Complex division, imag part

Average Error: 26.7 → 1.4

Time: 19.5s

Precision: binary64

Cost: 33152

Math

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

↓

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

FPCore

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

↓

(FPCore (a b c d)
 :precision binary64
 (fma
  (/ c (hypot c d))
  (/ b (hypot c d))
  (/ (/ (- d) (/ (hypot d c) a)) (hypot 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) {
	return fma((c / hypot(c, d)), (b / hypot(c, d)), ((-d / (hypot(d, c) / a)) / hypot(d, c)));
}

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)
	return fma(Float64(c / hypot(c, d)), Float64(b / hypot(c, d)), Float64(Float64(Float64(-d) / Float64(hypot(d, c) / a)) / hypot(d, c)))
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_] := N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] + N[(N[((-d) / N[(N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	26.7
Target	0.5
Herbie	1.4

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

Derivation

1. Initial program 26.7

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

2. Applied egg-rr 15.5

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

3. Applied egg-rr 1.5

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

4. Applied egg-rr 1.4

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

5. Final simplification 1.4

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

Alternatives

Alternative 1
Error	7.9
Cost	33288

\[\begin{array}{l} t_0 := \frac{c}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;d \leq -122006031482371650:\\ \;\;\;\;\mathsf{fma}\left(t_0, t_1, \frac{a}{\mathsf{hypot}\left(d, c\right)}\right)\\ \mathbf{elif}\;d \leq 7.395863073945087 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(t_0, t_1, \frac{d \cdot \left(-a\right)}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \end{array} \]

Alternative 2
Error	8.9
Cost	26500

\[\begin{array}{l} t_0 := \frac{\frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;d \leq -8941568.424361361:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{a}{\mathsf{hypot}\left(d, c\right)}\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-180}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 10^{-160}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c \cdot \frac{c}{d}}\\ \mathbf{elif}\;d \leq 1.5586786786398142 \cdot 10^{+46}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \end{array} \]

Alternative 3
Error	10.7
Cost	14160

\[\begin{array}{l} t_0 := \frac{\frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := \frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{if}\;d \leq -4.348610485428097 \cdot 10^{+35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-180}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 10^{-160}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c \cdot \frac{c}{d}}\\ \mathbf{elif}\;d \leq 1.5586786786398142 \cdot 10^{+46}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	12.4
Cost	13832

\[\begin{array}{l} t_0 := \frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{if}\;d \leq -122006031482371650:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, -a, c \cdot b\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-133}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \frac{-d}{c}, \frac{1}{c}, \frac{b}{c}\right)\\ \mathbf{elif}\;d \leq 1.5586786786398142 \cdot 10^{+46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, a, c \cdot b\right)}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	12.4
Cost	7824

\[\begin{array}{l} t_0 := d \cdot d + c \cdot c\\ t_1 := \frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{if}\;d \leq -122006031482371650:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{t_0}\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-133}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \frac{-d}{c}, \frac{1}{c}, \frac{b}{c}\right)\\ \mathbf{elif}\;d \leq 1.5586786786398142 \cdot 10^{+46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, a, c \cdot b\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	12.4
Cost	7564

\[\begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{d \cdot d + c \cdot c}\\ t_1 := \frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{if}\;d \leq -122006031482371650:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-160}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-133}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \frac{-d}{c}, \frac{1}{c}, \frac{b}{c}\right)\\ \mathbf{elif}\;d \leq 1.5586786786398142 \cdot 10^{+46}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	17.0
Cost	1496

\[\begin{array}{l} t_0 := \frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{if}\;d \leq -122006031482371650:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -4.1758521565422204 \cdot 10^{-48}:\\ \;\;\;\;\frac{d \cdot \left(-a\right)}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;d \leq -1.998814742675197 \cdot 10^{-75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -4.6 \cdot 10^{-135}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{elif}\;d \leq -1.8 \cdot 10^{-162}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq 7.605581508364986 \cdot 10^{-74}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c \cdot \frac{c}{d}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	12.6
Cost	1488

\[\begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{d \cdot d + c \cdot c}\\ t_1 := \frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{if}\;d \leq -122006031482371650:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-160}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-133}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c \cdot \frac{c}{d}}\\ \mathbf{elif}\;d \leq 1.5586786786398142 \cdot 10^{+46}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	16.3
Cost	1232

\[\begin{array}{l} \mathbf{if}\;c \leq -1.2088374622742729 \cdot 10^{+45}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c \cdot \frac{c}{a}}\\ \mathbf{elif}\;c \leq 3.337487325098142 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 7.916160279112307 \cdot 10^{+49}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c \cdot \frac{c}{d}}\\ \mathbf{elif}\;c \leq 4.9522731660807546 \cdot 10^{+101}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]

Alternative 10
Error	16.3
Cost	1232

\[\begin{array}{l} \mathbf{if}\;c \leq -1.2088374622742729 \cdot 10^{+45}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c \cdot \frac{c}{a}}\\ \mathbf{elif}\;c \leq 3.337487325098142 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 7.916160279112307 \cdot 10^{+49}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c \cdot \frac{c}{d}}\\ \mathbf{elif}\;c \leq 4.9522731660807546 \cdot 10^{+101}:\\ \;\;\;\;c \cdot \frac{\frac{b}{d}}{d} - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]

Alternative 11
Error	20.1
Cost	1104

\[\begin{array}{l} t_0 := \frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{if}\;c \leq -1.2088374622742729 \cdot 10^{+45}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 3.337487325098142 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 2.6689993466329684 \cdot 10^{-5}:\\ \;\;\;\;\frac{d \cdot \left(-a\right)}{c \cdot c}\\ \mathbf{elif}\;c \leq 627949049897513100:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.151460632922877 \cdot 10^{+60}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{+106}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 12
Error	16.0
Cost	1104

\[\begin{array}{l} t_0 := \frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{if}\;c \leq -1.2088374622742729 \cdot 10^{+45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 6.555413221717854 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 1.151460632922877 \cdot 10^{+60}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 4.9522731660807546 \cdot 10^{+101}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	16.2
Cost	1104

\[\begin{array}{l} t_0 := \frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{if}\;c \leq -1.2088374622742729 \cdot 10^{+45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 3.337487325098142 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 1.151460632922877 \cdot 10^{+60}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c \cdot \frac{c}{d}}\\ \mathbf{elif}\;c \leq 4.9522731660807546 \cdot 10^{+101}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 14
Error	16.5
Cost	1104

\[\begin{array}{l} \mathbf{if}\;c \leq -1.2088374622742729 \cdot 10^{+45}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c \cdot \frac{c}{a}}\\ \mathbf{elif}\;c \leq 3.337487325098142 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 1.151460632922877 \cdot 10^{+60}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c \cdot \frac{c}{d}}\\ \mathbf{elif}\;c \leq 4.9522731660807546 \cdot 10^{+101}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]

Alternative 15
Error	24.3
Cost	916

\[\begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;c \leq -1.2088374622742729 \cdot 10^{+45}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 3.337487325098142 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 2.6689993466329684 \cdot 10^{-5}:\\ \;\;\;\;\frac{d \cdot \left(-a\right)}{c \cdot c}\\ \mathbf{elif}\;c \leq 1.151460632922877 \cdot 10^{+60}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{+106}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 16
Error	23.8
Cost	784

\[\begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;c \leq -1.2088374622742729 \cdot 10^{+45}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 3.337487325098142 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.151460632922877 \cdot 10^{+60}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{+106}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 17
Error	37.4
Cost	192

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

Reproduce

herbie shell --seed 2022294 
(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))))