Complex division, real part

Percentage Accurate: 62.6% → 83.3%

Time: 8.7s

Alternatives: 7

Speedup: 1.2×

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: 62.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot a + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -2.45 \cdot 10^{+73}:\\ \;\;\;\;\frac{\frac{b}{c}}{\frac{c}{d}} + \frac{a}{c}\\ \mathbf{elif}\;c \leq -8.8 \cdot 10^{-94}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.8 \cdot 10^{-124}:\\ \;\;\;\;\frac{b + \frac{c \cdot a}{d}}{d}\\ \mathbf{elif}\;c \leq 3.2 \cdot 10^{+48}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* c a) (* b d)) (+ (* c c) (* d d)))))
   (if (<= c -2.45e+73)
     (+ (/ (/ b c) (/ c d)) (/ a c))
     (if (<= c -8.8e-94)
       t_0
       (if (<= c 1.8e-124)
         (/ (+ b (/ (* c a) d)) d)
         (if (<= c 3.2e+48) t_0 (/ (+ a (* b (/ d c))) c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * a) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -2.45e+73) {
		tmp = ((b / c) / (c / d)) + (a / c);
	} else if (c <= -8.8e-94) {
		tmp = t_0;
	} else if (c <= 1.8e-124) {
		tmp = (b + ((c * a) / d)) / d;
	} else if (c <= 3.2e+48) {
		tmp = t_0;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((c * a) + (b * d)) / ((c * c) + (d * d))
    if (c <= (-2.45d+73)) then
        tmp = ((b / c) / (c / d)) + (a / c)
    else if (c <= (-8.8d-94)) then
        tmp = t_0
    else if (c <= 1.8d-124) then
        tmp = (b + ((c * a) / d)) / d
    else if (c <= 3.2d+48) then
        tmp = t_0
    else
        tmp = (a + (b * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((c * a) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -2.45e+73) {
		tmp = ((b / c) / (c / d)) + (a / c);
	} else if (c <= -8.8e-94) {
		tmp = t_0;
	} else if (c <= 1.8e-124) {
		tmp = (b + ((c * a) / d)) / d;
	} else if (c <= 3.2e+48) {
		tmp = t_0;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * a) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -2.45e+73:
		tmp = ((b / c) / (c / d)) + (a / c)
	elif c <= -8.8e-94:
		tmp = t_0
	elif c <= 1.8e-124:
		tmp = (b + ((c * a) / d)) / d
	elif c <= 3.2e+48:
		tmp = t_0
	else:
		tmp = (a + (b * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * a) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -2.45e+73)
		tmp = Float64(Float64(Float64(b / c) / Float64(c / d)) + Float64(a / c));
	elseif (c <= -8.8e-94)
		tmp = t_0;
	elseif (c <= 1.8e-124)
		tmp = Float64(Float64(b + Float64(Float64(c * a) / d)) / d);
	elseif (c <= 3.2e+48)
		tmp = t_0;
	else
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c * a) + (b * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (c <= -2.45e+73)
		tmp = ((b / c) / (c / d)) + (a / c);
	elseif (c <= -8.8e-94)
		tmp = t_0;
	elseif (c <= 1.8e-124)
		tmp = (b + ((c * a) / d)) / d;
	elseif (c <= 3.2e+48)
		tmp = t_0;
	else
		tmp = (a + (b * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * a), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.45e+73], N[(N[(N[(b / c), $MachinePrecision] / N[(c / d), $MachinePrecision]), $MachinePrecision] + N[(a / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -8.8e-94], t$95$0, If[LessEqual[c, 1.8e-124], N[(N[(b + N[(N[(c * a), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 3.2e+48], t$95$0, N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.45e73

   1. Initial program 38.2%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a + \frac{b \cdot d}{c}\right), \color{blue}{c}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \left(\frac{b \cdot d}{c}\right)\right), c\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{/.f64}\left(\left(b \cdot d\right), c\right)\right), c\right) \]

      4. *-lowering-*.f64 76.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, d\right), c\right)\right), c\right) \]

   5. Simplified 76.7%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \left(b \cdot \frac{d}{c}\right)\right), c\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \left(\frac{d}{c} \cdot b\right)\right), c\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\left(\frac{d}{c}\right), b\right)\right), c\right) \]

      4. /-lowering-/.f64 82.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, c\right), b\right)\right), c\right) \]

   7. Applied egg-rr 82.4%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{d}{c} \cdot b + a\right), c\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{d}{c} \cdot b\right), a\right), c\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(b \cdot \frac{d}{c}\right), a\right), c\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(b \cdot \frac{1}{\frac{c}{d}}\right), a\right), c\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{b}{\frac{c}{d}}\right), a\right), c\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(b, \left(\frac{c}{d}\right)\right), a\right), c\right) \]

      7. /-lowering-/.f64 82.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(b, \mathsf{/.f64}\left(c, d\right)\right), a\right), c\right) \]

   9. Applied egg-rr 82.3%

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

       1. clear-num N/A

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

       2. +-commutative N/A

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

       3. associate-/r/ N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt-in N/A

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

       6. *-commutative N/A

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

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{b}{\frac{c}{d}} \cdot \frac{1}{c}\right), \color{blue}{\left(\frac{1}{c} \cdot a\right)}\right) \]

       8. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{b \cdot \frac{1}{c}}{\frac{c}{d}}\right), \left(\color{blue}{\frac{1}{c}} \cdot a\right)\right) \]

       9. div-inv N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{\frac{b}{c}}{\frac{c}{d}}\right), \left(\frac{\color{blue}{1}}{c} \cdot a\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{b}{c}\right), \left(\frac{c}{d}\right)\right), \left(\color{blue}{\frac{1}{c}} \cdot a\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(b, c\right), \left(\frac{c}{d}\right)\right), \left(\frac{\color{blue}{1}}{c} \cdot a\right)\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(b, c\right), \mathsf{/.f64}\left(c, d\right)\right), \left(\frac{1}{\color{blue}{c}} \cdot a\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(b, c\right), \mathsf{/.f64}\left(c, d\right)\right), \left(a \cdot \color{blue}{\frac{1}{c}}\right)\right) \]

       14. un-div-inv N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(b, c\right), \mathsf{/.f64}\left(c, d\right)\right), \left(\frac{a}{\color{blue}{c}}\right)\right) \]

       15. /-lowering-/.f64 82.4%

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(b, c\right), \mathsf{/.f64}\left(c, d\right)\right), \mathsf{/.f64}\left(a, \color{blue}{c}\right)\right) \]

   11. Applied egg-rr 82.4%

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

   if -2.45e73 < c < -8.80000000000000004e-94 or 1.80000000000000005e-124 < c < 3.2000000000000001e48

   1. Initial program 86.0%

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

   if -8.80000000000000004e-94 < c < 1.80000000000000005e-124

   1. Initial program 58.0%

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

   3. Taylor expanded in d around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \frac{a \cdot c}{d}\right), \color{blue}{d}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left(\frac{a \cdot c}{d}\right)\right), d\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{/.f64}\left(\left(a \cdot c\right), d\right)\right), d\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{/.f64}\left(\left(c \cdot a\right), d\right)\right), d\right) \]

      5. *-lowering-*.f64 94.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, a\right), d\right)\right), d\right) \]

   5. Simplified 94.4%

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

   if 3.2000000000000001e48 < c

   1. Initial program 35.7%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a + \frac{b \cdot d}{c}\right), \color{blue}{c}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \left(\frac{b \cdot d}{c}\right)\right), c\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{/.f64}\left(\left(b \cdot d\right), c\right)\right), c\right) \]

      4. *-lowering-*.f64 74.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, d\right), c\right)\right), c\right) \]

   5. Simplified 74.0%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \left(b \cdot \frac{d}{c}\right)\right), c\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \left(\frac{d}{c} \cdot b\right)\right), c\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\left(\frac{d}{c}\right), b\right)\right), c\right) \]

      4. /-lowering-/.f64 78.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, c\right), b\right)\right), c\right) \]

   7. Applied egg-rr 78.8%

      \[\leadsto \frac{a + \color{blue}{\frac{d}{c} \cdot b}}{c} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.45 \cdot 10^{+73}:\\ \;\;\;\;\frac{\frac{b}{c}}{\frac{c}{d}} + \frac{a}{c}\\ \mathbf{elif}\;c \leq -8.8 \cdot 10^{-94}:\\ \;\;\;\;\frac{c \cdot a + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.8 \cdot 10^{-124}:\\ \;\;\;\;\frac{b + \frac{c \cdot a}{d}}{d}\\ \mathbf{elif}\;c \leq 3.2 \cdot 10^{+48}:\\ \;\;\;\;\frac{c \cdot a + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3 \cdot 10^{+25}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{c}{d} \cdot a}{d}\\ \mathbf{elif}\;d \leq 1650000:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -3e+25)
   (+ (/ b d) (/ (* (/ c d) a) d))
   (if (<= d 1650000.0)
     (/ (+ a (* b (/ d c))) c)
     (+ (/ b d) (* (/ c d) (/ a d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -3e+25) {
		tmp = (b / d) + (((c / d) * a) / d);
	} else if (d <= 1650000.0) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = (b / d) + ((c / d) * (a / 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 <= (-3d+25)) then
        tmp = (b / d) + (((c / d) * a) / d)
    else if (d <= 1650000.0d0) then
        tmp = (a + (b * (d / c))) / c
    else
        tmp = (b / d) + ((c / d) * (a / d))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -3e+25) {
		tmp = (b / d) + (((c / d) * a) / d);
	} else if (d <= 1650000.0) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = (b / d) + ((c / d) * (a / d));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -3e+25:
		tmp = (b / d) + (((c / d) * a) / d)
	elif d <= 1650000.0:
		tmp = (a + (b * (d / c))) / c
	else:
		tmp = (b / d) + ((c / d) * (a / d))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -3e+25)
		tmp = Float64(Float64(b / d) + Float64(Float64(Float64(c / d) * a) / d));
	elseif (d <= 1650000.0)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	else
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -3e+25)
		tmp = (b / d) + (((c / d) * a) / d);
	elseif (d <= 1650000.0)
		tmp = (a + (b * (d / c))) / c;
	else
		tmp = (b / d) + ((c / d) * (a / d));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -3e+25], N[(N[(b / d), $MachinePrecision] + N[(N[(N[(c / d), $MachinePrecision] * a), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1650000.0], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.00000000000000006e25

   1. Initial program 34.4%

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

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{b}{d}\right), \color{blue}{\left(c \cdot \frac{a}{{d}^{2}}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \left(\color{blue}{c} \cdot \frac{a}{{d}^{2}}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \left(\frac{c \cdot a}{\color{blue}{{d}^{2}}}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \left(\frac{a \cdot c}{{\color{blue}{d}}^{2}}\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \left(a \cdot \color{blue}{\frac{c}{{d}^{2}}}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{c}{{d}^{2}}\right)}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \mathsf{*.f64}\left(a, \left(\frac{c}{d \cdot \color{blue}{d}}\right)\right)\right) \]

      10. associate-/r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \mathsf{*.f64}\left(a, \left(\frac{\frac{c}{d}}{\color{blue}{d}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\left(\frac{c}{d}\right), \color{blue}{d}\right)\right)\right) \]

      12. /-lowering-/.f64 74.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(c, d\right), d\right)\right)\right) \]

   5. Simplified 74.8%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \left(\frac{\frac{c}{d}}{d} \cdot \color{blue}{a}\right)\right) \]

      2. frac-2neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \left(\frac{\mathsf{neg}\left(\frac{c}{d}\right)}{\mathsf{neg}\left(d\right)} \cdot a\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \left(\frac{\left(\mathsf{neg}\left(\frac{c}{d}\right)\right) \cdot a}{\color{blue}{\mathsf{neg}\left(d\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\frac{c}{d}\right)\right) \cdot a\right), \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{neg}\left(\frac{c}{d}\right)\right), a\right), \left(\mathsf{neg}\left(\color{blue}{d}\right)\right)\right)\right) \]

      6. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(0 - \frac{c}{d}\right), a\right), \left(\mathsf{neg}\left(d\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{c}{d}\right)\right), a\right), \left(\mathsf{neg}\left(d\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, d\right)\right), a\right), \left(\mathsf{neg}\left(d\right)\right)\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, d\right)\right), a\right), \left(0 - \color{blue}{d}\right)\right)\right) \]

      10. --lowering--.f64 80.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, d\right)\right), a\right), \mathsf{\_.f64}\left(0, \color{blue}{d}\right)\right)\right) \]

   7. Applied egg-rr 80.8%

      \[\leadsto \frac{b}{d} + \color{blue}{\frac{\left(0 - \frac{c}{d}\right) \cdot a}{0 - d}} \]

   if -3.00000000000000006e25 < d < 1.65e6

   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

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a + \frac{b \cdot d}{c}\right), \color{blue}{c}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \left(\frac{b \cdot d}{c}\right)\right), c\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{/.f64}\left(\left(b \cdot d\right), c\right)\right), c\right) \]

      4. *-lowering-*.f64 85.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, d\right), c\right)\right), c\right) \]

   5. Simplified 85.0%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \left(b \cdot \frac{d}{c}\right)\right), c\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \left(\frac{d}{c} \cdot b\right)\right), c\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\left(\frac{d}{c}\right), b\right)\right), c\right) \]

      4. /-lowering-/.f64 85.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, c\right), b\right)\right), c\right) \]

   7. Applied egg-rr 85.8%

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

   if 1.65e6 < d

   1. Initial program 42.7%

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

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{b}{d}\right), \color{blue}{\left(c \cdot \frac{a}{{d}^{2}}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \left(\color{blue}{c} \cdot \frac{a}{{d}^{2}}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \left(\frac{c \cdot a}{\color{blue}{{d}^{2}}}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \left(\frac{a \cdot c}{{\color{blue}{d}}^{2}}\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \left(a \cdot \color{blue}{\frac{c}{{d}^{2}}}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{c}{{d}^{2}}\right)}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \mathsf{*.f64}\left(a, \left(\frac{c}{d \cdot \color{blue}{d}}\right)\right)\right) \]

      10. associate-/r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \mathsf{*.f64}\left(a, \left(\frac{\frac{c}{d}}{\color{blue}{d}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\left(\frac{c}{d}\right), \color{blue}{d}\right)\right)\right) \]

      12. /-lowering-/.f64 68.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(c, d\right), d\right)\right)\right) \]

   5. Simplified 68.3%

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

      1. associate-/l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \left(a \cdot \frac{c}{\color{blue}{d \cdot d}}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \left(\frac{a \cdot c}{\color{blue}{d \cdot d}}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \left(\frac{c \cdot a}{\color{blue}{d} \cdot d}\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \left(\frac{c}{d} \cdot \color{blue}{\frac{a}{d}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \mathsf{*.f64}\left(\left(\frac{c}{d}\right), \color{blue}{\left(\frac{a}{d}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(c, d\right), \left(\frac{\color{blue}{a}}{d}\right)\right)\right) \]

      7. /-lowering-/.f64 73.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(c, d\right), \mathsf{/.f64}\left(a, \color{blue}{d}\right)\right)\right) \]

   7. Applied egg-rr 73.7%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3 \cdot 10^{+25}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{c}{d} \cdot a}{d}\\ \mathbf{elif}\;d \leq 1650000:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{if}\;d \leq -1.12 \cdot 10^{+27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 290000000:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (/ b d) (* (/ c d) (/ a d)))))
   (if (<= d -1.12e+27)
     t_0
     (if (<= d 290000000.0) (/ (+ a (* b (/ d c))) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b / d) + ((c / d) * (a / d));
	double tmp;
	if (d <= -1.12e+27) {
		tmp = t_0;
	} else if (d <= 290000000.0) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (b / d) + ((c / d) * (a / d))
    if (d <= (-1.12d+27)) then
        tmp = t_0
    else if (d <= 290000000.0d0) then
        tmp = (a + (b * (d / c))) / c
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b / d) + ((c / d) * (a / d));
	double tmp;
	if (d <= -1.12e+27) {
		tmp = t_0;
	} else if (d <= 290000000.0) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b / d) + ((c / d) * (a / d))
	tmp = 0
	if d <= -1.12e+27:
		tmp = t_0
	elif d <= 290000000.0:
		tmp = (a + (b * (d / c))) / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)))
	tmp = 0.0
	if (d <= -1.12e+27)
		tmp = t_0;
	elseif (d <= 290000000.0)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b / d) + ((c / d) * (a / d));
	tmp = 0.0;
	if (d <= -1.12e+27)
		tmp = t_0;
	elseif (d <= 290000000.0)
		tmp = (a + (b * (d / c))) / c;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.12e+27], t$95$0, If[LessEqual[d, 290000000.0], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.12e27 or 2.9e8 < d

   1. Initial program 38.7%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{b}{d}\right), \color{blue}{\left(c \cdot \frac{a}{{d}^{2}}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \left(\color{blue}{c} \cdot \frac{a}{{d}^{2}}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \left(\frac{c \cdot a}{\color{blue}{{d}^{2}}}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \left(\frac{a \cdot c}{{\color{blue}{d}}^{2}}\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \left(a \cdot \color{blue}{\frac{c}{{d}^{2}}}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{c}{{d}^{2}}\right)}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \mathsf{*.f64}\left(a, \left(\frac{c}{d \cdot \color{blue}{d}}\right)\right)\right) \]

      10. associate-/r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \mathsf{*.f64}\left(a, \left(\frac{\frac{c}{d}}{\color{blue}{d}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\left(\frac{c}{d}\right), \color{blue}{d}\right)\right)\right) \]

      12. /-lowering-/.f64 71.4%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(c, d\right), d\right)\right)\right) \]

   5. Simplified 71.4%

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

      1. associate-/l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \left(a \cdot \frac{c}{\color{blue}{d \cdot d}}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \left(\frac{a \cdot c}{\color{blue}{d \cdot d}}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \left(\frac{c \cdot a}{\color{blue}{d} \cdot d}\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \left(\frac{c}{d} \cdot \color{blue}{\frac{a}{d}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \mathsf{*.f64}\left(\left(\frac{c}{d}\right), \color{blue}{\left(\frac{a}{d}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(c, d\right), \left(\frac{\color{blue}{a}}{d}\right)\right)\right) \]

      7. /-lowering-/.f64 77.1%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(b, d\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(c, d\right), \mathsf{/.f64}\left(a, \color{blue}{d}\right)\right)\right) \]

   7. Applied egg-rr 77.1%

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

   if -1.12e27 < d < 2.9e8

   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

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a + \frac{b \cdot d}{c}\right), \color{blue}{c}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \left(\frac{b \cdot d}{c}\right)\right), c\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{/.f64}\left(\left(b \cdot d\right), c\right)\right), c\right) \]

      4. *-lowering-*.f64 85.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, d\right), c\right)\right), c\right) \]

   5. Simplified 85.0%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \left(b \cdot \frac{d}{c}\right)\right), c\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \left(\frac{d}{c} \cdot b\right)\right), c\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\left(\frac{d}{c}\right), b\right)\right), c\right) \]

      4. /-lowering-/.f64 85.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, c\right), b\right)\right), c\right) \]

   7. Applied egg-rr 85.8%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.12 \cdot 10^{+27}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq 290000000:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b + \frac{c \cdot a}{d}}{d}\\ \mathbf{if}\;d \leq -6.8 \cdot 10^{+26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 185000000:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ b (/ (* c a) d)) d)))
   (if (<= d -6.8e+26)
     t_0
     (if (<= d 185000000.0) (/ (+ a (* b (/ d c))) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b + ((c * a) / d)) / d;
	double tmp;
	if (d <= -6.8e+26) {
		tmp = t_0;
	} else if (d <= 185000000.0) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (b + ((c * a) / d)) / d
    if (d <= (-6.8d+26)) then
        tmp = t_0
    else if (d <= 185000000.0d0) then
        tmp = (a + (b * (d / c))) / c
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b + ((c * a) / d)) / d;
	double tmp;
	if (d <= -6.8e+26) {
		tmp = t_0;
	} else if (d <= 185000000.0) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b + ((c * a) / d)) / d
	tmp = 0
	if d <= -6.8e+26:
		tmp = t_0
	elif d <= 185000000.0:
		tmp = (a + (b * (d / c))) / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b + Float64(Float64(c * a) / d)) / d)
	tmp = 0.0
	if (d <= -6.8e+26)
		tmp = t_0;
	elseif (d <= 185000000.0)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b + ((c * a) / d)) / d;
	tmp = 0.0;
	if (d <= -6.8e+26)
		tmp = t_0;
	elseif (d <= 185000000.0)
		tmp = (a + (b * (d / c))) / c;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b + N[(N[(c * a), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -6.8e+26], t$95$0, If[LessEqual[d, 185000000.0], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -6.8000000000000005e26 or 1.85e8 < d

   1. Initial program 38.7%

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

   3. Taylor expanded in d around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \frac{a \cdot c}{d}\right), \color{blue}{d}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left(\frac{a \cdot c}{d}\right)\right), d\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{/.f64}\left(\left(a \cdot c\right), d\right)\right), d\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{/.f64}\left(\left(c \cdot a\right), d\right)\right), d\right) \]

      5. *-lowering-*.f64 76.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, a\right), d\right)\right), d\right) \]

   5. Simplified 76.2%

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

   if -6.8000000000000005e26 < d < 1.85e8

   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

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a + \frac{b \cdot d}{c}\right), \color{blue}{c}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \left(\frac{b \cdot d}{c}\right)\right), c\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{/.f64}\left(\left(b \cdot d\right), c\right)\right), c\right) \]

      4. *-lowering-*.f64 85.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, d\right), c\right)\right), c\right) \]

   5. Simplified 85.0%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \left(b \cdot \frac{d}{c}\right)\right), c\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \left(\frac{d}{c} \cdot b\right)\right), c\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\left(\frac{d}{c}\right), b\right)\right), c\right) \]

      4. /-lowering-/.f64 85.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, c\right), b\right)\right), c\right) \]

   7. Applied egg-rr 85.8%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.8 \cdot 10^{+26}:\\ \;\;\;\;\frac{b + \frac{c \cdot a}{d}}{d}\\ \mathbf{elif}\;d \leq 185000000:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{c \cdot a}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.52 \cdot 10^{+29}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 5.9 \cdot 10^{+29}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.52e+29)
   (/ b d)
   (if (<= d 5.9e+29) (/ (+ a (* b (/ d c))) c) (/ b d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.52e+29) {
		tmp = b / d;
	} else if (d <= 5.9e+29) {
		tmp = (a + (b * (d / c))) / 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 <= (-1.52d+29)) then
        tmp = b / d
    else if (d <= 5.9d+29) then
        tmp = (a + (b * (d / c))) / 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 <= -1.52e+29) {
		tmp = b / d;
	} else if (d <= 5.9e+29) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -1.52e+29:
		tmp = b / d
	elif d <= 5.9e+29:
		tmp = (a + (b * (d / c))) / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.52e+29)
		tmp = Float64(b / d);
	elseif (d <= 5.9e+29)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -1.52e+29)
		tmp = b / d;
	elseif (d <= 5.9e+29)
		tmp = (a + (b * (d / c))) / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.52e+29], N[(b / d), $MachinePrecision], If[LessEqual[d, 5.9e+29], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.52e29 or 5.8999999999999999e29 < d

   1. Initial program 35.9%

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

   3. Taylor expanded in c around 0

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

      1. /-lowering-/.f64 69.4%

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{d}\right) \]

   5. Simplified 69.4%

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

   if -1.52e29 < d < 5.8999999999999999e29

   1. Initial program 76.3%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a + \frac{b \cdot d}{c}\right), \color{blue}{c}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \left(\frac{b \cdot d}{c}\right)\right), c\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{/.f64}\left(\left(b \cdot d\right), c\right)\right), c\right) \]

      4. *-lowering-*.f64 83.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, d\right), c\right)\right), c\right) \]

   5. Simplified 83.0%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \left(b \cdot \frac{d}{c}\right)\right), c\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \left(\frac{d}{c} \cdot b\right)\right), c\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\left(\frac{d}{c}\right), b\right)\right), c\right) \]

      4. /-lowering-/.f64 83.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, c\right), b\right)\right), c\right) \]

   7. Applied egg-rr 83.7%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.52 \cdot 10^{+29}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 5.9 \cdot 10^{+29}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 65000000000000:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -4.5e-16) (/ b d) (if (<= d 65000000000000.0) (/ a c) (/ b d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -4.5e-16) {
		tmp = b / d;
	} else if (d <= 65000000000000.0) {
		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 <= (-4.5d-16)) then
        tmp = b / d
    else if (d <= 65000000000000.0d0) 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 <= -4.5e-16) {
		tmp = b / d;
	} else if (d <= 65000000000000.0) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -4.5e-16:
		tmp = b / d
	elif d <= 65000000000000.0:
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -4.5e-16)
		tmp = Float64(b / d);
	elseif (d <= 65000000000000.0)
		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 <= -4.5e-16)
		tmp = b / d;
	elseif (d <= 65000000000000.0)
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -4.5e-16], N[(b / d), $MachinePrecision], If[LessEqual[d, 65000000000000.0], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -4.5000000000000002e-16 or 6.5e13 < d

   1. Initial program 40.1%

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

   3. Taylor expanded in c around 0

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

      1. /-lowering-/.f64 66.5%

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{d}\right) \]

   5. Simplified 66.5%

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

   if -4.5000000000000002e-16 < d < 6.5e13

   1. Initial program 76.2%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 66.4%

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{c}\right) \]

   5. Simplified 66.4%

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

Alternative 7: 42.8% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.0%

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

3. Taylor expanded in c around inf

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

   1. /-lowering-/.f64 40.9%

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{c}\right) \]

5. Simplified 40.9%

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

Developer Target 1: 99.3% accurate, 0.1× 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 2024150 
(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))))