Complex division, imag part

Percentage Accurate: 61.9% → 79.9%

Time: 9.1s

Alternatives: 9

Speedup: 1.5×

Specification

Math

\[\begin{array}{l} \\ \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \end{array} \]

FPCore

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

C

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

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

Java

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

Python

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

Julia

function code(a, b, c, d)
	return Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
end

MATLAB

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

Initial Program: 61.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \end{array} \]

FPCore

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

C

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

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

Java

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

Python

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

Julia

function code(a, b, c, d)
	return Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
end

MATLAB

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

Alternative 1: 79.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.05 \cdot 10^{+46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-d}{c}, a, b\right)}{c}\\ \mathbf{elif}\;c \leq -5.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{-26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, \frac{a}{c}, b\right)}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.05e+46)
   (/ (fma (/ (- d) c) a b) c)
   (if (<= c -5.2e-76)
     (/ (- (* b c) (* d a)) (+ (* d d) (* c c)))
     (if (<= c 3.4e-26)
       (/ (fma b (/ c d) (- a)) d)
       (/ (fma (- d) (/ a c) b) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.05e+46) {
		tmp = fma((-d / c), a, b) / c;
	} else if (c <= -5.2e-76) {
		tmp = ((b * c) - (d * a)) / ((d * d) + (c * c));
	} else if (c <= 3.4e-26) {
		tmp = fma(b, (c / d), -a) / d;
	} else {
		tmp = fma(-d, (a / c), b) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.05e+46)
		tmp = Float64(fma(Float64(Float64(-d) / c), a, b) / c);
	elseif (c <= -5.2e-76)
		tmp = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(Float64(d * d) + Float64(c * c)));
	elseif (c <= 3.4e-26)
		tmp = Float64(fma(b, Float64(c / d), Float64(-a)) / d);
	else
		tmp = Float64(fma(Float64(-d), Float64(a / c), b) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.05e+46], N[(N[(N[((-d) / c), $MachinePrecision] * a + b), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -5.2e-76], N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 3.4e-26], N[(N[(b * N[(c / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision], N[(N[((-d) * N[(a / c), $MachinePrecision] + b), $MachinePrecision] / c), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.05e46

   1. Initial program 49.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower-fma.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{-1 \cdot d}}{c}, a, b\right)}{c} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1 \cdot d}{c}}, a, b\right)}{c} \]

      11. mul-1-neg N/A

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

      12. lower-neg.f64 87.9

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

   5. Applied rewrites 87.9%

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

   if -1.05e46 < c < -5.1999999999999999e-76

   1. Initial program 88.7%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   if -5.1999999999999999e-76 < c < 3.40000000000000013e-26

   1. Initial program 64.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower-fma.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{-1 \cdot d}}{c}, a, b\right)}{c} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1 \cdot d}{c}}, a, b\right)}{c} \]

      11. mul-1-neg N/A

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

      12. lower-neg.f64 31.2

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

   5. Applied rewrites 31.2%

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

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{b \cdot c}{{d}^{2}} + -1 \cdot \frac{a}{d}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{b \cdot c}{{d}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{a}{d}\right)\right)} \]

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}}{d} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(b, \color{blue}{\frac{c}{d}}, -1 \cdot a\right)}{d} \]

      15. mul-1-neg N/A

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

      16. lower-neg.f64 83.7

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

   8. Applied rewrites 83.7%

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

   if 3.40000000000000013e-26 < c

   1. Initial program 43.6%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower-fma.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{-1 \cdot d}}{c}, a, b\right)}{c} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1 \cdot d}{c}}, a, b\right)}{c} \]

      11. mul-1-neg N/A

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

      12. lower-neg.f64 85.0

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1 \cdot d, \frac{a}{c}, b\right)}}{c} \]

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-/.f64 85.0

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

   8. Applied rewrites 85.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-d, \frac{a}{c}, b\right)}{c}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.05 \cdot 10^{+46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-d}{c}, a, b\right)}{c}\\ \mathbf{elif}\;c \leq -5.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{-26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, \frac{a}{c}, b\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.2 \cdot 10^{+156}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -3.4 \cdot 10^{-151}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{elif}\;c \leq 1.8 \cdot 10^{-26}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 3.2 \cdot 10^{+93}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3.2e+156)
   (/ b c)
   (if (<= c -3.4e-151)
     (* (/ c (fma d d (* c c))) b)
     (if (<= c 1.8e-26)
       (/ (- a) d)
       (if (<= c 3.2e+93) (/ (- (* b c) (* d a)) (* c c)) (/ b c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.2e+156) {
		tmp = b / c;
	} else if (c <= -3.4e-151) {
		tmp = (c / fma(d, d, (c * c))) * b;
	} else if (c <= 1.8e-26) {
		tmp = -a / d;
	} else if (c <= 3.2e+93) {
		tmp = ((b * c) - (d * a)) / (c * c);
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3.2e+156)
		tmp = Float64(b / c);
	elseif (c <= -3.4e-151)
		tmp = Float64(Float64(c / fma(d, d, Float64(c * c))) * b);
	elseif (c <= 1.8e-26)
		tmp = Float64(Float64(-a) / d);
	elseif (c <= 3.2e+93)
		tmp = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(c * c));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3.2e+156], N[(b / c), $MachinePrecision], If[LessEqual[c, -3.4e-151], N[(N[(c / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[c, 1.8e-26], N[((-a) / d), $MachinePrecision], If[LessEqual[c, 3.2e+93], N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(b / c), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -3.20000000000000002e156 or 3.2000000000000001e93 < c

   1. Initial program 35.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b}{c}} \]
   4. Step-by-step derivation

      1. lower-/.f64 75.4

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

   5. Applied rewrites 75.4%

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

   if -3.20000000000000002e156 < c < -3.4000000000000003e-151

   1. Initial program 73.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower-fma.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{-1 \cdot d}}{c}, a, b\right)}{c} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1 \cdot d}{c}}, a, b\right)}{c} \]

      11. mul-1-neg N/A

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

      12. lower-neg.f64 52.2

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b \cdot c}{{c}^{2} + {d}^{2}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 54.5

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

   8. Applied rewrites 54.5%

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

   if -3.4000000000000003e-151 < c < 1.8000000000000001e-26

   1. Initial program 73.1%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(a\right)}}{d} \]

      4. lower-neg.f64 69.2

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

   5. Applied rewrites 69.2%

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

   if 1.8000000000000001e-26 < c < 3.2000000000000001e93

   1. Initial program 75.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{c}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 52.4

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

   5. Applied rewrites 52.4%

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{c \cdot c}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.2 \cdot 10^{+156}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -3.4 \cdot 10^{-151}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{elif}\;c \leq 1.8 \cdot 10^{-26}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 3.2 \cdot 10^{+93}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 99.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (fabs(d) < fabs(c)) {
		tmp = (b - (a * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (-a + (b * (c / d))) / (d + (c * (c / 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
    real(8) :: tmp
    if (abs(d) < abs(c)) then
        tmp = (b - (a * (d / c))) / (c + (d * (d / c)))
    else
        tmp = (-a + (b * (c / d))) / (d + (c * (c / d)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (Math.abs(d) < Math.abs(c)) {
		tmp = (b - (a * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (-a + (b * (c / d))) / (d + (c * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if math.fabs(d) < math.fabs(c):
		tmp = (b - (a * (d / c))) / (c + (d * (d / c)))
	else:
		tmp = (-a + (b * (c / d))) / (d + (c * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (abs(d) < abs(c))
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / Float64(c + Float64(d * Float64(d / c))));
	else
		tmp = Float64(Float64(Float64(-a) + Float64(b * Float64(c / d))) / Float64(d + Float64(c * Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (abs(d) < abs(c))
		tmp = (b - (a * (d / c))) / (c + (d * (d / c)));
	else
		tmp = (-a + (b * (c / d))) / (d + (c * (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Less[N[Abs[d], $MachinePrecision], N[Abs[c], $MachinePrecision]], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c + N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-a) + N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d + N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024230 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :alt
  (! :herbie-platform default (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))))