Complex division, real part

Percentage Accurate: 61.8% → 83.4%

Time: 7.2s

Alternatives: 11

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

double code(double a, double b, double c, double d) {
	return ((a * c) + (b * 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 = ((a * c) + (b * d)) / ((c * c) + (d * d))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Initial Program: 61.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c, double d) {
	return ((a * c) + (b * 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 = ((a * c) + (b * d)) / ((c * c) + (d * d))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Alternative 1: 83.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ \mathbf{if}\;d \leq -2.9 \cdot 10^{+112}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{elif}\;d \leq -6.2 \cdot 10^{-156}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{c}{t\_0}, d \cdot \frac{b}{t\_0}\right)\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{-154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{elif}\;d \leq 4.1 \cdot 10^{+94}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))))
   (if (<= d -2.9e+112)
     (/ (fma (/ a d) c b) d)
     (if (<= d -6.2e-156)
       (fma a (/ c t_0) (* d (/ b t_0)))
       (if (<= d 6.2e-154)
         (/ (fma (/ d c) b a) c)
         (if (<= d 4.1e+94)
           (/ (fma d b (* c a)) t_0)
           (/ (fma a (/ c d) b) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double tmp;
	if (d <= -2.9e+112) {
		tmp = fma((a / d), c, b) / d;
	} else if (d <= -6.2e-156) {
		tmp = fma(a, (c / t_0), (d * (b / t_0)));
	} else if (d <= 6.2e-154) {
		tmp = fma((d / c), b, a) / c;
	} else if (d <= 4.1e+94) {
		tmp = fma(d, b, (c * a)) / t_0;
	} else {
		tmp = fma(a, (c / d), b) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	tmp = 0.0
	if (d <= -2.9e+112)
		tmp = Float64(fma(Float64(a / d), c, b) / d);
	elseif (d <= -6.2e-156)
		tmp = fma(a, Float64(c / t_0), Float64(d * Float64(b / t_0)));
	elseif (d <= 6.2e-154)
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	elseif (d <= 4.1e+94)
		tmp = Float64(fma(d, b, Float64(c * a)) / t_0);
	else
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.9e+112], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -6.2e-156], N[(a * N[(c / t$95$0), $MachinePrecision] + N[(d * N[(b / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6.2e-154], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 4.1e+94], N[(N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -2.9000000000000002e112

   1. Initial program 32.2%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 80.2

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

   5. Applied rewrites 80.2%

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

   if -2.9000000000000002e112 < d < -6.1999999999999996e-156

   1. Initial program 81.9%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 87.7

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 87.7

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

   4. Applied rewrites 87.7%

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

   if -6.1999999999999996e-156 < d < 6.19999999999999963e-154

   1. Initial program 72.2%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 68.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 68.9

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

   4. Applied rewrites 68.9%

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

   5. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 92.9

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

   7. Applied rewrites 92.9%

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

   if 6.19999999999999963e-154 < d < 4.10000000000000031e94

   1. Initial program 80.9%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 80.9

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 80.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 80.9

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

   4. Applied rewrites 80.9%

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

   if 4.10000000000000031e94 < d

   1. Initial program 31.6%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 86.3

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

   5. Applied rewrites 86.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}} \]
   6. Step-by-step derivation

   7. Applied rewrites 86.5%

      \[\leadsto \frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.9 \cdot 10^{+112}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{elif}\;d \leq -6.2 \cdot 10^{-156}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}, d \cdot \frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{-154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{elif}\;d \leq 4.1 \cdot 10^{+94}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{d \cdot d}\\ \mathbf{if}\;d \leq -1.9 \cdot 10^{+152}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -2.65 \cdot 10^{-87}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-156}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot a\\ \mathbf{elif}\;d \leq 4.8 \cdot 10^{-115}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a, d \cdot b\right)}{c \cdot c}\\ \mathbf{elif}\;d \leq 1.06 \cdot 10^{+97}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma d b (* c a)) (* d d))))
   (if (<= d -1.9e+152)
     (/ b d)
     (if (<= d -2.65e-87)
       t_0
       (if (<= d -2.2e-156)
         (* (/ c (fma d d (* c c))) a)
         (if (<= d 4.8e-115)
           (/ (fma c a (* d b)) (* c c))
           (if (<= d 1.06e+97) t_0 (/ b d))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, b, (c * a)) / (d * d);
	double tmp;
	if (d <= -1.9e+152) {
		tmp = b / d;
	} else if (d <= -2.65e-87) {
		tmp = t_0;
	} else if (d <= -2.2e-156) {
		tmp = (c / fma(d, d, (c * c))) * a;
	} else if (d <= 4.8e-115) {
		tmp = fma(c, a, (d * b)) / (c * c);
	} else if (d <= 1.06e+97) {
		tmp = t_0;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(d, b, Float64(c * a)) / Float64(d * d))
	tmp = 0.0
	if (d <= -1.9e+152)
		tmp = Float64(b / d);
	elseif (d <= -2.65e-87)
		tmp = t_0;
	elseif (d <= -2.2e-156)
		tmp = Float64(Float64(c / fma(d, d, Float64(c * c))) * a);
	elseif (d <= 4.8e-115)
		tmp = Float64(fma(c, a, Float64(d * b)) / Float64(c * c));
	elseif (d <= 1.06e+97)
		tmp = t_0;
	else
		tmp = Float64(b / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.9e+152], N[(b / d), $MachinePrecision], If[LessEqual[d, -2.65e-87], t$95$0, If[LessEqual[d, -2.2e-156], N[(N[(c / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[d, 4.8e-115], N[(N[(c * a + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.06e+97], t$95$0, N[(b / d), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.9e152 or 1.05999999999999994e97 < d

   1. Initial program 27.0%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 66.3

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

   5. Applied rewrites 66.3%

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

   if -1.9e152 < d < -2.64999999999999993e-87 or 4.80000000000000042e-115 < d < 1.05999999999999994e97

   1. Initial program 80.5%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 80.5

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 80.5

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 80.5

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

   4. Applied rewrites 80.5%

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

   5. Taylor expanded in c around 0

      \[\leadsto \frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\color{blue}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 63.4

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

   7. Applied rewrites 63.4%

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

   if -2.64999999999999993e-87 < d < -2.1999999999999999e-156

   1. Initial program 72.3%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 92.4

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 92.4

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

   4. Applied rewrites 92.4%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 78.8

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

   7. Applied rewrites 78.8%

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

   if -2.1999999999999999e-156 < d < 4.80000000000000042e-115

   1. Initial program 74.7%

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

   3. Taylor expanded in c around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 72.2

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

   5. Applied rewrites 72.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 72.2

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 72.2

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

   7. Applied rewrites 72.2%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{+152}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -2.65 \cdot 10^{-87}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{d \cdot d}\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-156}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot a\\ \mathbf{elif}\;d \leq 4.8 \cdot 10^{-115}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a, d \cdot b\right)}{c \cdot c}\\ \mathbf{elif}\;d \leq 1.06 \cdot 10^{+97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ \mathbf{if}\;d \leq -3.6 \cdot 10^{+68}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{elif}\;d \leq -6.2 \cdot 10^{-156}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{a}, c\right)}{t\_0} \cdot a\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{-154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{elif}\;d \leq 4.1 \cdot 10^{+94}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))))
   (if (<= d -3.6e+68)
     (/ (fma (/ a d) c b) d)
     (if (<= d -6.2e-156)
       (* (/ (fma b (/ d a) c) t_0) a)
       (if (<= d 6.2e-154)
         (/ (fma (/ d c) b a) c)
         (if (<= d 4.1e+94)
           (/ (fma d b (* c a)) t_0)
           (/ (fma a (/ c d) b) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double tmp;
	if (d <= -3.6e+68) {
		tmp = fma((a / d), c, b) / d;
	} else if (d <= -6.2e-156) {
		tmp = (fma(b, (d / a), c) / t_0) * a;
	} else if (d <= 6.2e-154) {
		tmp = fma((d / c), b, a) / c;
	} else if (d <= 4.1e+94) {
		tmp = fma(d, b, (c * a)) / t_0;
	} else {
		tmp = fma(a, (c / d), b) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	tmp = 0.0
	if (d <= -3.6e+68)
		tmp = Float64(fma(Float64(a / d), c, b) / d);
	elseif (d <= -6.2e-156)
		tmp = Float64(Float64(fma(b, Float64(d / a), c) / t_0) * a);
	elseif (d <= 6.2e-154)
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	elseif (d <= 4.1e+94)
		tmp = Float64(fma(d, b, Float64(c * a)) / t_0);
	else
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.6e+68], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -6.2e-156], N[(N[(N[(b * N[(d / a), $MachinePrecision] + c), $MachinePrecision] / t$95$0), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[d, 6.2e-154], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 4.1e+94], N[(N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -3.5999999999999999e68

   1. Initial program 41.5%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 79.3

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

   5. Applied rewrites 79.3%

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

   if -3.5999999999999999e68 < d < -6.1999999999999996e-156

   1. Initial program 83.8%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 88.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 88.9

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

   4. Applied rewrites 88.9%

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

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(\frac{c}{{c}^{2} + {d}^{2}} + \frac{b \cdot d}{a \cdot \left({c}^{2} + {d}^{2}\right)}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 89.1

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

   7. Applied rewrites 89.1%

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

   if -6.1999999999999996e-156 < d < 6.19999999999999963e-154

   1. Initial program 72.2%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 68.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 68.9

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

   4. Applied rewrites 68.9%

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

   5. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 92.9

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

   7. Applied rewrites 92.9%

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

   if 6.19999999999999963e-154 < d < 4.10000000000000031e94

   1. Initial program 80.9%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 80.9

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 80.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 80.9

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

   4. Applied rewrites 80.9%

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

   if 4.10000000000000031e94 < d

   1. Initial program 31.6%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 86.3

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

   5. Applied rewrites 86.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}} \]
   6. Step-by-step derivation

   7. Applied rewrites 86.5%

      \[\leadsto \frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.6 \cdot 10^{+68}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{elif}\;d \leq -6.2 \cdot 10^{-156}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{a}, c\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot a\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{-154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{elif}\;d \leq 4.1 \cdot 10^{+94}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{if}\;d \leq -9.5 \cdot 10^{+75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{elif}\;d \leq -1.9 \cdot 10^{-132}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{-154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{elif}\;d \leq 4.1 \cdot 10^{+94}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma d b (* c a)) (fma d d (* c c)))))
   (if (<= d -9.5e+75)
     (/ (fma (/ a d) c b) d)
     (if (<= d -1.9e-132)
       t_0
       (if (<= d 6.2e-154)
         (/ (fma (/ d c) b a) c)
         (if (<= d 4.1e+94) t_0 (/ (fma a (/ c d) b) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, b, (c * a)) / fma(d, d, (c * c));
	double tmp;
	if (d <= -9.5e+75) {
		tmp = fma((a / d), c, b) / d;
	} else if (d <= -1.9e-132) {
		tmp = t_0;
	} else if (d <= 6.2e-154) {
		tmp = fma((d / c), b, a) / c;
	} else if (d <= 4.1e+94) {
		tmp = t_0;
	} else {
		tmp = fma(a, (c / d), b) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(d, b, Float64(c * a)) / fma(d, d, Float64(c * c)))
	tmp = 0.0
	if (d <= -9.5e+75)
		tmp = Float64(fma(Float64(a / d), c, b) / d);
	elseif (d <= -1.9e-132)
		tmp = t_0;
	elseif (d <= 6.2e-154)
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	elseif (d <= 4.1e+94)
		tmp = t_0;
	else
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -9.5e+75], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -1.9e-132], t$95$0, If[LessEqual[d, 6.2e-154], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 4.1e+94], t$95$0, N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -9.50000000000000061e75

   1. Initial program 40.5%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 78.9

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

   5. Applied rewrites 78.9%

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

   if -9.50000000000000061e75 < d < -1.8999999999999998e-132 or 6.19999999999999963e-154 < d < 4.10000000000000031e94

   1. Initial program 83.7%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 83.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 83.7

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 83.7

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

   4. Applied rewrites 83.7%

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

   if -1.8999999999999998e-132 < d < 6.19999999999999963e-154

   1. Initial program 71.0%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 70.7

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 70.7

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

   4. Applied rewrites 70.7%

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

   5. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 90.6

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

   7. Applied rewrites 90.6%

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

   if 4.10000000000000031e94 < d

   1. Initial program 31.6%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 86.3

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

   5. Applied rewrites 86.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}} \]
   6. Step-by-step derivation

   7. Applied rewrites 86.5%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.5 \cdot 10^{+75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{elif}\;d \leq -1.9 \cdot 10^{-132}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{-154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{elif}\;d \leq 4.1 \cdot 10^{+94}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ \mathbf{if}\;d \leq -1.1 \cdot 10^{-25}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-156}:\\ \;\;\;\;\frac{c}{t\_0} \cdot a\\ \mathbf{elif}\;d \leq 1.16 \cdot 10^{-104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a, d \cdot b\right)}{c \cdot c}\\ \mathbf{elif}\;d \leq 6.9 \cdot 10^{+137}:\\ \;\;\;\;\frac{d}{t\_0} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))))
   (if (<= d -1.1e-25)
     (/ b d)
     (if (<= d -2.2e-156)
       (* (/ c t_0) a)
       (if (<= d 1.16e-104)
         (/ (fma c a (* d b)) (* c c))
         (if (<= d 6.9e+137) (* (/ d t_0) b) (/ b d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double tmp;
	if (d <= -1.1e-25) {
		tmp = b / d;
	} else if (d <= -2.2e-156) {
		tmp = (c / t_0) * a;
	} else if (d <= 1.16e-104) {
		tmp = fma(c, a, (d * b)) / (c * c);
	} else if (d <= 6.9e+137) {
		tmp = (d / t_0) * b;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	tmp = 0.0
	if (d <= -1.1e-25)
		tmp = Float64(b / d);
	elseif (d <= -2.2e-156)
		tmp = Float64(Float64(c / t_0) * a);
	elseif (d <= 1.16e-104)
		tmp = Float64(fma(c, a, Float64(d * b)) / Float64(c * c));
	elseif (d <= 6.9e+137)
		tmp = Float64(Float64(d / t_0) * b);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.1e-25], N[(b / d), $MachinePrecision], If[LessEqual[d, -2.2e-156], N[(N[(c / t$95$0), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[d, 1.16e-104], N[(N[(c * a + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6.9e+137], N[(N[(d / t$95$0), $MachinePrecision] * b), $MachinePrecision], N[(b / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.1000000000000001e-25 or 6.90000000000000039e137 < d

   1. Initial program 45.0%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 62.2

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

   5. Applied rewrites 62.2%

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

   if -1.1000000000000001e-25 < d < -2.1999999999999999e-156

   1. Initial program 79.5%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 94.4

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 94.4

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

   4. Applied rewrites 94.4%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 74.2

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

   7. Applied rewrites 74.2%

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

   if -2.1999999999999999e-156 < d < 1.16000000000000001e-104

   1. Initial program 75.3%

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

   3. Taylor expanded in c around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 70.5

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

   5. Applied rewrites 70.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 70.5

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 70.5

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

   7. Applied rewrites 70.5%

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

   if 1.16000000000000001e-104 < d < 6.90000000000000039e137

   1. Initial program 71.4%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 74.0

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 74.0

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

   4. Applied rewrites 74.0%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 54.0

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

   7. Applied rewrites 54.0%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{-25}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-156}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot a\\ \mathbf{elif}\;d \leq 1.16 \cdot 10^{-104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a, d \cdot b\right)}{c \cdot c}\\ \mathbf{elif}\;d \leq 6.9 \cdot 10^{+137}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{+152}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -6.8 \cdot 10^{-46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{d \cdot d}\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{+63}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.9e+152)
   (/ b d)
   (if (<= d -6.8e-46)
     (/ (fma d b (* c a)) (* d d))
     (if (<= d 1.55e+63) (/ (fma (/ d c) b a) c) (/ b d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.9e+152) {
		tmp = b / d;
	} else if (d <= -6.8e-46) {
		tmp = fma(d, b, (c * a)) / (d * d);
	} else if (d <= 1.55e+63) {
		tmp = fma((d / c), b, a) / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.9e+152)
		tmp = Float64(b / d);
	elseif (d <= -6.8e-46)
		tmp = Float64(fma(d, b, Float64(c * a)) / Float64(d * d));
	elseif (d <= 1.55e+63)
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.9e+152], N[(b / d), $MachinePrecision], If[LessEqual[d, -6.8e-46], N[(N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.55e+63], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.9e152 or 1.55e63 < d

   1. Initial program 32.2%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 65.4

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

   5. Applied rewrites 65.4%

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

   if -1.9e152 < d < -6.79999999999999992e-46

   1. Initial program 81.4%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 81.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 81.4

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 81.4

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

   4. Applied rewrites 81.4%

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

   5. Taylor expanded in c around 0

      \[\leadsto \frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\color{blue}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 69.9

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

   7. Applied rewrites 69.9%

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

   if -6.79999999999999992e-46 < d < 1.55e63

   1. Initial program 74.9%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 74.2

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 74.2

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

   4. Applied rewrites 74.2%

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

   5. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 77.4

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

   7. Applied rewrites 77.4%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{+152}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -6.8 \cdot 10^{-46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{d \cdot d}\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{+63}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -6.8 \cdot 10^{-46} \lor \neg \left(d \leq 4.5 \cdot 10^{+62}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -6.8e-46) (not (<= d 4.5e+62)))
   (/ (fma a (/ c d) b) d)
   (/ (fma (/ d c) b a) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -6.8e-46) || !(d <= 4.5e+62)) {
		tmp = fma(a, (c / d), b) / d;
	} else {
		tmp = fma((d / c), b, a) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -6.8e-46) || !(d <= 4.5e+62))
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	else
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -6.8e-46], N[Not[LessEqual[d, 4.5e+62]], $MachinePrecision]], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -6.79999999999999992e-46 or 4.49999999999999999e62 < d

   1. Initial program 49.4%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 79.6

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

   5. Applied rewrites 79.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}} \]
   6. Step-by-step derivation

   7. Applied rewrites 80.3%

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

   if -6.79999999999999992e-46 < d < 4.49999999999999999e62

   1. Initial program 74.9%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 74.2

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 74.2

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

   4. Applied rewrites 74.2%

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

   5. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 77.4

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

   7. Applied rewrites 77.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.8 \cdot 10^{-46} \lor \neg \left(d \leq 4.5 \cdot 10^{+62}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.65 \cdot 10^{-22}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{elif}\;c \leq 9500:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.65e-22)
   (/ (fma (/ d c) b a) c)
   (if (<= c 9500.0) (/ (fma a (/ c d) b) d) (/ (fma (/ b c) d a) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.65e-22) {
		tmp = fma((d / c), b, a) / c;
	} else if (c <= 9500.0) {
		tmp = fma(a, (c / d), b) / d;
	} else {
		tmp = fma((b / c), d, a) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.65e-22)
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	elseif (c <= 9500.0)
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	else
		tmp = Float64(fma(Float64(b / c), d, a) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.65e-22], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 9500.0], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], N[(N[(N[(b / c), $MachinePrecision] * d + a), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.65e-22

   1. Initial program 51.2%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 56.6

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 56.6

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

   4. Applied rewrites 56.6%

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

   5. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 74.4

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

   7. Applied rewrites 74.4%

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

   if -1.65e-22 < c < 9500

   1. Initial program 73.7%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 80.1

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

   5. Applied rewrites 80.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}} \]
   6. Step-by-step derivation

   7. Applied rewrites 84.2%

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

   if 9500 < c

   1. Initial program 47.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 72.6

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

   5. Applied rewrites 72.6%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.65 \cdot 10^{-22}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{elif}\;c \leq 9500:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{-25}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -1.6 \cdot 10^{-155}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot a\\ \mathbf{elif}\;d \leq 1.32 \cdot 10^{+19}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.1e-25)
   (/ b d)
   (if (<= d -1.6e-155)
     (* (/ c (fma d d (* c c))) a)
     (if (<= d 1.32e+19) (/ a c) (/ b d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.1e-25) {
		tmp = b / d;
	} else if (d <= -1.6e-155) {
		tmp = (c / fma(d, d, (c * c))) * a;
	} else if (d <= 1.32e+19) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.1e-25)
		tmp = Float64(b / d);
	elseif (d <= -1.6e-155)
		tmp = Float64(Float64(c / fma(d, d, Float64(c * c))) * a);
	elseif (d <= 1.32e+19)
		tmp = Float64(a / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.1e-25], N[(b / d), $MachinePrecision], If[LessEqual[d, -1.6e-155], N[(N[(c / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[d, 1.32e+19], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.1000000000000001e-25 or 1.32e19 < d

   1. Initial program 49.1%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 60.3

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

   5. Applied rewrites 60.3%

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

   if -1.1000000000000001e-25 < d < -1.60000000000000006e-155

   1. Initial program 79.5%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 94.4

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 94.4

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

   4. Applied rewrites 94.4%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 74.2

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

   7. Applied rewrites 74.2%

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

   if -1.60000000000000006e-155 < d < 1.32e19

   1. Initial program 75.7%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 62.6

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

   5. Applied rewrites 62.6%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{-25}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -1.6 \cdot 10^{-155}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot a\\ \mathbf{elif}\;d \leq 1.32 \cdot 10^{+19}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2 \cdot 10^{-32} \lor \neg \left(c \leq 3.1 \cdot 10^{-40}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2e-32) (not (<= c 3.1e-40))) (/ a c) (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2e-32) || !(c <= 3.1e-40)) {
		tmp = a / c;
	} 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
    real(8) :: tmp
    if ((c <= (-2d-32)) .or. (.not. (c <= 3.1d-40))) then
        tmp = a / c
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2e-32) || !(c <= 3.1e-40)) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -2e-32) or not (c <= 3.1e-40):
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2e-32) || !(c <= 3.1e-40))
		tmp = Float64(a / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -2e-32) || ~((c <= 3.1e-40)))
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2e-32], N[Not[LessEqual[c, 3.1e-40]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.00000000000000011e-32 or 3.10000000000000011e-40 < c

   1. Initial program 52.7%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 57.4

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

   5. Applied rewrites 57.4%

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

   if -2.00000000000000011e-32 < c < 3.10000000000000011e-40

   1. Initial program 72.8%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 61.7

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

   5. Applied rewrites 61.7%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2 \cdot 10^{-32} \lor \neg \left(c \leq 3.1 \cdot 10^{-40}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{a}{c} \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (/ a c))

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_, d_] := N[(a / c), $MachinePrecision]

Derivation

1. Initial program 62.1%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 38.4

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

5. Applied rewrites 38.4%

   \[\leadsto \color{blue}{\frac{a}{c}} \]
6. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (< (fabs d) (fabs c))
   (/ (+ a (* b (/ d c))) (+ c (* d (/ d c))))
   (/ (+ b (* a (/ 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 = (a + (b * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (b + (a * (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 = (a + (b * (d / c))) / (c + (d * (d / c)))
    else
        tmp = (b + (a * (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 = (a + (b * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (b + (a * (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 = (a + (b * (d / c))) / (c + (d * (d / c)))
	else:
		tmp = (b + (a * (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(a + Float64(b * Float64(d / c))) / Float64(c + Float64(d * Float64(d / c))));
	else
		tmp = Float64(Float64(b + Float64(a * 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 = (a + (b * (d / c))) / (c + (d * (d / c)));
	else
		tmp = (b + (a * (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[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c + N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d + N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024332 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

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