Complex division, real part

Percentage Accurate: 61.6% → 78.9%

Time: 8.3s

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: 61.6% 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: 78.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(d, \frac{b}{c}, a\right)}{c}\\ \mathbf{if}\;c \leq -3.7 \cdot 10^{+59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.85 \cdot 10^{-125}:\\ \;\;\;\;\frac{b + \frac{\mathsf{fma}\left(a, c, \frac{c \cdot \left(c \cdot b\right)}{-d}\right)}{d}}{d}\\ \mathbf{elif}\;c \leq 6.2 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma d (/ b c) a) c)))
   (if (<= c -3.7e+59)
     t_0
     (if (<= c 1.85e-125)
       (/ (+ b (/ (fma a c (/ (* c (* c b)) (- d))) d)) d)
       (if (<= c 6.2e+48)
         (* (fma a c (* b d)) (/ 1.0 (fma c c (* d d))))
         t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, (b / c), a) / c;
	double tmp;
	if (c <= -3.7e+59) {
		tmp = t_0;
	} else if (c <= 1.85e-125) {
		tmp = (b + (fma(a, c, ((c * (c * b)) / -d)) / d)) / d;
	} else if (c <= 6.2e+48) {
		tmp = fma(a, c, (b * d)) * (1.0 / fma(c, c, (d * d)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(d, Float64(b / c), a) / c)
	tmp = 0.0
	if (c <= -3.7e+59)
		tmp = t_0;
	elseif (c <= 1.85e-125)
		tmp = Float64(Float64(b + Float64(fma(a, c, Float64(Float64(c * Float64(c * b)) / Float64(-d))) / d)) / d);
	elseif (c <= 6.2e+48)
		tmp = Float64(fma(a, c, Float64(b * d)) * Float64(1.0 / fma(c, c, Float64(d * d))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(d * N[(b / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -3.7e+59], t$95$0, If[LessEqual[c, 1.85e-125], N[(N[(b + N[(N[(a * c + N[(N[(c * N[(c * b), $MachinePrecision]), $MachinePrecision] / (-d)), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 6.2e+48], N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.69999999999999997e59 or 6.20000000000000011e48 < c

   1. Initial program 44.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 75.4

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   7. 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. lower-fma.f64 N/A

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

      6. lower-/.f64 83.3

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

   8. Applied rewrites 83.3%

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

   if -3.69999999999999997e59 < c < 1.85e-125

   1. Initial program 65.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 18.9

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

   5. Applied rewrites 18.9%

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

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   7. 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. lower-fma.f64 N/A

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

      6. lower-/.f64 24.6

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

   8. Applied rewrites 24.6%

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

   10. Applied rewrites 25.4%

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

   11. Taylor expanded in d around inf

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

       1. +-commutative N/A

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

       2. mul-1-neg N/A

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

       3. unsub-neg N/A

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

       4. unpow2 N/A

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

       5. associate-/r* N/A

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

       6. div-sub N/A

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

       7. unsub-neg N/A

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

       8. mul-1-neg N/A

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

       9. +-commutative N/A

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

       10. lower-/.f64 N/A

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

   13. Applied rewrites 85.0%

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

   if 1.85e-125 < c < 6.20000000000000011e48

   1. Initial program 84.3%

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

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. clear-num N/A

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

      11. flip-+ N/A

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

      12. lift-+.f64 N/A

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

      13. lower-/.f64 84.3

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 84.4

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

   4. Applied rewrites 84.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{\mathsf{fma}\left(c, c, d \cdot d\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.7 \cdot 10^{+59}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, \frac{b}{c}, a\right)}{c}\\ \mathbf{elif}\;c \leq 1.85 \cdot 10^{-125}:\\ \;\;\;\;\frac{b + \frac{\mathsf{fma}\left(a, c, \frac{c \cdot \left(c \cdot b\right)}{-d}\right)}{d}}{d}\\ \mathbf{elif}\;c \leq 6.2 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, \frac{b}{c}, a\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(d, \frac{b}{c}, a\right)}{c}\\ \mathbf{if}\;c \leq -1.25 \cdot 10^{+60}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.85 \cdot 10^{-125}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;c \leq 6.2 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma d (/ b c) a) c)))
   (if (<= c -1.25e+60)
     t_0
     (if (<= c 1.85e-125)
       (/ (fma a (/ c d) b) d)
       (if (<= c 6.2e+48)
         (* (fma a c (* b d)) (/ 1.0 (fma c c (* d d))))
         t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, (b / c), a) / c;
	double tmp;
	if (c <= -1.25e+60) {
		tmp = t_0;
	} else if (c <= 1.85e-125) {
		tmp = fma(a, (c / d), b) / d;
	} else if (c <= 6.2e+48) {
		tmp = fma(a, c, (b * d)) * (1.0 / fma(c, c, (d * d)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(d, Float64(b / c), a) / c)
	tmp = 0.0
	if (c <= -1.25e+60)
		tmp = t_0;
	elseif (c <= 1.85e-125)
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	elseif (c <= 6.2e+48)
		tmp = Float64(fma(a, c, Float64(b * d)) * Float64(1.0 / fma(c, c, Float64(d * d))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(d * N[(b / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -1.25e+60], t$95$0, If[LessEqual[c, 1.85e-125], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 6.2e+48], N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.24999999999999994e60 or 6.20000000000000011e48 < c

   1. Initial program 44.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 75.4

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   7. 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. lower-fma.f64 N/A

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

      6. lower-/.f64 83.3

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

   8. Applied rewrites 83.3%

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

   if -1.24999999999999994e60 < c < 1.85e-125

   1. Initial program 65.0%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 84.4

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

   5. Applied rewrites 84.4%

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

   if 1.85e-125 < c < 6.20000000000000011e48

   1. Initial program 84.3%

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

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. clear-num N/A

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

      11. flip-+ N/A

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

      12. lift-+.f64 N/A

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

      13. lower-/.f64 84.3

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 84.4

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

   4. Applied rewrites 84.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{\mathsf{fma}\left(c, c, d \cdot d\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 80.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(d, \frac{b}{c}, a\right)}{c}\\ \mathbf{if}\;c \leq -1.25 \cdot 10^{+60}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.85 \cdot 10^{-125}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;c \leq 6.2 \cdot 10^{+48}:\\ \;\;\;\;\frac{b \cdot d + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma d (/ b c) a) c)))
   (if (<= c -1.25e+60)
     t_0
     (if (<= c 1.85e-125)
       (/ (fma a (/ c d) b) d)
       (if (<= c 6.2e+48) (/ (+ (* b d) (* c a)) (+ (* d d) (* c c))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, (b / c), a) / c;
	double tmp;
	if (c <= -1.25e+60) {
		tmp = t_0;
	} else if (c <= 1.85e-125) {
		tmp = fma(a, (c / d), b) / d;
	} else if (c <= 6.2e+48) {
		tmp = ((b * d) + (c * a)) / ((d * d) + (c * c));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(d, Float64(b / c), a) / c)
	tmp = 0.0
	if (c <= -1.25e+60)
		tmp = t_0;
	elseif (c <= 1.85e-125)
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	elseif (c <= 6.2e+48)
		tmp = Float64(Float64(Float64(b * d) + Float64(c * a)) / Float64(Float64(d * d) + Float64(c * c)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(d * N[(b / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -1.25e+60], t$95$0, If[LessEqual[c, 1.85e-125], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 6.2e+48], N[(N[(N[(b * d), $MachinePrecision] + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.24999999999999994e60 or 6.20000000000000011e48 < c

   1. Initial program 44.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 75.4

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   7. 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. lower-fma.f64 N/A

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

      6. lower-/.f64 83.3

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

   8. Applied rewrites 83.3%

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

   if -1.24999999999999994e60 < c < 1.85e-125

   1. Initial program 65.0%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 84.4

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

   5. Applied rewrites 84.4%

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

   if 1.85e-125 < c < 6.20000000000000011e48

   1. Initial program 84.3%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.25 \cdot 10^{+60}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, \frac{b}{c}, a\right)}{c}\\ \mathbf{elif}\;c \leq 1.85 \cdot 10^{-125}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;c \leq 6.2 \cdot 10^{+48}:\\ \;\;\;\;\frac{b \cdot d + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, \frac{b}{c}, a\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.25 \cdot 10^{+60}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, \frac{b}{c}, a\right)}{c}\\ \mathbf{elif}\;c \leq 3.1 \cdot 10^{-52}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{+118}:\\ \;\;\;\;\frac{a}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.25e+60)
   (/ (fma d (/ b c) a) c)
   (if (<= c 3.1e-52)
     (/ (fma a (/ c d) b) d)
     (if (<= c 4.5e+118)
       (/ a (/ (fma c c (* d d)) c))
       (/ (fma b (/ d c) a) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.25e+60) {
		tmp = fma(d, (b / c), a) / c;
	} else if (c <= 3.1e-52) {
		tmp = fma(a, (c / d), b) / d;
	} else if (c <= 4.5e+118) {
		tmp = a / (fma(c, c, (d * d)) / c);
	} else {
		tmp = fma(b, (d / c), a) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.25e+60)
		tmp = Float64(fma(d, Float64(b / c), a) / c);
	elseif (c <= 3.1e-52)
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	elseif (c <= 4.5e+118)
		tmp = Float64(a / Float64(fma(c, c, Float64(d * d)) / c));
	else
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.25e+60], N[(N[(d * N[(b / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 3.1e-52], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 4.5e+118], N[(a / N[(N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.24999999999999994e60

   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 inf

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

      1. lower-/.f64 76.0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   7. 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. lower-fma.f64 N/A

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

      6. lower-/.f64 86.3

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

   8. Applied rewrites 86.3%

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

   if -1.24999999999999994e60 < c < 3.0999999999999999e-52

   1. Initial program 65.2%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 82.1

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

   5. Applied rewrites 82.1%

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

   if 3.0999999999999999e-52 < c < 4.50000000000000002e118

   1. Initial program 79.5%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 60.2

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

   5. Applied rewrites 60.2%

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

   7. Applied rewrites 79.9%

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

   if 4.50000000000000002e118 < c

   1. Initial program 39.4%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   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. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 81.4

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

   5. Applied rewrites 81.4%

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

Alternative 5: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.25 \cdot 10^{+60}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, \frac{b}{c}, a\right)}{c}\\ \mathbf{elif}\;c \leq 3.1 \cdot 10^{-52}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{+118}:\\ \;\;\;\;a \cdot \frac{c}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.25e+60)
   (/ (fma d (/ b c) a) c)
   (if (<= c 3.1e-52)
     (/ (fma a (/ c d) b) d)
     (if (<= c 4.5e+118)
       (* a (/ c (fma c c (* d d))))
       (/ (fma b (/ d c) a) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.25e+60) {
		tmp = fma(d, (b / c), a) / c;
	} else if (c <= 3.1e-52) {
		tmp = fma(a, (c / d), b) / d;
	} else if (c <= 4.5e+118) {
		tmp = a * (c / fma(c, c, (d * d)));
	} else {
		tmp = fma(b, (d / c), a) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.25e+60)
		tmp = Float64(fma(d, Float64(b / c), a) / c);
	elseif (c <= 3.1e-52)
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	elseif (c <= 4.5e+118)
		tmp = Float64(a * Float64(c / fma(c, c, Float64(d * d))));
	else
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.25e+60], N[(N[(d * N[(b / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 3.1e-52], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 4.5e+118], N[(a * N[(c / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.24999999999999994e60

   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 inf

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

      1. lower-/.f64 76.0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   7. 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. lower-fma.f64 N/A

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

      6. lower-/.f64 86.3

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

   8. Applied rewrites 86.3%

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

   if -1.24999999999999994e60 < c < 3.0999999999999999e-52

   1. Initial program 65.2%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 82.1

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

   5. Applied rewrites 82.1%

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

   if 3.0999999999999999e-52 < c < 4.50000000000000002e118

   1. Initial program 79.5%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 60.2

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

   5. Applied rewrites 60.2%

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

   7. Applied rewrites 79.7%

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

   if 4.50000000000000002e118 < c

   1. Initial program 39.4%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   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. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 81.4

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

   5. Applied rewrites 81.4%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.25 \cdot 10^{+60}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, \frac{b}{c}, a\right)}{c}\\ \mathbf{elif}\;c \leq 3.1 \cdot 10^{-52}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{+118}:\\ \;\;\;\;a \cdot \frac{c}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{if}\;c \leq -1.25 \cdot 10^{+60}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 3.1 \cdot 10^{-52}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{+118}:\\ \;\;\;\;a \cdot \frac{c}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma b (/ d c) a) c)))
   (if (<= c -1.25e+60)
     t_0
     (if (<= c 3.1e-52)
       (/ (fma a (/ c d) b) d)
       (if (<= c 4.5e+118) (* a (/ c (fma c c (* d d)))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(b, (d / c), a) / c;
	double tmp;
	if (c <= -1.25e+60) {
		tmp = t_0;
	} else if (c <= 3.1e-52) {
		tmp = fma(a, (c / d), b) / d;
	} else if (c <= 4.5e+118) {
		tmp = a * (c / fma(c, c, (d * d)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(b, Float64(d / c), a) / c)
	tmp = 0.0
	if (c <= -1.25e+60)
		tmp = t_0;
	elseif (c <= 3.1e-52)
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	elseif (c <= 4.5e+118)
		tmp = Float64(a * Float64(c / fma(c, c, Float64(d * d))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -1.25e+60], t$95$0, If[LessEqual[c, 3.1e-52], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 4.5e+118], N[(a * N[(c / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.24999999999999994e60 or 4.50000000000000002e118 < c

   1. Initial program 42.6%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 82.3

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

   5. Applied rewrites 82.3%

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

   if -1.24999999999999994e60 < c < 3.0999999999999999e-52

   1. Initial program 65.2%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 82.1

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

   5. Applied rewrites 82.1%

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

   if 3.0999999999999999e-52 < c < 4.50000000000000002e118

   1. Initial program 79.5%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 60.2

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

   5. Applied rewrites 60.2%

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

   7. Applied rewrites 79.7%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.25 \cdot 10^{+60}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;c \leq 3.1 \cdot 10^{-52}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{+118}:\\ \;\;\;\;a \cdot \frac{c}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.25 \cdot 10^{+60}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 3.1 \cdot 10^{-52}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{+164}:\\ \;\;\;\;a \cdot \frac{c}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.25e+60)
   (/ a c)
   (if (<= c 3.1e-52)
     (/ (fma a (/ c d) b) d)
     (if (<= c 1.3e+164) (* a (/ c (fma c c (* d d)))) (/ a c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.25e+60) {
		tmp = a / c;
	} else if (c <= 3.1e-52) {
		tmp = fma(a, (c / d), b) / d;
	} else if (c <= 1.3e+164) {
		tmp = a * (c / fma(c, c, (d * d)));
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.25e+60)
		tmp = Float64(a / c);
	elseif (c <= 3.1e-52)
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	elseif (c <= 1.3e+164)
		tmp = Float64(a * Float64(c / fma(c, c, Float64(d * d))));
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.25e+60], N[(a / c), $MachinePrecision], If[LessEqual[c, 3.1e-52], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 1.3e+164], N[(a * N[(c / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.24999999999999994e60 or 1.3e164 < c

   1. Initial program 40.7%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 76.9

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

   5. Applied rewrites 76.9%

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

   if -1.24999999999999994e60 < c < 3.0999999999999999e-52

   1. Initial program 65.2%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 82.1

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

   5. Applied rewrites 82.1%

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

   if 3.0999999999999999e-52 < c < 1.3e164

   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 a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 55.0

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

   5. Applied rewrites 55.0%

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

   7. Applied rewrites 75.2%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.25 \cdot 10^{+60}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 3.1 \cdot 10^{-52}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{+164}:\\ \;\;\;\;a \cdot \frac{c}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.8 \cdot 10^{+57}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 7.8 \cdot 10^{-53}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{+164}:\\ \;\;\;\;a \cdot \frac{c}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3.8e+57)
   (/ a c)
   (if (<= c 7.8e-53)
     (/ b d)
     (if (<= c 1.3e+164) (* a (/ c (fma c c (* d d)))) (/ a c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.8e+57) {
		tmp = a / c;
	} else if (c <= 7.8e-53) {
		tmp = b / d;
	} else if (c <= 1.3e+164) {
		tmp = a * (c / fma(c, c, (d * d)));
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3.8e+57)
		tmp = Float64(a / c);
	elseif (c <= 7.8e-53)
		tmp = Float64(b / d);
	elseif (c <= 1.3e+164)
		tmp = Float64(a * Float64(c / fma(c, c, Float64(d * d))));
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3.8e+57], N[(a / c), $MachinePrecision], If[LessEqual[c, 7.8e-53], N[(b / d), $MachinePrecision], If[LessEqual[c, 1.3e+164], N[(a * N[(c / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.7999999999999999e57 or 1.3e164 < c

   1. Initial program 41.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 75.3

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

   5. Applied rewrites 75.3%

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

   if -3.7999999999999999e57 < c < 7.8000000000000004e-53

   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 c around 0

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

      1. lower-/.f64 67.8

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

   5. Applied rewrites 67.8%

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

   if 7.8000000000000004e-53 < c < 1.3e164

   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 a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 55.0

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

   5. Applied rewrites 55.0%

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

   7. Applied rewrites 75.2%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.8 \cdot 10^{+57}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 7.8 \cdot 10^{-53}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{+164}:\\ \;\;\;\;a \cdot \frac{c}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.8 \cdot 10^{+57}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 3.1 \cdot 10^{-25}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{c}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3.8e+57) (/ a c) (if (<= c 3.1e-25) (/ b d) (/ 1.0 (/ c a)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.8e+57) {
		tmp = a / c;
	} else if (c <= 3.1e-25) {
		tmp = b / d;
	} else {
		tmp = 1.0 / (c / a);
	}
	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 <= (-3.8d+57)) then
        tmp = a / c
    else if (c <= 3.1d-25) then
        tmp = b / d
    else
        tmp = 1.0d0 / (c / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.8e+57) {
		tmp = a / c;
	} else if (c <= 3.1e-25) {
		tmp = b / d;
	} else {
		tmp = 1.0 / (c / a);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -3.8e+57:
		tmp = a / c
	elif c <= 3.1e-25:
		tmp = b / d
	else:
		tmp = 1.0 / (c / a)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3.8e+57)
		tmp = Float64(a / c);
	elseif (c <= 3.1e-25)
		tmp = Float64(b / d);
	else
		tmp = Float64(1.0 / Float64(c / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -3.8e+57)
		tmp = a / c;
	elseif (c <= 3.1e-25)
		tmp = b / d;
	else
		tmp = 1.0 / (c / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3.8e+57], N[(a / c), $MachinePrecision], If[LessEqual[c, 3.1e-25], N[(b / d), $MachinePrecision], N[(1.0 / N[(c / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.7999999999999999e57

   1. Initial program 45.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 73.6

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

   5. Applied rewrites 73.6%

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

   if -3.7999999999999999e57 < c < 3.09999999999999995e-25

   1. Initial program 66.0%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 66.1

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

   5. Applied rewrites 66.1%

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

   if 3.09999999999999995e-25 < c

   1. Initial program 56.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 70.4

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

   5. Applied rewrites 70.4%

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

   7. Applied rewrites 70.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{c}{a}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 64.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.8 \cdot 10^{+57}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 3.1 \cdot 10^{-25}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3.8e+57) (/ a c) (if (<= c 3.1e-25) (/ b d) (/ a c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.8e+57) {
		tmp = a / c;
	} else if (c <= 3.1e-25) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	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 <= (-3.8d+57)) then
        tmp = a / c
    else if (c <= 3.1d-25) then
        tmp = b / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.8e+57) {
		tmp = a / c;
	} else if (c <= 3.1e-25) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -3.8e+57:
		tmp = a / c
	elif c <= 3.1e-25:
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3.8e+57)
		tmp = Float64(a / c);
	elseif (c <= 3.1e-25)
		tmp = Float64(b / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -3.8e+57)
		tmp = a / c;
	elseif (c <= 3.1e-25)
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3.8e+57], N[(a / c), $MachinePrecision], If[LessEqual[c, 3.1e-25], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -3.7999999999999999e57 or 3.09999999999999995e-25 < c

   1. Initial program 51.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 71.9

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

   5. Applied rewrites 71.9%

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

   if -3.7999999999999999e57 < c < 3.09999999999999995e-25

   1. Initial program 66.0%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 66.1

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

   5. Applied rewrites 66.1%

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

Alternative 11: 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 58.7%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 45.9

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

5. Applied rewrites 45.9%

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

Developer Target 1: 99.3% accurate, 0.6× speedup

Math

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

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 2024226 
(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))))