Complex division, real part

Average Error: 26.6 → 14.5

Time: 10.9s

Precision: binary64

Cost: 7568

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{b}{d} + c \cdot \frac{a}{{d}^{2}}\\ \mathbf{if}\;d \leq -3.6 \cdot 10^{+28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -3.1 \cdot 10^{-156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{-143}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{\frac{{c}^{2}}{d}}\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{+87}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))
        (t_1 (+ (/ b d) (* c (/ a (pow d 2.0))))))
   (if (<= d -3.6e+28)
     t_1
     (if (<= d -3.1e-156)
       t_0
       (if (<= d 1.65e-143)
         (+ (/ a c) (/ b (/ (pow c 2.0) d)))
         (if (<= d 4.2e+87) t_0 t_1))))))

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 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = (b / d) + (c * (a / pow(d, 2.0)));
	double tmp;
	if (d <= -3.6e+28) {
		tmp = t_1;
	} else if (d <= -3.1e-156) {
		tmp = t_0;
	} else if (d <= 1.65e-143) {
		tmp = (a / c) + (b / (pow(c, 2.0) / d));
	} else if (d <= 4.2e+87) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = ((a * c) + (b * d)) / ((c * c) + (d * d))
end function

↓

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
    t_1 = (b / d) + (c * (a / (d ** 2.0d0)))
    if (d <= (-3.6d+28)) then
        tmp = t_1
    else if (d <= (-3.1d-156)) then
        tmp = t_0
    else if (d <= 1.65d-143) then
        tmp = (a / c) + (b / ((c ** 2.0d0) / d))
    else if (d <= 4.2d+87) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	return ((a * c) + (b * d)) / ((c * c) + (d * d));
}

↓

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = (b / d) + (c * (a / Math.pow(d, 2.0)));
	double tmp;
	if (d <= -3.6e+28) {
		tmp = t_1;
	} else if (d <= -3.1e-156) {
		tmp = t_0;
	} else if (d <= 1.65e-143) {
		tmp = (a / c) + (b / (Math.pow(c, 2.0) / d));
	} else if (d <= 4.2e+87) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

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

↓

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	t_1 = (b / d) + (c * (a / math.pow(d, 2.0)))
	tmp = 0
	if d <= -3.6e+28:
		tmp = t_1
	elif d <= -3.1e-156:
		tmp = t_0
	elif d <= 1.65e-143:
		tmp = (a / c) + (b / (math.pow(c, 2.0) / d))
	elif d <= 4.2e+87:
		tmp = t_0
	else:
		tmp = t_1
	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(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(b / d) + Float64(c * Float64(a / (d ^ 2.0))))
	tmp = 0.0
	if (d <= -3.6e+28)
		tmp = t_1;
	elseif (d <= -3.1e-156)
		tmp = t_0;
	elseif (d <= 1.65e-143)
		tmp = Float64(Float64(a / c) + Float64(b / Float64((c ^ 2.0) / d)));
	elseif (d <= 4.2e+87)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	t_1 = (b / d) + (c * (a / (d ^ 2.0)));
	tmp = 0.0;
	if (d <= -3.6e+28)
		tmp = t_1;
	elseif (d <= -3.1e-156)
		tmp = t_0;
	elseif (d <= 1.65e-143)
		tmp = (a / c) + (b / ((c ^ 2.0) / d));
	elseif (d <= 4.2e+87)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = 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), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b / d), $MachinePrecision] + N[(c * N[(a / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.6e+28], t$95$1, If[LessEqual[d, -3.1e-156], t$95$0, If[LessEqual[d, 1.65e-143], N[(N[(a / c), $MachinePrecision] + N[(b / N[(N[Power[c, 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.2e+87], t$95$0, t$95$1]]]]]]

Target

Original	26.6
Target	0.5
Herbie	14.5

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

Derivation

1. Split input into 3 regimes

2. if d < -3.5999999999999999e28 or 4.2e87 < d

   1. Initial program 36.7

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

   2. Taylor expanded in c around 0 17.8

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

   3. Simplified 16.1

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

      [Start] 17.8	\[ \frac{b}{d} + \frac{c \cdot a}{{d}^{2}} \]
      rational.json-simplify-2 [<=] 17.8	\[ \frac{b}{d} + \frac{\color{blue}{a \cdot c}}{{d}^{2}} \]
      rational.json-simplify-49 [=>] 16.1	\[ \frac{b}{d} + \color{blue}{c \cdot \frac{a}{{d}^{2}}} \]

   if -3.5999999999999999e28 < d < -3.0999999999999998e-156 or 1.65e-143 < d < 4.2e87

   1. Initial program 14.5

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

   if -3.0999999999999998e-156 < d < 1.65e-143

   1. Initial program 24.4

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

   2. Taylor expanded in c around inf 9.9

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

   3. Simplified 11.1

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

      [Start] 9.9	\[ \frac{a}{c} + \frac{d \cdot b}{{c}^{2}} \]
      rational.json-simplify-49 [=>] 11.1	\[ \frac{a}{c} + \color{blue}{b \cdot \frac{d}{{c}^{2}}} \]

   4. Applied egg-rr 11.2

      \[\leadsto \frac{a}{c} + \color{blue}{\frac{b}{\frac{{c}^{2}}{d}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 14.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.6 \cdot 10^{+28}:\\ \;\;\;\;\frac{b}{d} + c \cdot \frac{a}{{d}^{2}}\\ \mathbf{elif}\;d \leq -3.1 \cdot 10^{-156}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{-143}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{\frac{{c}^{2}}{d}}\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{+87}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + c \cdot \frac{a}{{d}^{2}}\\ \end{array} \]

Alternatives

Alternative 1
Error	14.9
Cost	7436

\[\begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -2.3 \cdot 10^{+141}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -7 \cdot 10^{-151}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{-143}:\\ \;\;\;\;\frac{a}{c} + b \cdot \frac{d}{{c}^{2}}\\ \mathbf{elif}\;d \leq 4.8 \cdot 10^{+87}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 2
Error	14.9
Cost	7436

\[\begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -1.7 \cdot 10^{+137}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -1.75 \cdot 10^{-156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-143}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{\frac{{c}^{2}}{d}}\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+85}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 3
Error	23.0
Cost	1496

\[\begin{array}{l} t_0 := c \cdot c + d \cdot d\\ t_1 := \frac{b}{t_0} \cdot d\\ \mathbf{if}\;d \leq -5.8 \cdot 10^{+46}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -1.65 \cdot 10^{-154}:\\ \;\;\;\;\frac{c}{t_0} \cdot a\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-103}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 6500000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{+25}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{+114}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 4
Error	22.5
Cost	1496

\[\begin{array}{l} t_0 := c \cdot c + d \cdot d\\ t_1 := \frac{d}{t_0} \cdot b\\ \mathbf{if}\;d \leq -1.55 \cdot 10^{+47}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -2.1 \cdot 10^{-150}:\\ \;\;\;\;\frac{c}{t_0} \cdot a\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-90}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 330000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 2.35 \cdot 10^{+25}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+142}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 5
Error	22.5
Cost	1496

\[\begin{array}{l} t_0 := c \cdot c + d \cdot d\\ t_1 := \frac{b}{\frac{t_0}{d}}\\ \mathbf{if}\;d \leq -6.6 \cdot 10^{+46}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -2.25 \cdot 10^{-150}:\\ \;\;\;\;\frac{c}{t_0} \cdot a\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{-93}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 310000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 2.35 \cdot 10^{+25}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{+142}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 6
Error	16.4
Cost	1488

\[\begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -9.2 \cdot 10^{+141}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -1.08 \cdot 10^{-259}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 10^{-143}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 4.8 \cdot 10^{+87}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 7
Error	22.3
Cost	1364

\[\begin{array}{l} t_0 := \frac{b}{c \cdot c + d \cdot d} \cdot d\\ \mathbf{if}\;d \leq -1.3 \cdot 10^{+27}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-91}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 80000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 2.35 \cdot 10^{+25}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{+113}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 8
Error	23.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{+26}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 2.35 \cdot 10^{+25}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{d}{b}}\\ \end{array} \]

Alternative 9
Error	22.9
Cost	456

\[\begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{+26}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 2.35 \cdot 10^{+25}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 10
Error	37.9
Cost	192

\[\frac{a}{c} \]

Reproduce

herbie shell --seed 2023073 
(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))))