *-commutative [=>] 32.1	\[ \frac{b}{c} - \frac{\color{blue}{d \cdot a}}{{c}^{2}} \]
      unpow2 [=>] 32.1	\[ \frac{b}{c} - \frac{d \cdot a}{\color{blue}{c \cdot c}} \]
      times-frac [=>] 30.1	\[ \frac{b}{c} - \color{blue}{\frac{d}{c} \cdot \frac{a}{c}} \]

   if -2.0000000000000001e-59 < d < -1.7999999999999999e-197 or 6.90000000000000039e-152 < d < 2.39999999999999983e137

   1. Initial program 17.2

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

   2. Applied egg-rr 11.6

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

   if -1.7999999999999999e-197 < d < 6.90000000000000039e-152

   1. Initial program 24.8

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

   2. Taylor expanded in c around inf 25.0

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

   3. Simplified 25.0

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

      [Start] 25.0	\[ \frac{b \cdot c - a \cdot d}{{c}^{2}} \]
      unpow2 [=>] 25.0	\[ \frac{b \cdot c - a \cdot d}{\color{blue}{c \cdot c}} \]

   4. Applied egg-rr 46.4

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(b \cdot c - a \cdot d\right) \cdot {c}^{-2}\right)} - 1} \]

   5. Simplified 6.4

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

      [Start] 46.4	\[ e^{\mathsf{log1p}\left(\left(b \cdot c - a \cdot d\right) \cdot {c}^{-2}\right)} - 1 \]
      expm1-def [=>] 34.1	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(b \cdot c - a \cdot d\right) \cdot {c}^{-2}\right)\right)} \]
      expm1-log1p [=>] 25.2	\[ \color{blue}{\left(b \cdot c - a \cdot d\right) \cdot {c}^{-2}} \]
      sqr-pow [=>] 25.3	\[ \left(b \cdot c - a \cdot d\right) \cdot \color{blue}{\left({c}^{\left(\frac{-2}{2}\right)} \cdot {c}^{\left(\frac{-2}{2}\right)}\right)} \]
      associate-*r* [=>] 16.5	\[ \color{blue}{\left(\left(b \cdot c - a \cdot d\right) \cdot {c}^{\left(\frac{-2}{2}\right)}\right) \cdot {c}^{\left(\frac{-2}{2}\right)}} \]
      metadata-eval [=>] 16.5	\[ \left(\left(b \cdot c - a \cdot d\right) \cdot {c}^{\color{blue}{-1}}\right) \cdot {c}^{\left(\frac{-2}{2}\right)} \]
      unpow-1 [=>] 16.5	\[ \left(\left(b \cdot c - a \cdot d\right) \cdot \color{blue}{\frac{1}{c}}\right) \cdot {c}^{\left(\frac{-2}{2}\right)} \]
      associate-*r/ [=>] 16.4	\[ \color{blue}{\frac{\left(b \cdot c - a \cdot d\right) \cdot 1}{c}} \cdot {c}^{\left(\frac{-2}{2}\right)} \]
      associate-*l/ [<=] 16.4	\[ \color{blue}{\left(\frac{b \cdot c - a \cdot d}{c} \cdot 1\right)} \cdot {c}^{\left(\frac{-2}{2}\right)} \]
      *-rgt-identity [=>] 16.4	\[ \color{blue}{\frac{b \cdot c - a \cdot d}{c}} \cdot {c}^{\left(\frac{-2}{2}\right)} \]
      metadata-eval [=>] 16.4	\[ \frac{b \cdot c - a \cdot d}{c} \cdot {c}^{\color{blue}{-1}} \]
      unpow-1 [=>] 16.4	\[ \frac{b \cdot c - a \cdot d}{c} \cdot \color{blue}{\frac{1}{c}} \]
      associate-*r/ [=>] 16.3	\[ \color{blue}{\frac{\frac{b \cdot c - a \cdot d}{c} \cdot 1}{c}} \]
      *-commutative [<=] 16.3	\[ \frac{\color{blue}{1 \cdot \frac{b \cdot c - a \cdot d}{c}}}{c} \]
      *-lft-identity [=>] 16.3	\[ \frac{\color{blue}{\frac{b \cdot c - a \cdot d}{c}}}{c} \]

3. Recombined 4 regimes into one program.

4. Final simplification 11.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.3 \cdot 10^{+33}:\\ \;\;\;\;\frac{b}{d} \cdot \frac{c}{d} - \frac{a}{d}\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-59}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{elif}\;d \leq -1.8 \cdot 10^{-197}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq 6.9 \cdot 10^{-152}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{+137}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} \cdot \frac{c}{d} - \frac{a}{d}\\ \end{array} \]

Alternatives

Alternative 1
Error	13.8
Cost	13768

\[\begin{array}{l} t_0 := \frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{if}\;c \leq -9 \cdot 10^{+23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -0.000155:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(d, c\right) \cdot \frac{-1}{\frac{d}{\mathsf{hypot}\left(d, c\right)}}}\\ \mathbf{elif}\;c \leq -1.65 \cdot 10^{-29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 4.2 \cdot 10^{-122}:\\ \;\;\;\;\frac{b}{d} \cdot \frac{c}{d} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 10^{+90}:\\ \;\;\;\;\frac{\left(b \cdot c + \mathsf{fma}\left(-d, a, d \cdot a\right)\right) - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	13.6
Cost	7628

\[\begin{array}{l} t_0 := \frac{b}{d} \cdot \frac{c}{d} - \frac{a}{d}\\ \mathbf{if}\;d \leq -4.5 \cdot 10^{+30}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 3.9 \cdot 10^{-149}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{+137}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	13.6
Cost	1356

\[\begin{array}{l} t_0 := \frac{b}{d} \cdot \frac{c}{d} - \frac{a}{d}\\ \mathbf{if}\;d \leq -9.3 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{-149}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{+137}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	15.1
Cost	969

\[\begin{array}{l} \mathbf{if}\;d \leq -2.5 \cdot 10^{+27} \lor \neg \left(d \leq 0.0024\right):\\ \;\;\;\;\frac{b}{d} \cdot \frac{c}{d} - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]

Alternative 5
Error	23.6
Cost	908

\[\begin{array}{l} t_0 := -\frac{a}{d}\\ \mathbf{if}\;d \leq -1.3 \cdot 10^{+40}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 3.05 \cdot 10^{-89}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 680000:\\ \;\;\;\;\frac{d}{c} \cdot \left(-\frac{a}{c}\right)\\ \mathbf{elif}\;d \leq 10^{+36}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	18.7
Cost	841

\[\begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{+40} \lor \neg \left(d \leq 7.8 \cdot 10^{+34}\right):\\ \;\;\;\;-\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]

Alternative 7
Error	15.3
Cost	841

\[\begin{array}{l} \mathbf{if}\;d \leq -1.4 \cdot 10^{+30} \lor \neg \left(d \leq 0.00041\right):\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]

Alternative 8
Error	22.6
Cost	521

\[\begin{array}{l} \mathbf{if}\;d \leq -1.52 \cdot 10^{+38} \lor \neg \left(d \leq 1.9 \cdot 10^{-27}\right):\\ \;\;\;\;-\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 9
Error	37.9
Cost	192

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

Reproduce

herbie shell --seed 2023060 
(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))))