Complex division, real part

Average Error: 25.7 → 15.2

Time: 13.9s

Precision: binary64

Cost: 7436

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -4.7 \cdot 10^{+123}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 3.4 \cdot 10^{-101}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot b}{{c}^{2}}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+109}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

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)))))
   (if (<= d -4.7e+123)
     (/ b d)
     (if (<= d -4e-28)
       t_0
       (if (<= d 3.4e-101)
         (+ (/ a c) (/ (* d b) (pow c 2.0)))
         (if (<= d 1.7e+109) t_0 (/ b d)))))))

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 tmp;
	if (d <= -4.7e+123) {
		tmp = b / d;
	} else if (d <= -4e-28) {
		tmp = t_0;
	} else if (d <= 3.4e-101) {
		tmp = (a / c) + ((d * b) / pow(c, 2.0));
	} else if (d <= 1.7e+109) {
		tmp = t_0;
	} else {
		tmp = b / d;
	}
	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) :: tmp
    t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
    if (d <= (-4.7d+123)) then
        tmp = b / d
    else if (d <= (-4d-28)) then
        tmp = t_0
    else if (d <= 3.4d-101) then
        tmp = (a / c) + ((d * b) / (c ** 2.0d0))
    else if (d <= 1.7d+109) then
        tmp = t_0
    else
        tmp = b / d
    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 tmp;
	if (d <= -4.7e+123) {
		tmp = b / d;
	} else if (d <= -4e-28) {
		tmp = t_0;
	} else if (d <= 3.4e-101) {
		tmp = (a / c) + ((d * b) / Math.pow(c, 2.0));
	} else if (d <= 1.7e+109) {
		tmp = t_0;
	} else {
		tmp = b / d;
	}
	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))
	tmp = 0
	if d <= -4.7e+123:
		tmp = b / d
	elif d <= -4e-28:
		tmp = t_0
	elif d <= 3.4e-101:
		tmp = (a / c) + ((d * b) / math.pow(c, 2.0))
	elif d <= 1.7e+109:
		tmp = t_0
	else:
		tmp = b / d
	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)))
	tmp = 0.0
	if (d <= -4.7e+123)
		tmp = Float64(b / d);
	elseif (d <= -4e-28)
		tmp = t_0;
	elseif (d <= 3.4e-101)
		tmp = Float64(Float64(a / c) + Float64(Float64(d * b) / (c ^ 2.0)));
	elseif (d <= 1.7e+109)
		tmp = t_0;
	else
		tmp = Float64(b / d);
	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));
	tmp = 0.0;
	if (d <= -4.7e+123)
		tmp = b / d;
	elseif (d <= -4e-28)
		tmp = t_0;
	elseif (d <= 3.4e-101)
		tmp = (a / c) + ((d * b) / (c ^ 2.0));
	elseif (d <= 1.7e+109)
		tmp = t_0;
	else
		tmp = b / d;
	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]}, If[LessEqual[d, -4.7e+123], N[(b / d), $MachinePrecision], If[LessEqual[d, -4e-28], t$95$0, If[LessEqual[d, 3.4e-101], N[(N[(a / c), $MachinePrecision] + N[(N[(d * b), $MachinePrecision] / N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.7e+109], t$95$0, N[(b / d), $MachinePrecision]]]]]]

Target

Original	25.7
Target	0.5
Herbie	15.2

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

Derivation

1. Split input into 3 regimes

2. if d < -4.69999999999999979e123 or 1.70000000000000003e109 < d

   1. Initial program 40.5

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

   2. Taylor expanded in c around 0 15.5

      \[\leadsto \color{blue}{\frac{b}{d}} \]

   if -4.69999999999999979e123 < d < -3.99999999999999988e-28 or 3.39999999999999989e-101 < d < 1.70000000000000003e109

   1. Initial program 17.4

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

   if -3.99999999999999988e-28 < d < 3.39999999999999989e-101

   1. Initial program 18.5

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

   2. Taylor expanded in c around inf 13.0

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

4. Final simplification 15.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.7 \cdot 10^{+123}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-28}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 3.4 \cdot 10^{-101}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot b}{{c}^{2}}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+109}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternatives

Alternative 1
Error	17.7
Cost	1752

\[\begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -2.5 \cdot 10^{+125}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -6.5 \cdot 10^{-112}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq -2.4 \cdot 10^{-234}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{-101}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{+108}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 2
Error	22.9
Cost	1100

\[\begin{array}{l} \mathbf{if}\;d \leq -5.5 \cdot 10^{+64}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -1.05 \cdot 10^{+35}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq -4.2 \cdot 10^{-28}:\\ \;\;\;\;\frac{d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 3
Error	23.3
Cost	720

\[\begin{array}{l} \mathbf{if}\;d \leq -5.8 \cdot 10^{+63}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -1.6 \cdot 10^{+20}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq -4.2 \cdot 10^{-28}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 4
Error	37.6
Cost	192

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

Reproduce

herbie shell --seed 2023077 
(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))))