Complex division, real part

Percentage Accurate: 62.2% → 83.8%

Time: 7.1s

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: 62.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = ((a * c) + (b * d)) / ((c * c) + (d * d))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{if}\;d \leq -1.28 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -1.85 \cdot 10^{-66}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{d}{t\_0}, \frac{a}{t\_0} \cdot c\right)\\ \mathbf{elif}\;d \leq 1.14 \cdot 10^{-58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+111}:\\ \;\;\;\;\frac{1}{\frac{t\_0}{\mathsf{fma}\left(d, b, c \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))) (t_1 (/ (fma (/ a d) c b) d)))
   (if (<= d -1.28e+154)
     t_1
     (if (<= d -1.85e-66)
       (fma b (/ d t_0) (* (/ a t_0) c))
       (if (<= d 1.14e-58)
         (/ (fma (/ d c) b a) c)
         (if (<= d 5.5e+111) (/ 1.0 (/ t_0 (fma d b (* c a)))) t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double t_1 = fma((a / d), c, b) / d;
	double tmp;
	if (d <= -1.28e+154) {
		tmp = t_1;
	} else if (d <= -1.85e-66) {
		tmp = fma(b, (d / t_0), ((a / t_0) * c));
	} else if (d <= 1.14e-58) {
		tmp = fma((d / c), b, a) / c;
	} else if (d <= 5.5e+111) {
		tmp = 1.0 / (t_0 / fma(d, b, (c * a)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = Float64(fma(Float64(a / d), c, b) / d)
	tmp = 0.0
	if (d <= -1.28e+154)
		tmp = t_1;
	elseif (d <= -1.85e-66)
		tmp = fma(b, Float64(d / t_0), Float64(Float64(a / t_0) * c));
	elseif (d <= 1.14e-58)
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	elseif (d <= 5.5e+111)
		tmp = Float64(1.0 / Float64(t_0 / fma(d, b, Float64(c * a))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -1.28e+154], t$95$1, If[LessEqual[d, -1.85e-66], N[(b * N[(d / t$95$0), $MachinePrecision] + N[(N[(a / t$95$0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.14e-58], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 5.5e+111], N[(1.0 / N[(t$95$0 / N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.2800000000000001e154 or 5.4999999999999998e111 < d

   1. Initial program 40.2%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 86.7

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

   5. Applied rewrites 86.7%

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

   if -1.2800000000000001e154 < d < -1.8500000000000001e-66

   1. Initial program 71.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 75.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 75.8

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

   4. Applied rewrites 75.8%

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

   if -1.8500000000000001e-66 < d < 1.14e-58

   1. Initial program 66.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 66.2

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 66.2

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

   4. Applied rewrites 66.2%

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

   5. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 90.8

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

   7. Applied rewrites 90.8%

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

   if 1.14e-58 < d < 5.4999999999999998e111

   1. Initial program 83.7%

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

      1. lift-/.f64 N/A

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

      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 83.8

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

      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 83.9

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

      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 83.9

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

      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 83.9

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

   4. Applied rewrites 83.9%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.28 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{elif}\;d \leq -1.85 \cdot 10^{-66}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{a}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot c\right)\\ \mathbf{elif}\;d \leq 1.14 \cdot 10^{-58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+111}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(d, d, c \cdot c\right)}{\mathsf{fma}\left(d, b, c \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, b, c \cdot a\right)\\ t_1 := \frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{if}\;d \leq -2.5 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -1.85 \cdot 10^{-66}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq 1.14 \cdot 10^{-58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+111}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(d, d, c \cdot c\right)}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d b (* c a))) (t_1 (/ (fma (/ a d) c b) d)))
   (if (<= d -2.5e+91)
     t_1
     (if (<= d -1.85e-66)
       (/ t_0 (fma c c (* d d)))
       (if (<= d 1.14e-58)
         (/ (fma (/ d c) b a) c)
         (if (<= d 5.5e+111) (/ 1.0 (/ (fma d d (* c c)) t_0)) t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, b, (c * a));
	double t_1 = fma((a / d), c, b) / d;
	double tmp;
	if (d <= -2.5e+91) {
		tmp = t_1;
	} else if (d <= -1.85e-66) {
		tmp = t_0 / fma(c, c, (d * d));
	} else if (d <= 1.14e-58) {
		tmp = fma((d / c), b, a) / c;
	} else if (d <= 5.5e+111) {
		tmp = 1.0 / (fma(d, d, (c * c)) / t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, b, Float64(c * a))
	t_1 = Float64(fma(Float64(a / d), c, b) / d)
	tmp = 0.0
	if (d <= -2.5e+91)
		tmp = t_1;
	elseif (d <= -1.85e-66)
		tmp = Float64(t_0 / fma(c, c, Float64(d * d)));
	elseif (d <= 1.14e-58)
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	elseif (d <= 5.5e+111)
		tmp = Float64(1.0 / Float64(fma(d, d, Float64(c * c)) / t_0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -2.5e+91], t$95$1, If[LessEqual[d, -1.85e-66], N[(t$95$0 / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.14e-58], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 5.5e+111], N[(1.0 / N[(N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.5000000000000001e91 or 5.4999999999999998e111 < d

   1. Initial program 43.7%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 83.7

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

   5. Applied rewrites 83.7%

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

   if -2.5000000000000001e91 < d < -1.8500000000000001e-66

   1. Initial program 76.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 76.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 76.7

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

   4. Applied rewrites 76.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 76.7

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

   6. Applied rewrites 76.7%

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

   if -1.8500000000000001e-66 < d < 1.14e-58

   1. Initial program 66.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 66.2

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 66.2

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

   4. Applied rewrites 66.2%

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

   5. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 90.8

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

   7. Applied rewrites 90.8%

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

   if 1.14e-58 < d < 5.4999999999999998e111

   1. Initial program 83.7%

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

      1. lift-/.f64 N/A

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

      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 83.8

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

      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 83.9

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

      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 83.9

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

      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 83.9

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

   4. Applied rewrites 83.9%

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

Alternative 3: 82.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ t_1 := \frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{if}\;d \leq -2.5 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -1.85 \cdot 10^{-66}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{+106}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma d b (* c a)) (fma c c (* d d))))
        (t_1 (/ (fma (/ a d) c b) d)))
   (if (<= d -2.5e+91)
     t_1
     (if (<= d -1.85e-66)
       t_0
       (if (<= d 1.2e-57)
         (/ (fma (/ d c) b a) c)
         (if (<= d 4.5e+106) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, b, (c * a)) / fma(c, c, (d * d));
	double t_1 = fma((a / d), c, b) / d;
	double tmp;
	if (d <= -2.5e+91) {
		tmp = t_1;
	} else if (d <= -1.85e-66) {
		tmp = t_0;
	} else if (d <= 1.2e-57) {
		tmp = fma((d / c), b, a) / c;
	} else if (d <= 4.5e+106) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(d, b, Float64(c * a)) / fma(c, c, Float64(d * d)))
	t_1 = Float64(fma(Float64(a / d), c, b) / d)
	tmp = 0.0
	if (d <= -2.5e+91)
		tmp = t_1;
	elseif (d <= -1.85e-66)
		tmp = t_0;
	elseif (d <= 1.2e-57)
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	elseif (d <= 4.5e+106)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -2.5e+91], t$95$1, If[LessEqual[d, -1.85e-66], t$95$0, If[LessEqual[d, 1.2e-57], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 4.5e+106], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.5000000000000001e91 or 4.4999999999999997e106 < d

   1. Initial program 44.9%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 84.0

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

   5. Applied rewrites 84.0%

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

   if -2.5000000000000001e91 < d < -1.8500000000000001e-66 or 1.20000000000000003e-57 < d < 4.4999999999999997e106

   1. Initial program 80.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 80.0

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 80.0

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

   4. Applied rewrites 80.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 80.0

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

   6. Applied rewrites 80.0%

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

   if -1.8500000000000001e-66 < d < 1.20000000000000003e-57

   1. Initial program 66.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 66.5

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 66.5

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

   4. Applied rewrites 66.5%

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

   5. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 90.9

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

   7. Applied rewrites 90.9%

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

Alternative 4: 71.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -8 \cdot 10^{+104}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 135:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{elif}\;c \leq 2.1 \cdot 10^{+152}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot c\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -8e+104)
   (/ a c)
   (if (<= c 135.0)
     (/ (fma (/ a d) c b) d)
     (if (<= c 2.1e+152) (* (/ a (fma d d (* c c))) c) (/ a c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -8e+104) {
		tmp = a / c;
	} else if (c <= 135.0) {
		tmp = fma((a / d), c, b) / d;
	} else if (c <= 2.1e+152) {
		tmp = (a / fma(d, d, (c * c))) * c;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -8e+104)
		tmp = Float64(a / c);
	elseif (c <= 135.0)
		tmp = Float64(fma(Float64(a / d), c, b) / d);
	elseif (c <= 2.1e+152)
		tmp = Float64(Float64(a / fma(d, d, Float64(c * c))) * c);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -8e+104], N[(a / c), $MachinePrecision], If[LessEqual[c, 135.0], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 2.1e+152], N[(N[(a / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], N[(a / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -8e104 or 2.1000000000000002e152 < c

   1. Initial program 35.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 84.2

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

   5. Applied rewrites 84.2%

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

   if -8e104 < c < 135

   1. Initial program 74.9%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 77.0

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

   5. Applied rewrites 77.0%

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

   if 135 < c < 2.1000000000000002e152

   1. Initial program 68.3%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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 66.3

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

   5. Applied rewrites 66.3%

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

Alternative 5: 64.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.5 \cdot 10^{+61}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{-49}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{+110}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.5e+61)
   (/ b d)
   (if (<= d 6.5e-49)
     (/ a c)
     (if (<= d 1.8e+110) (/ (fma d b (* c a)) (* d d)) (/ b d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.5e+61) {
		tmp = b / d;
	} else if (d <= 6.5e-49) {
		tmp = a / c;
	} else if (d <= 1.8e+110) {
		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 <= -1.5e+61)
		tmp = Float64(b / d);
	elseif (d <= 6.5e-49)
		tmp = Float64(a / c);
	elseif (d <= 1.8e+110)
		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, -1.5e+61], N[(b / d), $MachinePrecision], If[LessEqual[d, 6.5e-49], N[(a / c), $MachinePrecision], If[LessEqual[d, 1.8e+110], N[(N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.5e61 or 1.7999999999999998e110 < d

   1. Initial program 46.8%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 74.6

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

   5. Applied rewrites 74.6%

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

   if -1.5e61 < d < 6.49999999999999968e-49

   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 69.7

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

   5. Applied rewrites 69.7%

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

   if 6.49999999999999968e-49 < d < 1.7999999999999998e110

   1. Initial program 82.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 82.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 82.4

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

   4. Applied rewrites 82.4%

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

   5. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.2

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

   7. Applied rewrites 59.2%

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

Alternative 6: 65.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.5 \cdot 10^{+61}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 5.6 \cdot 10^{-49}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 2.55 \cdot 10^{+138}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.5e+61)
   (/ b d)
   (if (<= d 5.6e-49)
     (/ a c)
     (if (<= d 2.55e+138) (* (/ d (fma c c (* d d))) b) (/ b d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.5e+61) {
		tmp = b / d;
	} else if (d <= 5.6e-49) {
		tmp = a / c;
	} else if (d <= 2.55e+138) {
		tmp = (d / fma(c, c, (d * d))) * b;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.5e+61)
		tmp = Float64(b / d);
	elseif (d <= 5.6e-49)
		tmp = Float64(a / c);
	elseif (d <= 2.55e+138)
		tmp = Float64(Float64(d / fma(c, c, Float64(d * d))) * b);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.5e+61], N[(b / d), $MachinePrecision], If[LessEqual[d, 5.6e-49], N[(a / c), $MachinePrecision], If[LessEqual[d, 2.55e+138], N[(N[(d / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[(b / d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.5e61 or 2.5499999999999999e138 < d

   1. Initial program 46.0%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 75.2

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

   5. Applied rewrites 75.2%

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

   if -1.5e61 < d < 5.59999999999999995e-49

   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 69.7

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

   5. Applied rewrites 69.7%

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

   if 5.59999999999999995e-49 < d < 2.5499999999999999e138

   1. Initial program 79.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 79.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 79.7

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

   4. Applied rewrites 79.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 79.7

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

   6. Applied rewrites 79.7%

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

   7. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 58.1

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

   9. Applied rewrites 58.1%

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

Alternative 7: 78.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{if}\;d \leq -1.55 \cdot 10^{+61}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (/ a d) c b) d)))
   (if (<= d -1.55e+61) t_0 (if (<= d 2e+34) (/ (fma (/ d c) b a) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((a / d), c, b) / d;
	double tmp;
	if (d <= -1.55e+61) {
		tmp = t_0;
	} else if (d <= 2e+34) {
		tmp = fma((d / c), b, a) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(a / d), c, b) / d)
	tmp = 0.0
	if (d <= -1.55e+61)
		tmp = t_0;
	elseif (d <= 2e+34)
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -1.55e+61], t$95$0, If[LessEqual[d, 2e+34], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.55e61 or 1.99999999999999989e34 < d

   1. Initial program 52.3%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 80.4

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

   5. Applied rewrites 80.4%

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

   if -1.55e61 < d < 1.99999999999999989e34

   1. Initial program 69.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 69.5

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 69.5

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

   4. Applied rewrites 69.5%

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

   5. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 81.0

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

   7. Applied rewrites 81.0%

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

Alternative 8: 75.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \mathbf{if}\;c \leq -9.8 \cdot 10^{+101}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 2.65 \cdot 10^{-46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (/ b c) d a) c)))
   (if (<= c -9.8e+101) t_0 (if (<= c 2.65e-46) (/ (fma (/ c d) a b) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((b / c), d, a) / c;
	double tmp;
	if (c <= -9.8e+101) {
		tmp = t_0;
	} else if (c <= 2.65e-46) {
		tmp = fma((c / d), a, b) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(b / c), d, a) / c)
	tmp = 0.0
	if (c <= -9.8e+101)
		tmp = t_0;
	elseif (c <= 2.65e-46)
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b / c), $MachinePrecision] * d + a), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -9.8e+101], t$95$0, If[LessEqual[c, 2.65e-46], N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -9.79999999999999965e101 or 2.65000000000000009e-46 < c

   1. Initial program 46.5%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   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 80.2

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

   5. Applied rewrites 80.2%

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

   if -9.79999999999999965e101 < c < 2.65000000000000009e-46

   1. Initial program 75.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 75.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 75.4

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

   4. Applied rewrites 75.4%

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

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{b + \frac{a \cdot c}{d}}{d}} \]
   6. 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

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 80.4

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

   7. Applied rewrites 80.4%

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

Alternative 9: 74.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \mathbf{if}\;c \leq -9.8 \cdot 10^{+101}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 2.65 \cdot 10^{-46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (/ b c) d a) c)))
   (if (<= c -9.8e+101) t_0 (if (<= c 2.65e-46) (/ (fma (/ a d) c b) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((b / c), d, a) / c;
	double tmp;
	if (c <= -9.8e+101) {
		tmp = t_0;
	} else if (c <= 2.65e-46) {
		tmp = fma((a / d), c, b) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(b / c), d, a) / c)
	tmp = 0.0
	if (c <= -9.8e+101)
		tmp = t_0;
	elseif (c <= 2.65e-46)
		tmp = Float64(fma(Float64(a / d), c, b) / d);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b / c), $MachinePrecision] * d + a), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -9.8e+101], t$95$0, If[LessEqual[c, 2.65e-46], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -9.79999999999999965e101 or 2.65000000000000009e-46 < c

   1. Initial program 46.5%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   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 80.2

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

   5. Applied rewrites 80.2%

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

   if -9.79999999999999965e101 < c < 2.65000000000000009e-46

   1. Initial program 75.4%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 79.0

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

   5. Applied rewrites 79.0%

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

Alternative 10: 64.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.5 \cdot 10^{+61}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+34}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.5e+61) (/ b d) (if (<= d 2e+34) (/ a c) (/ b d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.5e+61) {
		tmp = b / d;
	} else if (d <= 2e+34) {
		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 <= (-1.5d+61)) then
        tmp = b / d
    else if (d <= 2d+34) 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 <= -1.5e+61) {
		tmp = b / d;
	} else if (d <= 2e+34) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -1.5e+61:
		tmp = b / d
	elif d <= 2e+34:
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.5e+61)
		tmp = Float64(b / d);
	elseif (d <= 2e+34)
		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 <= -1.5e+61)
		tmp = b / d;
	elseif (d <= 2e+34)
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.5e+61], N[(b / d), $MachinePrecision], If[LessEqual[d, 2e+34], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.5e61 or 1.99999999999999989e34 < d

   1. Initial program 52.3%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 71.6

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

   5. Applied rewrites 71.6%

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

   if -1.5e61 < d < 1.99999999999999989e34

   1. Initial program 69.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 65.3

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

   5. Applied rewrites 65.3%

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

Alternative 11: 43.7% 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 61.7%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 45.2

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

5. Applied rewrites 45.2%

   \[\leadsto \color{blue}{\frac{a}{c}} \]
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 2024298 
(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))))