Complex division, real part

Percentage Accurate: 62.3% → 82.2%

Time: 6.9s

Alternatives: 9

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: 62.3% 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: 82.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.66 \cdot 10^{+97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{b \cdot d + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{-159}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(d, d, c \cdot c\right)}{\mathsf{fma}\left(d, b, c \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.66e+97)
   (/ (fma (/ a d) c b) d)
   (if (<= d -3.5e-57)
     (/ (+ (* b d) (* c a)) (+ (* d d) (* c c)))
     (if (<= d 1.25e-159)
       (/ (fma (/ b c) d a) c)
       (if (<= d 7.5e+146)
         (/ 1.0 (/ (fma d d (* c c)) (fma d b (* c a))))
         (/ (fma a (/ c d) b) d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.66e+97) {
		tmp = fma((a / d), c, b) / d;
	} else if (d <= -3.5e-57) {
		tmp = ((b * d) + (c * a)) / ((d * d) + (c * c));
	} else if (d <= 1.25e-159) {
		tmp = fma((b / c), d, a) / c;
	} else if (d <= 7.5e+146) {
		tmp = 1.0 / (fma(d, d, (c * c)) / fma(d, b, (c * a)));
	} else {
		tmp = fma(a, (c / d), b) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.66e+97)
		tmp = Float64(fma(Float64(a / d), c, b) / d);
	elseif (d <= -3.5e-57)
		tmp = Float64(Float64(Float64(b * d) + Float64(c * a)) / Float64(Float64(d * d) + Float64(c * c)));
	elseif (d <= 1.25e-159)
		tmp = Float64(fma(Float64(b / c), d, a) / c);
	elseif (d <= 7.5e+146)
		tmp = Float64(1.0 / Float64(fma(d, d, Float64(c * c)) / fma(d, b, Float64(c * a))));
	else
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.66e+97], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -3.5e-57], N[(N[(N[(b * d), $MachinePrecision] + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.25e-159], N[(N[(N[(b / c), $MachinePrecision] * d + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 7.5e+146], N[(1.0 / N[(N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.6599999999999999e97

   1. Initial program 38.7%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 90.6

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

   5. Applied rewrites 90.6%

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

   if -1.6599999999999999e97 < d < -3.49999999999999991e-57

   1. Initial program 77.5%

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

   if -3.49999999999999991e-57 < d < 1.25000000000000008e-159

   1. Initial program 66.8%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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 94.1

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

   5. Applied rewrites 94.1%

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

   if 1.25000000000000008e-159 < d < 7.49999999999999983e146

   1. Initial program 84.8%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 84.8

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 84.8

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 84.8

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 84.8

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

   4. Applied rewrites 84.8%

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

   if 7.49999999999999983e146 < d

   1. Initial program 29.7%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 93.9

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

   5. Applied rewrites 93.9%

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

   7. Applied rewrites 93.9%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.66 \cdot 10^{+97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{b \cdot d + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{-159}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(d, d, c \cdot c\right)}{\mathsf{fma}\left(d, b, c \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot d + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{if}\;d \leq -1.66 \cdot 10^{+97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{-159}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+146}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* b d) (* c a)) (+ (* d d) (* c c)))))
   (if (<= d -1.66e+97)
     (/ (fma (/ a d) c b) d)
     (if (<= d -3.5e-57)
       t_0
       (if (<= d 1.25e-159)
         (/ (fma (/ b c) d a) c)
         (if (<= d 7.5e+146) t_0 (/ (fma a (/ c d) b) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * d) + (c * a)) / ((d * d) + (c * c));
	double tmp;
	if (d <= -1.66e+97) {
		tmp = fma((a / d), c, b) / d;
	} else if (d <= -3.5e-57) {
		tmp = t_0;
	} else if (d <= 1.25e-159) {
		tmp = fma((b / c), d, a) / c;
	} else if (d <= 7.5e+146) {
		tmp = t_0;
	} else {
		tmp = fma(a, (c / d), b) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * d) + Float64(c * a)) / Float64(Float64(d * d) + Float64(c * c)))
	tmp = 0.0
	if (d <= -1.66e+97)
		tmp = Float64(fma(Float64(a / d), c, b) / d);
	elseif (d <= -3.5e-57)
		tmp = t_0;
	elseif (d <= 1.25e-159)
		tmp = Float64(fma(Float64(b / c), d, a) / c);
	elseif (d <= 7.5e+146)
		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[(N[(b * d), $MachinePrecision] + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.66e+97], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -3.5e-57], t$95$0, If[LessEqual[d, 1.25e-159], N[(N[(N[(b / c), $MachinePrecision] * d + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 7.5e+146], t$95$0, N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.6599999999999999e97

   1. Initial program 38.7%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 90.6

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

   5. Applied rewrites 90.6%

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

   if -1.6599999999999999e97 < d < -3.49999999999999991e-57 or 1.25000000000000008e-159 < d < 7.49999999999999983e146

   1. Initial program 82.3%

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

   if -3.49999999999999991e-57 < d < 1.25000000000000008e-159

   1. Initial program 66.8%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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 94.1

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

   5. Applied rewrites 94.1%

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

   if 7.49999999999999983e146 < d

   1. Initial program 29.7%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 93.9

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

   5. Applied rewrites 93.9%

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

   7. Applied rewrites 93.9%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.66 \cdot 10^{+97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{b \cdot d + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{-159}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{b \cdot d + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{+188}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -5.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{a}{d} \cdot c}{d}\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-56}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot d\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{+100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -2.3e+188)
   (/ b d)
   (if (<= d -5.2e+102)
     (/ (* (/ a d) c) d)
     (if (<= d -3.5e-56)
       (* (/ b (fma d d (* c c))) d)
       (if (<= d 6.5e-52)
         (/ a c)
         (if (<= d 1.35e+100) (/ (fma d b (* c a)) (* d d)) (/ b d)))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2.3e+188) {
		tmp = b / d;
	} else if (d <= -5.2e+102) {
		tmp = ((a / d) * c) / d;
	} else if (d <= -3.5e-56) {
		tmp = (b / fma(d, d, (c * c))) * d;
	} else if (d <= 6.5e-52) {
		tmp = a / c;
	} else if (d <= 1.35e+100) {
		tmp = fma(d, b, (c * a)) / (d * d);
	} else {
		tmp = b / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -2.3e+188)
		tmp = Float64(b / d);
	elseif (d <= -5.2e+102)
		tmp = Float64(Float64(Float64(a / d) * c) / d);
	elseif (d <= -3.5e-56)
		tmp = Float64(Float64(b / fma(d, d, Float64(c * c))) * d);
	elseif (d <= 6.5e-52)
		tmp = Float64(a / c);
	elseif (d <= 1.35e+100)
		tmp = Float64(fma(d, b, Float64(c * a)) / Float64(d * d));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -2.3e+188], N[(b / d), $MachinePrecision], If[LessEqual[d, -5.2e+102], N[(N[(N[(a / d), $MachinePrecision] * c), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -3.5e-56], N[(N[(b / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision], If[LessEqual[d, 6.5e-52], N[(a / c), $MachinePrecision], If[LessEqual[d, 1.35e+100], N[(N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -2.30000000000000011e188 or 1.34999999999999999e100 < d

   1. Initial program 36.8%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 81.4

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

   5. Applied rewrites 81.4%

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

   if -2.30000000000000011e188 < d < -5.20000000000000013e102

   1. Initial program 58.3%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in a around inf

      \[\leadsto \frac{\frac{a \cdot c}{d}}{d} \]
   7. Step-by-step derivation

   8. Applied rewrites 78.8%

      \[\leadsto \frac{\frac{a \cdot c}{d}}{d} \]
   9. Step-by-step derivation

   10. Applied rewrites 79.1%

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

   if -5.20000000000000013e102 < d < -3.4999999999999998e-56

   1. Initial program 76.1%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 62.3

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

   5. Applied rewrites 62.3%

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

   if -3.4999999999999998e-56 < d < 6.5e-52

   1. Initial program 68.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 76.4

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

   5. Applied rewrites 76.4%

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

   if 6.5e-52 < d < 1.34999999999999999e100

   1. Initial program 91.7%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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 31.4

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

   5. Applied rewrites 31.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 31.4

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

   7. Applied rewrites 31.4%

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

   8. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 63.7

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

   10. Applied rewrites 63.7%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{+188}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -5.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{a}{d} \cdot c}{d}\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-56}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot d\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{+100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot d\\ \mathbf{if}\;d \leq -2.3 \cdot 10^{+188}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -5.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{a}{d} \cdot c}{d}\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-56}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{-48}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+104}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* (/ b (fma d d (* c c))) d)))
   (if (<= d -2.3e+188)
     (/ b d)
     (if (<= d -5.2e+102)
       (/ (* (/ a d) c) d)
       (if (<= d -3.5e-56)
         t_0
         (if (<= d 1.7e-48) (/ a c) (if (<= d 5e+104) t_0 (/ b d))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b / fma(d, d, (c * c))) * d;
	double tmp;
	if (d <= -2.3e+188) {
		tmp = b / d;
	} else if (d <= -5.2e+102) {
		tmp = ((a / d) * c) / d;
	} else if (d <= -3.5e-56) {
		tmp = t_0;
	} else if (d <= 1.7e-48) {
		tmp = a / c;
	} else if (d <= 5e+104) {
		tmp = t_0;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b / fma(d, d, Float64(c * c))) * d)
	tmp = 0.0
	if (d <= -2.3e+188)
		tmp = Float64(b / d);
	elseif (d <= -5.2e+102)
		tmp = Float64(Float64(Float64(a / d) * c) / d);
	elseif (d <= -3.5e-56)
		tmp = t_0;
	elseif (d <= 1.7e-48)
		tmp = Float64(a / c);
	elseif (d <= 5e+104)
		tmp = t_0;
	else
		tmp = Float64(b / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]}, If[LessEqual[d, -2.3e+188], N[(b / d), $MachinePrecision], If[LessEqual[d, -5.2e+102], N[(N[(N[(a / d), $MachinePrecision] * c), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -3.5e-56], t$95$0, If[LessEqual[d, 1.7e-48], N[(a / c), $MachinePrecision], If[LessEqual[d, 5e+104], t$95$0, N[(b / d), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.30000000000000011e188 or 4.9999999999999997e104 < d

   1. Initial program 37.3%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 82.6

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

   5. Applied rewrites 82.6%

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

   if -2.30000000000000011e188 < d < -5.20000000000000013e102

   1. Initial program 58.3%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in a around inf

      \[\leadsto \frac{\frac{a \cdot c}{d}}{d} \]
   7. Step-by-step derivation

   8. Applied rewrites 78.8%

      \[\leadsto \frac{\frac{a \cdot c}{d}}{d} \]
   9. Step-by-step derivation

   10. Applied rewrites 79.1%

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

   if -5.20000000000000013e102 < d < -3.4999999999999998e-56 or 1.70000000000000014e-48 < d < 4.9999999999999997e104

   1. Initial program 82.5%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 60.1

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

   5. Applied rewrites 60.1%

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

   if -3.4999999999999998e-56 < d < 1.70000000000000014e-48

   1. Initial program 68.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 75.8

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

   5. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\frac{a}{c}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 64.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot d\\ \mathbf{if}\;d \leq -2.3 \cdot 10^{+188}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -5.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{a}{d}}{d} \cdot c\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-56}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{-48}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+104}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* (/ b (fma d d (* c c))) d)))
   (if (<= d -2.3e+188)
     (/ b d)
     (if (<= d -5.2e+102)
       (* (/ (/ a d) d) c)
       (if (<= d -3.5e-56)
         t_0
         (if (<= d 1.7e-48) (/ a c) (if (<= d 5e+104) t_0 (/ b d))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b / fma(d, d, (c * c))) * d;
	double tmp;
	if (d <= -2.3e+188) {
		tmp = b / d;
	} else if (d <= -5.2e+102) {
		tmp = ((a / d) / d) * c;
	} else if (d <= -3.5e-56) {
		tmp = t_0;
	} else if (d <= 1.7e-48) {
		tmp = a / c;
	} else if (d <= 5e+104) {
		tmp = t_0;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b / fma(d, d, Float64(c * c))) * d)
	tmp = 0.0
	if (d <= -2.3e+188)
		tmp = Float64(b / d);
	elseif (d <= -5.2e+102)
		tmp = Float64(Float64(Float64(a / d) / d) * c);
	elseif (d <= -3.5e-56)
		tmp = t_0;
	elseif (d <= 1.7e-48)
		tmp = Float64(a / c);
	elseif (d <= 5e+104)
		tmp = t_0;
	else
		tmp = Float64(b / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]}, If[LessEqual[d, -2.3e+188], N[(b / d), $MachinePrecision], If[LessEqual[d, -5.2e+102], N[(N[(N[(a / d), $MachinePrecision] / d), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[d, -3.5e-56], t$95$0, If[LessEqual[d, 1.7e-48], N[(a / c), $MachinePrecision], If[LessEqual[d, 5e+104], t$95$0, N[(b / d), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.30000000000000011e188 or 4.9999999999999997e104 < d

   1. Initial program 37.3%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 82.6

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

   5. Applied rewrites 82.6%

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

   if -2.30000000000000011e188 < d < -5.20000000000000013e102

   1. Initial program 58.3%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 52.8

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

   5. Applied rewrites 52.8%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 66.0%

      \[\leadsto \frac{\frac{a}{d}}{d} \cdot c \]

   if -5.20000000000000013e102 < d < -3.4999999999999998e-56 or 1.70000000000000014e-48 < d < 4.9999999999999997e104

   1. Initial program 82.5%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 60.1

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

   5. Applied rewrites 60.1%

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

   if -3.4999999999999998e-56 < d < 1.70000000000000014e-48

   1. Initial program 68.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 75.8

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

   5. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\frac{a}{c}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 73.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.8 \cdot 10^{+93}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -2.6 \cdot 10^{-21}:\\ \;\;\;\;\frac{c \cdot a}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;c \leq 1.08 \cdot 10^{+83}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.8e+93)
   (/ a c)
   (if (<= c -2.6e-21)
     (/ (* c a) (fma d d (* c c)))
     (if (<= c 1.08e+83) (/ (fma a (/ c d) b) d) (/ a c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.8e+93) {
		tmp = a / c;
	} else if (c <= -2.6e-21) {
		tmp = (c * a) / fma(d, d, (c * c));
	} else if (c <= 1.08e+83) {
		tmp = fma(a, (c / d), b) / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.8e+93)
		tmp = Float64(a / c);
	elseif (c <= -2.6e-21)
		tmp = Float64(Float64(c * a) / fma(d, d, Float64(c * c)));
	elseif (c <= 1.08e+83)
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.8e+93], N[(a / c), $MachinePrecision], If[LessEqual[c, -2.6e-21], N[(N[(c * a), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.08e+83], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.8e93 or 1.08e83 < c

   1. Initial program 42.4%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 77.0

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

   5. Applied rewrites 77.0%

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

   if -1.8e93 < c < -2.60000000000000017e-21

   1. Initial program 80.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 63.8

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

   5. Applied rewrites 63.8%

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

   7. Applied rewrites 64.3%

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

   if -2.60000000000000017e-21 < c < 1.08e83

   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 d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 78.0

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 79.4%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.8 \cdot 10^{+93}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -2.6 \cdot 10^{-21}:\\ \;\;\;\;\frac{c \cdot a}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;c \leq 1.08 \cdot 10^{+83}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.6 \cdot 10^{-53}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{-52}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -3.6e-53)
   (/ (fma (/ a d) c b) d)
   (if (<= d 7.2e-52) (/ (fma (/ b c) d a) c) (/ (fma a (/ c d) b) d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -3.6e-53) {
		tmp = fma((a / d), c, b) / d;
	} else if (d <= 7.2e-52) {
		tmp = fma((b / c), d, a) / c;
	} else {
		tmp = fma(a, (c / d), b) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -3.6e-53)
		tmp = Float64(fma(Float64(a / d), c, b) / d);
	elseif (d <= 7.2e-52)
		tmp = Float64(fma(Float64(b / c), d, a) / c);
	else
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -3.6e-53], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 7.2e-52], N[(N[(N[(b / c), $MachinePrecision] * d + a), $MachinePrecision] / c), $MachinePrecision], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.5999999999999999e-53

   1. Initial program 56.3%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 74.4

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

   5. Applied rewrites 74.4%

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

   if -3.5999999999999999e-53 < d < 7.19999999999999976e-52

   1. Initial program 68.2%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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 91.3

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

   5. Applied rewrites 91.3%

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

   if 7.19999999999999976e-52 < d

   1. Initial program 65.4%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 74.8

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

   5. Applied rewrites 74.8%

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

   7. Applied rewrites 74.9%

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

Alternative 8: 63.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{-53}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-43}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -2e-53) (/ b d) (if (<= d 1.15e-43) (/ a c) (/ b d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2e-53) {
		tmp = b / d;
	} else if (d <= 1.15e-43) {
		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 (d <= (-2d-53)) then
        tmp = b / d
    else if (d <= 1.15d-43) 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 (d <= -2e-53) {
		tmp = b / d;
	} else if (d <= 1.15e-43) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -2e-53:
		tmp = b / d
	elif d <= 1.15e-43:
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -2e-53)
		tmp = Float64(b / d);
	elseif (d <= 1.15e-43)
		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 (d <= -2e-53)
		tmp = b / d;
	elseif (d <= 1.15e-43)
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -2e-53], N[(b / d), $MachinePrecision], If[LessEqual[d, 1.15e-43], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -2.00000000000000006e-53 or 1.1499999999999999e-43 < d

   1. Initial program 60.7%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 60.4

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

   5. Applied rewrites 60.4%

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

   if -2.00000000000000006e-53 < d < 1.1499999999999999e-43

   1. Initial program 68.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 75.8

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

   5. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\frac{a}{c}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 42.9% 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 63.9%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 43.0

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

5. Applied rewrites 43.0%

   \[\leadsto \color{blue}{\frac{a}{c}} \]
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 2024288 
(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))))