Complex division, real part

Average Error: 26.3 → 11.1

Time: 5.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;c \leq -3.9 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(\frac{d}{c}, \frac{b}{c}, \frac{a}{c}\right)\\ \mathbf{elif}\;c \leq -3.7 \cdot 10^{-27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 10^{-284}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{d}, \frac{a}{d}, \frac{b}{d}\right)\\ \mathbf{elif}\;c \leq 1.25 \cdot 10^{+110}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \mathsf{fma}\left(\frac{d}{c}, b, a\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (/ (fma a c (* d b)) (hypot c d)) (hypot c d))))
   (if (<= c -3.9e+155)
     (fma (/ d c) (/ b c) (/ a c))
     (if (<= c -3.7e-27)
       t_0
       (if (<= c 1e-284)
         (fma (/ c d) (/ a d) (/ b d))
         (if (<= c 1.25e+110)
           t_0
           (* (/ 1.0 (hypot c d)) (fma (/ d c) b a))))))))

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 t_0 = (fma(a, c, (d * b)) / hypot(c, d)) / hypot(c, d);
	double tmp;
	if (c <= -3.9e+155) {
		tmp = fma((d / c), (b / c), (a / c));
	} else if (c <= -3.7e-27) {
		tmp = t_0;
	} else if (c <= 1e-284) {
		tmp = fma((c / d), (a / d), (b / d));
	} else if (c <= 1.25e+110) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(c, d)) * fma((d / c), b, a);
	}
	return tmp;
}

Julia

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)
	t_0 = Float64(Float64(fma(a, c, Float64(d * b)) / hypot(c, d)) / hypot(c, d))
	tmp = 0.0
	if (c <= -3.9e+155)
		tmp = fma(Float64(d / c), Float64(b / c), Float64(a / c));
	elseif (c <= -3.7e-27)
		tmp = t_0;
	elseif (c <= 1e-284)
		tmp = fma(Float64(c / d), Float64(a / d), Float64(b / d));
	elseif (c <= 1.25e+110)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * fma(Float64(d / c), b, a));
	end
	return tmp
end

Wolfram

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_] := Block[{t$95$0 = N[(N[(N[(a * c + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -3.9e+155], N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision] + N[(a / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -3.7e-27], t$95$0, If[LessEqual[c, 1e-284], N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision] + N[(b / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.25e+110], t$95$0, N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision]), $MachinePrecision]]]]]]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.3
Target	0.5
Herbie	11.1

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

Derivation

1. Split input into 4 regimes

2. if c < -3.8999999999999998e155

   1. Initial program 45.5

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

   2. Simplified 45.5

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

   3. Applied egg-rr 29.5

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

   4. Taylor expanded in c around inf 13.5

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

   5. Simplified 6.3

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

   if -3.8999999999999998e155 < c < -3.70000000000000029e-27 or 1.00000000000000004e-284 < c < 1.24999999999999995e110

   1. Initial program 18.5

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

   2. Simplified 18.5

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

   3. Applied egg-rr 12.3

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

   4. Applied egg-rr 12.5

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

   5. Applied egg-rr 12.1

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

   if -3.70000000000000029e-27 < c < 1.00000000000000004e-284

   1. Initial program 20.3

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

   2. Simplified 20.3

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

   3. Applied egg-rr 11.7

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

   4. Taylor expanded in c around 0 14.5

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

   5. Simplified 12.9

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

   if 1.24999999999999995e110 < c

   1. Initial program 40.8

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

   2. Simplified 40.8

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

   3. Applied egg-rr 27.3

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

   4. Taylor expanded in c around inf 13.5

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

   5. Simplified 9.7

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\mathsf{fma}\left(\frac{d}{c}, b, a\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 11.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.9 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(\frac{d}{c}, \frac{b}{c}, \frac{a}{c}\right)\\ \mathbf{elif}\;c \leq -3.7 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq 10^{-284}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{d}, \frac{a}{d}, \frac{b}{d}\right)\\ \mathbf{elif}\;c \leq 1.25 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \mathsf{fma}\left(\frac{d}{c}, b, a\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(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))))