Complex division, imag part

Average Error: 26.3 → 17.1

Time: 4.8s

Precision: binary64

Cost: 2148

Math

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

↓

\[\begin{array}{l} t_0 := -\frac{a}{d}\\ t_1 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -1.95 \cdot 10^{+123}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.32 \cdot 10^{-120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 6.6 \cdot 10^{-192}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 3.1 \cdot 10^{-73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-38}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{+22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{+50}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 3.75 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (/ a d))) (t_1 (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))))
   (if (<= d -1.95e+123)
     t_0
     (if (<= d -1.32e-120)
       t_1
       (if (<= d 6.6e-192)
         (/ b c)
         (if (<= d 3.1e-73)
           t_1
           (if (<= d 1.05e-38)
             (/ b c)
             (if (<= d 6.8e+22)
               t_1
               (if (<= d 2.9e+50)
                 (/ b c)
                 (if (<= d 3.75e+65) t_0 (if (<= d 1e+154) t_1 t_0)))))))))))

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) {
	double t_0 = -(a / d);
	double t_1 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -1.95e+123) {
		tmp = t_0;
	} else if (d <= -1.32e-120) {
		tmp = t_1;
	} else if (d <= 6.6e-192) {
		tmp = b / c;
	} else if (d <= 3.1e-73) {
		tmp = t_1;
	} else if (d <= 1.05e-38) {
		tmp = b / c;
	} else if (d <= 6.8e+22) {
		tmp = t_1;
	} else if (d <= 2.9e+50) {
		tmp = b / c;
	} else if (d <= 3.75e+65) {
		tmp = t_0;
	} else if (d <= 1e+154) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	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 = ((b * c) - (a * 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 / d)
    t_1 = ((b * c) - (a * d)) / ((c * c) + (d * d))
    if (d <= (-1.95d+123)) then
        tmp = t_0
    else if (d <= (-1.32d-120)) then
        tmp = t_1
    else if (d <= 6.6d-192) then
        tmp = b / c
    else if (d <= 3.1d-73) then
        tmp = t_1
    else if (d <= 1.05d-38) then
        tmp = b / c
    else if (d <= 6.8d+22) then
        tmp = t_1
    else if (d <= 2.9d+50) then
        tmp = b / c
    else if (d <= 3.75d+65) then
        tmp = t_0
    else if (d <= 1d+154) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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

↓

public static double code(double a, double b, double c, double d) {
	double t_0 = -(a / d);
	double t_1 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -1.95e+123) {
		tmp = t_0;
	} else if (d <= -1.32e-120) {
		tmp = t_1;
	} else if (d <= 6.6e-192) {
		tmp = b / c;
	} else if (d <= 3.1e-73) {
		tmp = t_1;
	} else if (d <= 1.05e-38) {
		tmp = b / c;
	} else if (d <= 6.8e+22) {
		tmp = t_1;
	} else if (d <= 2.9e+50) {
		tmp = b / c;
	} else if (d <= 3.75e+65) {
		tmp = t_0;
	} else if (d <= 1e+154) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(a, b, c, d):
	t_0 = -(a / d)
	t_1 = ((b * c) - (a * d)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -1.95e+123:
		tmp = t_0
	elif d <= -1.32e-120:
		tmp = t_1
	elif d <= 6.6e-192:
		tmp = b / c
	elif d <= 3.1e-73:
		tmp = t_1
	elif d <= 1.05e-38:
		tmp = b / c
	elif d <= 6.8e+22:
		tmp = t_1
	elif d <= 2.9e+50:
		tmp = b / c
	elif d <= 3.75e+65:
		tmp = t_0
	elif d <= 1e+154:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

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)
	t_0 = Float64(-Float64(a / d))
	t_1 = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -1.95e+123)
		tmp = t_0;
	elseif (d <= -1.32e-120)
		tmp = t_1;
	elseif (d <= 6.6e-192)
		tmp = Float64(b / c);
	elseif (d <= 3.1e-73)
		tmp = t_1;
	elseif (d <= 1.05e-38)
		tmp = Float64(b / c);
	elseif (d <= 6.8e+22)
		tmp = t_1;
	elseif (d <= 2.9e+50)
		tmp = Float64(b / c);
	elseif (d <= 3.75e+65)
		tmp = t_0;
	elseif (d <= 1e+154)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(a, b, c, d)
	t_0 = -(a / d);
	t_1 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (d <= -1.95e+123)
		tmp = t_0;
	elseif (d <= -1.32e-120)
		tmp = t_1;
	elseif (d <= 6.6e-192)
		tmp = b / c;
	elseif (d <= 3.1e-73)
		tmp = t_1;
	elseif (d <= 1.05e-38)
		tmp = b / c;
	elseif (d <= 6.8e+22)
		tmp = t_1;
	elseif (d <= 2.9e+50)
		tmp = b / c;
	elseif (d <= 3.75e+65)
		tmp = t_0;
	elseif (d <= 1e+154)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
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_] := Block[{t$95$0 = (-N[(a / d), $MachinePrecision])}, Block[{t$95$1 = N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.95e+123], t$95$0, If[LessEqual[d, -1.32e-120], t$95$1, If[LessEqual[d, 6.6e-192], N[(b / c), $MachinePrecision], If[LessEqual[d, 3.1e-73], t$95$1, If[LessEqual[d, 1.05e-38], N[(b / c), $MachinePrecision], If[LessEqual[d, 6.8e+22], t$95$1, If[LessEqual[d, 2.9e+50], N[(b / c), $MachinePrecision], If[LessEqual[d, 3.75e+65], t$95$0, If[LessEqual[d, 1e+154], t$95$1, t$95$0]]]]]]]]]]]

Target

Original	26.3
Target	0.4
Herbie	17.1

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

Derivation

1. Split input into 3 regimes

2. if d < -1.94999999999999996e123 or 2.9e50 < d < 3.75000000000000003e65 or 1.00000000000000004e154 < d

   1. Initial program 42.3

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

   2. Taylor expanded in c around 0 15.4

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]

   3. Simplified 15.4

      \[\leadsto \color{blue}{-\frac{a}{d}} \]
      Proof

      [Start] 15.4	\[ -1 \cdot \frac{a}{d} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 15.4	\[ \color{blue}{\frac{a}{d} \cdot -1} \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 15.4	\[ \color{blue}{-\frac{a}{d}} \]

   if -1.94999999999999996e123 < d < -1.32000000000000004e-120 or 6.59999999999999978e-192 < d < 3.09999999999999969e-73 or 1.05000000000000006e-38 < d < 6.8e22 or 3.75000000000000003e65 < d < 1.00000000000000004e154

   1. Initial program 17.5

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

   if -1.32000000000000004e-120 < d < 6.59999999999999978e-192 or 3.09999999999999969e-73 < d < 1.05000000000000006e-38 or 6.8e22 < d < 2.9e50

   1. Initial program 22.6

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

   2. Taylor expanded in c around inf 18.4

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

4. Final simplification 17.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.95 \cdot 10^{+123}:\\ \;\;\;\;-\frac{a}{d}\\ \mathbf{elif}\;d \leq -1.32 \cdot 10^{-120}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 6.6 \cdot 10^{-192}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 3.1 \cdot 10^{-73}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-38}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{+22}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{+50}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 3.75 \cdot 10^{+65}:\\ \;\;\;\;-\frac{a}{d}\\ \mathbf{elif}\;d \leq 10^{+154}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;-\frac{a}{d}\\ \end{array} \]

Alternatives

Alternative 1
Error	22.9
Cost	520

\[\begin{array}{l} \mathbf{if}\;c \leq -2.45 \cdot 10^{+38}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 2.75 \cdot 10^{-33}:\\ \;\;\;\;-\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 2
Error	37.3
Cost	192

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

Reproduce

herbie shell --seed 2023090 
(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))))