Complex division, real part

Percentage Accurate: 61.0% → 84.2%

Time: 15.9s

Alternatives: 9

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.0% 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: 84.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \mathsf{fma}\left(b, \frac{d}{t\_0}, \frac{a \cdot c}{t\_0}\right)\\ t_2 := \frac{\mathsf{fma}\left(c, a \cdot \frac{1}{d}, b\right)}{d}\\ \mathbf{if}\;d \leq -2 \cdot 10^{+102}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;d \leq -9.6 \cdot 10^{-141}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 6 \cdot 10^{-139}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{+104}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c)))
        (t_1 (fma b (/ d t_0) (/ (* a c) t_0)))
        (t_2 (/ (fma c (* a (/ 1.0 d)) b) d)))
   (if (<= d -2e+102)
     t_2
     (if (<= d -9.6e-141)
       t_1
       (if (<= d 6e-139)
         (/ (fma b (/ d c) a) c)
         (if (<= d 4.5e+104) t_1 t_2))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double t_1 = fma(b, (d / t_0), ((a * c) / t_0));
	double t_2 = fma(c, (a * (1.0 / d)), b) / d;
	double tmp;
	if (d <= -2e+102) {
		tmp = t_2;
	} else if (d <= -9.6e-141) {
		tmp = t_1;
	} else if (d <= 6e-139) {
		tmp = fma(b, (d / c), a) / c;
	} else if (d <= 4.5e+104) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = fma(b, Float64(d / t_0), Float64(Float64(a * c) / t_0))
	t_2 = Float64(fma(c, Float64(a * Float64(1.0 / d)), b) / d)
	tmp = 0.0
	if (d <= -2e+102)
		tmp = t_2;
	elseif (d <= -9.6e-141)
		tmp = t_1;
	elseif (d <= 6e-139)
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	elseif (d <= 4.5e+104)
		tmp = t_1;
	else
		tmp = t_2;
	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[(b * N[(d / t$95$0), $MachinePrecision] + N[(N[(a * c), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * N[(a * N[(1.0 / d), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -2e+102], t$95$2, If[LessEqual[d, -9.6e-141], t$95$1, If[LessEqual[d, 6e-139], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 4.5e+104], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.99999999999999995e102 or 4.4999999999999998e104 < d

   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 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 81.1

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

   5. Applied rewrites 81.1%

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

   7. Applied rewrites 82.6%

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

   if -1.99999999999999995e102 < d < -9.6000000000000004e-141 or 5.9999999999999998e-139 < d < 4.4999999999999998e104

   1. Initial program 84.7%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 86.4

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

   5. Applied rewrites 86.4%

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

   if -9.6000000000000004e-141 < d < 5.9999999999999998e-139

   1. Initial program 75.0%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   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 96.3

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

   5. Applied rewrites 96.3%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{+102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a \cdot \frac{1}{d}, b\right)}{d}\\ \mathbf{elif}\;d \leq -9.6 \cdot 10^{-141}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{a \cdot c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)\\ \mathbf{elif}\;d \leq 6 \cdot 10^{-139}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{+104}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{a \cdot c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a \cdot \frac{1}{d}, b\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ t_1 := \frac{\mathsf{fma}\left(c, a \cdot \frac{1}{d}, b\right)}{d}\\ \mathbf{if}\;d \leq -1.5 \cdot 10^{+102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -2.6 \cdot 10^{-141}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 7.6 \cdot 10^{-139}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{+79}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma a c (* d b)) (fma c c (* d d))))
        (t_1 (/ (fma c (* a (/ 1.0 d)) b) d)))
   (if (<= d -1.5e+102)
     t_1
     (if (<= d -2.6e-141)
       t_0
       (if (<= d 7.6e-139)
         (/ (fma b (/ d c) a) c)
         (if (<= d 2.1e+79) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(a, c, (d * b)) / fma(c, c, (d * d));
	double t_1 = fma(c, (a * (1.0 / d)), b) / d;
	double tmp;
	if (d <= -1.5e+102) {
		tmp = t_1;
	} else if (d <= -2.6e-141) {
		tmp = t_0;
	} else if (d <= 7.6e-139) {
		tmp = fma(b, (d / c), a) / c;
	} else if (d <= 2.1e+79) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(a, c, Float64(d * b)) / fma(c, c, Float64(d * d)))
	t_1 = Float64(fma(c, Float64(a * Float64(1.0 / d)), b) / d)
	tmp = 0.0
	if (d <= -1.5e+102)
		tmp = t_1;
	elseif (d <= -2.6e-141)
		tmp = t_0;
	elseif (d <= 7.6e-139)
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	elseif (d <= 2.1e+79)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a * c + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * N[(a * N[(1.0 / d), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -1.5e+102], t$95$1, If[LessEqual[d, -2.6e-141], t$95$0, If[LessEqual[d, 7.6e-139], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2.1e+79], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.4999999999999999e102 or 2.10000000000000008e79 < d

   1. Initial program 37.2%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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 79.9

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

   5. Applied rewrites 79.9%

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

   7. Applied rewrites 81.4%

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

   if -1.4999999999999999e102 < d < -2.60000000000000011e-141 or 7.60000000000000015e-139 < d < 2.10000000000000008e79

   1. Initial program 85.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 85.9

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 85.9

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

   4. Applied rewrites 85.9%

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

   if -2.60000000000000011e-141 < d < 7.60000000000000015e-139

   1. Initial program 75.0%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   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 96.3

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

   5. Applied rewrites 96.3%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a \cdot \frac{1}{d}, b\right)}{d}\\ \mathbf{elif}\;d \leq -2.6 \cdot 10^{-141}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq 7.6 \cdot 10^{-139}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{+79}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a \cdot \frac{1}{d}, b\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(c, a \cdot \frac{1}{d}, b\right)}{d}\\ \mathbf{if}\;d \leq -3000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{-106}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{+103}:\\ \;\;\;\;b \cdot \frac{d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma c (* a (/ 1.0 d)) b) d)))
   (if (<= d -3000000.0)
     t_0
     (if (<= d 2.1e-106)
       (/ (fma b (/ d c) a) c)
       (if (<= d 1.4e+103) (* b (/ d (fma c c (* d d)))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(c, (a * (1.0 / d)), b) / d;
	double tmp;
	if (d <= -3000000.0) {
		tmp = t_0;
	} else if (d <= 2.1e-106) {
		tmp = fma(b, (d / c), a) / c;
	} else if (d <= 1.4e+103) {
		tmp = b * (d / fma(c, c, (d * d)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(c, Float64(a * Float64(1.0 / d)), b) / d)
	tmp = 0.0
	if (d <= -3000000.0)
		tmp = t_0;
	elseif (d <= 2.1e-106)
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	elseif (d <= 1.4e+103)
		tmp = Float64(b * Float64(d / 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[(c * N[(a * N[(1.0 / d), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -3000000.0], t$95$0, If[LessEqual[d, 2.1e-106], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.4e+103], N[(b * N[(d / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3e6 or 1.40000000000000004e103 < 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 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 76.2

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

   5. Applied rewrites 76.2%

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

   7. Applied rewrites 77.4%

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

   if -3e6 < d < 2.10000000000000003e-106

   1. Initial program 77.1%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   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 86.3

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

   5. Applied rewrites 86.3%

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

   if 2.10000000000000003e-106 < d < 1.40000000000000004e103

   1. Initial program 87.6%

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

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

      3. lower-fma.f64 87.6

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 87.6

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

   4. Applied rewrites 87.6%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 68.8

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

   7. Applied rewrites 68.8%

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

   9. Applied rewrites 73.7%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a \cdot \frac{1}{d}, b\right)}{d}\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{-106}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{+103}:\\ \;\;\;\;b \cdot \frac{d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a \cdot \frac{1}{d}, b\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{if}\;d \leq -3000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{-106}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 1.46 \cdot 10^{+103}:\\ \;\;\;\;b \cdot \frac{d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma a (/ c d) b) d)))
   (if (<= d -3000000.0)
     t_0
     (if (<= d 2.1e-106)
       (/ (fma b (/ d c) a) c)
       (if (<= d 1.46e+103) (* b (/ d (fma c c (* d d)))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(a, (c / d), b) / d;
	double tmp;
	if (d <= -3000000.0) {
		tmp = t_0;
	} else if (d <= 2.1e-106) {
		tmp = fma(b, (d / c), a) / c;
	} else if (d <= 1.46e+103) {
		tmp = b * (d / fma(c, c, (d * d)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(a, Float64(c / d), b) / d)
	tmp = 0.0
	if (d <= -3000000.0)
		tmp = t_0;
	elseif (d <= 2.1e-106)
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	elseif (d <= 1.46e+103)
		tmp = Float64(b * Float64(d / 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[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -3000000.0], t$95$0, If[LessEqual[d, 2.1e-106], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.46e+103], N[(b * N[(d / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3e6 or 1.45999999999999998e103 < 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 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 76.2

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

   5. Applied rewrites 76.2%

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

   if -3e6 < d < 2.10000000000000003e-106

   1. Initial program 77.1%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   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 86.3

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

   5. Applied rewrites 86.3%

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

   if 2.10000000000000003e-106 < d < 1.45999999999999998e103

   1. Initial program 87.6%

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

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

      3. lower-fma.f64 87.6

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 87.6

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

   4. Applied rewrites 87.6%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 68.8

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

   7. Applied rewrites 68.8%

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

   9. Applied rewrites 73.7%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{-106}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 1.46 \cdot 10^{+103}:\\ \;\;\;\;b \cdot \frac{d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{if}\;d \leq -2050000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-134}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 1.46 \cdot 10^{+103}:\\ \;\;\;\;b \cdot \frac{d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma a (/ c d) b) d)))
   (if (<= d -2050000.0)
     t_0
     (if (<= d 1.45e-134)
       (/ a c)
       (if (<= d 1.46e+103) (* b (/ d (fma c c (* d d)))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(a, (c / d), b) / d;
	double tmp;
	if (d <= -2050000.0) {
		tmp = t_0;
	} else if (d <= 1.45e-134) {
		tmp = a / c;
	} else if (d <= 1.46e+103) {
		tmp = b * (d / fma(c, c, (d * d)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(a, Float64(c / d), b) / d)
	tmp = 0.0
	if (d <= -2050000.0)
		tmp = t_0;
	elseif (d <= 1.45e-134)
		tmp = Float64(a / c);
	elseif (d <= 1.46e+103)
		tmp = Float64(b * Float64(d / 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[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -2050000.0], t$95$0, If[LessEqual[d, 1.45e-134], N[(a / c), $MachinePrecision], If[LessEqual[d, 1.46e+103], N[(b * N[(d / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.05e6 or 1.45999999999999998e103 < 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 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 76.2

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

   5. Applied rewrites 76.2%

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

   if -2.05e6 < d < 1.44999999999999997e-134

   1. Initial program 75.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 68.9

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

   5. Applied rewrites 68.9%

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

   if 1.44999999999999997e-134 < d < 1.45999999999999998e103

   1. Initial program 89.1%

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

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

      3. lower-fma.f64 89.1

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 89.1

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

   4. Applied rewrites 89.1%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 67.9

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

   7. Applied rewrites 67.9%

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

   9. Applied rewrites 72.3%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2050000:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-134}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 1.46 \cdot 10^{+103}:\\ \;\;\;\;b \cdot \frac{d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2750000:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-134}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+103}:\\ \;\;\;\;b \cdot \frac{d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -2750000.0)
   (/ b d)
   (if (<= d 1.45e-134)
     (/ a c)
     (if (<= d 7.5e+103) (* b (/ d (fma c c (* d d)))) (/ b d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2750000.0) {
		tmp = b / d;
	} else if (d <= 1.45e-134) {
		tmp = a / c;
	} else if (d <= 7.5e+103) {
		tmp = b * (d / fma(c, c, (d * d)));
	} else {
		tmp = b / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -2750000.0)
		tmp = Float64(b / d);
	elseif (d <= 1.45e-134)
		tmp = Float64(a / c);
	elseif (d <= 7.5e+103)
		tmp = Float64(b * Float64(d / fma(c, c, Float64(d * d))));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -2750000.0], N[(b / d), $MachinePrecision], If[LessEqual[d, 1.45e-134], N[(a / c), $MachinePrecision], If[LessEqual[d, 7.5e+103], N[(b * N[(d / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.75e6 or 7.49999999999999922e103 < 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 66.2

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

   5. Applied rewrites 66.2%

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

   if -2.75e6 < d < 1.44999999999999997e-134

   1. Initial program 75.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 68.9

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

   5. Applied rewrites 68.9%

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

   if 1.44999999999999997e-134 < d < 7.49999999999999922e103

   1. Initial program 89.1%

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

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

      3. lower-fma.f64 89.1

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 89.1

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

   4. Applied rewrites 89.1%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 67.9

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

   7. Applied rewrites 67.9%

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

   9. Applied rewrites 72.3%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2750000:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-134}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+103}:\\ \;\;\;\;b \cdot \frac{d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2750000:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{-134}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{+103}:\\ \;\;\;\;d \cdot \frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -2750000.0)
   (/ b d)
   (if (<= d 1.5e-134)
     (/ a c)
     (if (<= d 2.4e+103) (* d (/ b (fma d d (* c c)))) (/ b d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2750000.0) {
		tmp = b / d;
	} else if (d <= 1.5e-134) {
		tmp = a / c;
	} else if (d <= 2.4e+103) {
		tmp = d * (b / fma(d, d, (c * c)));
	} else {
		tmp = b / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -2750000.0)
		tmp = Float64(b / d);
	elseif (d <= 1.5e-134)
		tmp = Float64(a / c);
	elseif (d <= 2.4e+103)
		tmp = Float64(d * Float64(b / fma(d, d, Float64(c * c))));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -2750000.0], N[(b / d), $MachinePrecision], If[LessEqual[d, 1.5e-134], N[(a / c), $MachinePrecision], If[LessEqual[d, 2.4e+103], N[(d * N[(b / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.75e6 or 2.3999999999999998e103 < 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 66.2

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

   5. Applied rewrites 66.2%

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

   if -2.75e6 < d < 1.5e-134

   1. Initial program 75.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 68.9

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

   5. Applied rewrites 68.9%

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

   if 1.5e-134 < d < 2.3999999999999998e103

   1. Initial program 89.1%

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

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

      3. lower-fma.f64 89.1

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 89.1

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

   4. Applied rewrites 89.1%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 67.9

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

   7. Applied rewrites 67.9%

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

Alternative 8: 62.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2750000:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{-107}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -2750000.0) (/ b d) (if (<= d 6.2e-107) (/ a c) (/ b d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2750000.0) {
		tmp = b / d;
	} else if (d <= 6.2e-107) {
		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 <= (-2750000.0d0)) then
        tmp = b / d
    else if (d <= 6.2d-107) 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 <= -2750000.0) {
		tmp = b / d;
	} else if (d <= 6.2e-107) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -2750000.0:
		tmp = b / d
	elif d <= 6.2e-107:
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -2750000.0)
		tmp = Float64(b / d);
	elseif (d <= 6.2e-107)
		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 <= -2750000.0)
		tmp = b / d;
	elseif (d <= 6.2e-107)
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -2750000.0], N[(b / d), $MachinePrecision], If[LessEqual[d, 6.2e-107], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -2.75e6 or 6.20000000000000043e-107 < d

   1. Initial program 58.6%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 62.2

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

   5. Applied rewrites 62.2%

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

   if -2.75e6 < d < 6.20000000000000043e-107

   1. Initial program 77.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 67.1

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

   5. Applied rewrites 67.1%

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

Alternative 9: 42.7% 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 66.6%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 42.2

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

5. Applied rewrites 42.2%

   \[\leadsto \color{blue}{\frac{a}{c}} \]
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 2024221 
(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))))