Complex division, real part

Average Error: 26.5 → 14.6

Time: 9.9s

Precision: binary64

Cost: 7436

Math

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

↓

\[\begin{array}{l} t_0 := c \cdot c + d \cdot d\\ t_1 := a \cdot c + b \cdot d\\ \mathbf{if}\;d \leq -8.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{b}{d} + c \cdot \frac{a}{{d}^{2}}\\ \mathbf{elif}\;d \leq -4.8 \cdot 10^{-127}:\\ \;\;\;\;\frac{1}{t_0} \cdot t_1\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-104}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{\frac{{c}^{2}}{d}}\\ \mathbf{elif}\;d \leq 4 \cdot 10^{+136}:\\ \;\;\;\;\frac{t_1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

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 (+ (* c c) (* d d))) (t_1 (+ (* a c) (* b d))))
   (if (<= d -8.5e+102)
     (+ (/ b d) (* c (/ a (pow d 2.0))))
     (if (<= d -4.8e-127)
       (* (/ 1.0 t_0) t_1)
       (if (<= d 1.6e-104)
         (+ (/ a c) (/ b (/ (pow c 2.0) d)))
         (if (<= d 4e+136) (/ t_1 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 = (c * c) + (d * d);
	double t_1 = (a * c) + (b * d);
	double tmp;
	if (d <= -8.5e+102) {
		tmp = (b / d) + (c * (a / pow(d, 2.0)));
	} else if (d <= -4.8e-127) {
		tmp = (1.0 / t_0) * t_1;
	} else if (d <= 1.6e-104) {
		tmp = (a / c) + (b / (pow(c, 2.0) / d));
	} else if (d <= 4e+136) {
		tmp = t_1 / 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) :: t_1
    real(8) :: tmp
    t_0 = (c * c) + (d * d)
    t_1 = (a * c) + (b * d)
    if (d <= (-8.5d+102)) then
        tmp = (b / d) + (c * (a / (d ** 2.0d0)))
    else if (d <= (-4.8d-127)) then
        tmp = (1.0d0 / t_0) * t_1
    else if (d <= 1.6d-104) then
        tmp = (a / c) + (b / ((c ** 2.0d0) / d))
    else if (d <= 4d+136) then
        tmp = t_1 / 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 = (c * c) + (d * d);
	double t_1 = (a * c) + (b * d);
	double tmp;
	if (d <= -8.5e+102) {
		tmp = (b / d) + (c * (a / Math.pow(d, 2.0)));
	} else if (d <= -4.8e-127) {
		tmp = (1.0 / t_0) * t_1;
	} else if (d <= 1.6e-104) {
		tmp = (a / c) + (b / (Math.pow(c, 2.0) / d));
	} else if (d <= 4e+136) {
		tmp = t_1 / 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 = (c * c) + (d * d)
	t_1 = (a * c) + (b * d)
	tmp = 0
	if d <= -8.5e+102:
		tmp = (b / d) + (c * (a / math.pow(d, 2.0)))
	elif d <= -4.8e-127:
		tmp = (1.0 / t_0) * t_1
	elif d <= 1.6e-104:
		tmp = (a / c) + (b / (math.pow(c, 2.0) / d))
	elif d <= 4e+136:
		tmp = t_1 / 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(c * c) + Float64(d * d))
	t_1 = Float64(Float64(a * c) + Float64(b * d))
	tmp = 0.0
	if (d <= -8.5e+102)
		tmp = Float64(Float64(b / d) + Float64(c * Float64(a / (d ^ 2.0))));
	elseif (d <= -4.8e-127)
		tmp = Float64(Float64(1.0 / t_0) * t_1);
	elseif (d <= 1.6e-104)
		tmp = Float64(Float64(a / c) + Float64(b / Float64((c ^ 2.0) / d)));
	elseif (d <= 4e+136)
		tmp = Float64(t_1 / 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 = (c * c) + (d * d);
	t_1 = (a * c) + (b * d);
	tmp = 0.0;
	if (d <= -8.5e+102)
		tmp = (b / d) + (c * (a / (d ^ 2.0)));
	elseif (d <= -4.8e-127)
		tmp = (1.0 / t_0) * t_1;
	elseif (d <= 1.6e-104)
		tmp = (a / c) + (b / ((c ^ 2.0) / d));
	elseif (d <= 4e+136)
		tmp = t_1 / 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[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -8.5e+102], N[(N[(b / d), $MachinePrecision] + N[(c * N[(a / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4.8e-127], N[(N[(1.0 / t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[d, 1.6e-104], N[(N[(a / c), $MachinePrecision] + N[(b / N[(N[Power[c, 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4e+136], N[(t$95$1 / t$95$0), $MachinePrecision], N[(b / d), $MachinePrecision]]]]]]]

Target

Original	26.5
Target	0.5
Herbie	14.6

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

Derivation

1. Split input into 5 regimes

2. if d < -8.4999999999999996e102

   1. Initial program 41.1

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

   2. Taylor expanded in c around 0 15.4

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

   3. Simplified 13.9

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

      [Start] 15.4	\[ \frac{b}{d} + \frac{c \cdot a}{{d}^{2}} \]
      rational.json-simplify-2 [<=] 15.4	\[ \frac{b}{d} + \frac{\color{blue}{a \cdot c}}{{d}^{2}} \]
      rational.json-simplify-49 [=>] 13.9	\[ \frac{b}{d} + \color{blue}{c \cdot \frac{a}{{d}^{2}}} \]

   if -8.4999999999999996e102 < d < -4.79999999999999964e-127

   1. Initial program 17.4

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

   2. Applied egg-rr 17.5

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

   if -4.79999999999999964e-127 < d < 1.59999999999999994e-104

   1. Initial program 22.0

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

   2. Taylor expanded in c around inf 10.7

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

   3. Simplified 14.1

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

      [Start] 10.7	\[ \frac{a}{c} + \frac{d \cdot b}{{c}^{2}} \]
      rational.json-simplify-2 [=>] 10.7	\[ \frac{a}{c} + \frac{\color{blue}{b \cdot d}}{{c}^{2}} \]
      rational.json-simplify-49 [=>] 14.1	\[ \frac{a}{c} + \color{blue}{d \cdot \frac{b}{{c}^{2}}} \]

   4. Applied egg-rr 11.8

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

   if 1.59999999999999994e-104 < d < 4.00000000000000023e136

   1. Initial program 17.0

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

   if 4.00000000000000023e136 < d

   1. Initial program 42.3

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

   2. Taylor expanded in c around 0 13.9

      \[\leadsto \color{blue}{\frac{b}{d}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 14.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{b}{d} + c \cdot \frac{a}{{d}^{2}}\\ \mathbf{elif}\;d \leq -4.8 \cdot 10^{-127}:\\ \;\;\;\;\frac{1}{c \cdot c + d \cdot d} \cdot \left(a \cdot c + b \cdot d\right)\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-104}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{\frac{{c}^{2}}{d}}\\ \mathbf{elif}\;d \leq 4 \cdot 10^{+136}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternatives

Alternative 1
Error	15.6
Cost	7436

\[\begin{array}{l} t_0 := c \cdot c + d \cdot d\\ t_1 := a \cdot c + b \cdot d\\ \mathbf{if}\;d \leq -1.25 \cdot 10^{+104}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -4.2 \cdot 10^{-125}:\\ \;\;\;\;\frac{1}{t_0} \cdot t_1\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{-118}:\\ \;\;\;\;\frac{a}{c} + d \cdot \frac{b}{{c}^{2}}\\ \mathbf{elif}\;d \leq 4 \cdot 10^{+136}:\\ \;\;\;\;\frac{t_1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 2
Error	15.0
Cost	7436

\[\begin{array}{l} t_0 := c \cdot c + d \cdot d\\ t_1 := a \cdot c + b \cdot d\\ \mathbf{if}\;d \leq -1.3 \cdot 10^{+104}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -2.9 \cdot 10^{-126}:\\ \;\;\;\;\frac{1}{t_0} \cdot t_1\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{-105}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{\frac{{c}^{2}}{d}}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{+136}:\\ \;\;\;\;\frac{t_1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 3
Error	14.7
Cost	7436

\[\begin{array}{l} t_0 := c \cdot c + d \cdot d\\ t_1 := a \cdot c + b \cdot d\\ \mathbf{if}\;d \leq -2.1 \cdot 10^{+102}:\\ \;\;\;\;\frac{b}{d} + a \cdot \frac{c}{{d}^{2}}\\ \mathbf{elif}\;d \leq -2.1 \cdot 10^{-126}:\\ \;\;\;\;\frac{1}{t_0} \cdot t_1\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{\frac{{c}^{2}}{d}}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{+136}:\\ \;\;\;\;\frac{t_1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 4
Error	16.1
Cost	1488

\[\begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -2.4 \cdot 10^{+104}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -1.7 \cdot 10^{-127}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{-105}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{+136}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 5
Error	16.2
Cost	1488

\[\begin{array}{l} t_0 := c \cdot c + d \cdot d\\ t_1 := a \cdot c + b \cdot d\\ \mathbf{if}\;d \leq -7.6 \cdot 10^{+103}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-125}:\\ \;\;\;\;\frac{1}{t_0} \cdot t_1\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{-105}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{+136}:\\ \;\;\;\;\frac{t_1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 6
Error	21.1
Cost	1232

\[\begin{array}{l} t_0 := \frac{d}{c \cdot c + d \cdot d} \cdot b\\ \mathbf{if}\;d \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -1.9 \cdot 10^{-125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{+136}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 7
Error	22.4
Cost	968

\[\begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{+64}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -8.2 \cdot 10^{-123}:\\ \;\;\;\;\frac{b}{c \cdot c + d \cdot d} \cdot d\\ \mathbf{elif}\;d \leq 10^{-13}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 8
Error	23.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;d \leq -3.9 \cdot 10^{-75}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-15}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 9
Error	37.0
Cost	192

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

Reproduce

herbie shell --seed 2023065 
(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))))