Complex division, imag part

Average Error: 25.9 → 14.4

Time: 5.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{\mathsf{fma}\left(d, -a, b \cdot c\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;t_0 \leq -5 \cdot 10^{-232}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{\mathsf{fma}\left(d, a, b \cdot c\right)}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{+285}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \end{array} \]

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)) (+ (* c c) (* d d))))
        (t_1 (/ (fma d (- a) (* b c)) (fma c c (* d d)))))
   (if (<= t_0 (- INFINITY))
     (/ (- a) d)
     (if (<= t_0 -5e-232)
       t_1
       (if (<= t_0 0.0)
         (* (/ 1.0 (hypot d c)) (/ (fma d a (* b c)) (hypot d c)))
         (if (<= t_0 4e+285) t_1 (- (/ b c) (* (/ a c) (/ d c)))))))))

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)) / ((c * c) + (d * d));
	double t_1 = fma(d, -a, (b * c)) / fma(c, c, (d * d));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -a / d;
	} else if (t_0 <= -5e-232) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (1.0 / hypot(d, c)) * (fma(d, a, (b * c)) / hypot(d, c));
	} else if (t_0 <= 4e+285) {
		tmp = t_1;
	} else {
		tmp = (b / c) - ((a / c) * (d / c));
	}
	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 * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(fma(d, Float64(-a), Float64(b * c)) / fma(c, c, Float64(d * d)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(-a) / d);
	elseif (t_0 <= -5e-232)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(1.0 / hypot(d, c)) * Float64(fma(d, a, Float64(b * c)) / hypot(d, c)));
	elseif (t_0 <= 4e+285)
		tmp = t_1;
	else
		tmp = Float64(Float64(b / c) - Float64(Float64(a / c) * Float64(d / c)));
	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[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(d * (-a) + N[(b * c), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[((-a) / d), $MachinePrecision], If[LessEqual[t$95$0, -5e-232], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(1.0 / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(d * a + N[(b * c), $MachinePrecision]), $MachinePrecision] / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+285], t$95$1, N[(N[(b / c), $MachinePrecision] - N[(N[(a / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	25.9
Target	0.5
Herbie	14.4

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

Derivation

1. Split input into 4 regimes

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

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   3. Taylor expanded in d around inf 37.1

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]

   4. Simplified 37.1

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

   if -inf.0 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < -4.9999999999999999e-232 or 0.0 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 3.9999999999999999e285

   1. Initial program 0.6

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

   2. Simplified 0.6

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

   if -4.9999999999999999e-232 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 0.0

   1. Initial program 25.5

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

   2. Simplified 25.5

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

   3. Applied egg-rr 16.0

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

   if 3.9999999999999999e285 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 62.7

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

   2. Simplified 62.7

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

   3. Taylor expanded in d around 0 39.4

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

   4. Simplified 34.5

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

   5. Applied egg-rr 32.1

      \[\leadsto \frac{b}{c} - \color{blue}{\frac{a}{c} \cdot \frac{d}{c}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 14.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq -\infty:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq -5 \cdot 10^{-232}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, -a, b \cdot c\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{\mathsf{fma}\left(d, a, b \cdot c\right)}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq 4 \cdot 10^{+285}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, -a, b \cdot c\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \end{array} \]

Reproduce

herbie shell --seed 2022190 
(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))))