Complex division, real part

Percentage Accurate: 61.1% → 80.1%

Time: 7.8s

Alternatives: 9

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: 61.1% 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: 80.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{d} + \frac{\frac{a}{d} \cdot c}{d}\\ \mathbf{if}\;d \leq -1.2 \cdot 10^{+83}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -1.2 \cdot 10^{-131}:\\ \;\;\;\;\frac{1}{c \cdot c + d \cdot d} \cdot \left(a \cdot c + d \cdot b\right)\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{+74}:\\ \;\;\;\;\frac{a + \left(d \cdot b\right) \cdot \frac{1}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (/ b d) (/ (* (/ a d) c) d))))
   (if (<= d -1.2e+83)
     t_0
     (if (<= d -1.2e-131)
       (* (/ 1.0 (+ (* c c) (* d d))) (+ (* a c) (* d b)))
       (if (<= d 9.2e+74) (/ (+ a (* (* d b) (/ 1.0 c))) c) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b / d) + (((a / d) * c) / d);
	double tmp;
	if (d <= -1.2e+83) {
		tmp = t_0;
	} else if (d <= -1.2e-131) {
		tmp = (1.0 / ((c * c) + (d * d))) * ((a * c) + (d * b));
	} else if (d <= 9.2e+74) {
		tmp = (a + ((d * b) * (1.0 / 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) + (((a / d) * c) / d)
    if (d <= (-1.2d+83)) then
        tmp = t_0
    else if (d <= (-1.2d-131)) then
        tmp = (1.0d0 / ((c * c) + (d * d))) * ((a * c) + (d * b))
    else if (d <= 9.2d+74) then
        tmp = (a + ((d * b) * (1.0d0 / 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) + (((a / d) * c) / d);
	double tmp;
	if (d <= -1.2e+83) {
		tmp = t_0;
	} else if (d <= -1.2e-131) {
		tmp = (1.0 / ((c * c) + (d * d))) * ((a * c) + (d * b));
	} else if (d <= 9.2e+74) {
		tmp = (a + ((d * b) * (1.0 / c))) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b / d) + (((a / d) * c) / d)
	tmp = 0
	if d <= -1.2e+83:
		tmp = t_0
	elif d <= -1.2e-131:
		tmp = (1.0 / ((c * c) + (d * d))) * ((a * c) + (d * b))
	elif d <= 9.2e+74:
		tmp = (a + ((d * b) * (1.0 / c))) / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b / d) + Float64(Float64(Float64(a / d) * c) / d))
	tmp = 0.0
	if (d <= -1.2e+83)
		tmp = t_0;
	elseif (d <= -1.2e-131)
		tmp = Float64(Float64(1.0 / Float64(Float64(c * c) + Float64(d * d))) * Float64(Float64(a * c) + Float64(d * b)));
	elseif (d <= 9.2e+74)
		tmp = Float64(Float64(a + Float64(Float64(d * b) * Float64(1.0 / c))) / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b / d) + (((a / d) * c) / d);
	tmp = 0.0;
	if (d <= -1.2e+83)
		tmp = t_0;
	elseif (d <= -1.2e-131)
		tmp = (1.0 / ((c * c) + (d * d))) * ((a * c) + (d * b));
	elseif (d <= 9.2e+74)
		tmp = (a + ((d * b) * (1.0 / 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[(N[(a / d), $MachinePrecision] * c), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.2e+83], t$95$0, If[LessEqual[d, -1.2e-131], N[(N[(1.0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(a * c), $MachinePrecision] + N[(d * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 9.2e+74], N[(N[(a + N[(N[(d * b), $MachinePrecision] * N[(1.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.19999999999999996e83 or 9.1999999999999994e74 < d

   1. Initial program 35.4%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. /-lowering-/.f64 N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 81.6%

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

   5. Simplified 81.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 84.2%

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

   7. Applied egg-rr 84.2%

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

   if -1.19999999999999996e83 < d < -1.2e-131

   1. Initial program 88.3%

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

      1. div-inv N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. *-commutative N/A

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

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

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

      6. clear-num N/A

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

      7. flip-+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. *-lowering-*.f64 88.3%

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

   4. Applied egg-rr 88.3%

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

   if -1.2e-131 < d < 9.1999999999999994e74

   1. Initial program 66.1%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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. /-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 86.1%

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

   5. Simplified 86.1%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 86.2%

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

   7. Applied egg-rr 86.2%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.2 \cdot 10^{+83}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a}{d} \cdot c}{d}\\ \mathbf{elif}\;d \leq -1.2 \cdot 10^{-131}:\\ \;\;\;\;\frac{1}{c \cdot c + d \cdot d} \cdot \left(a \cdot c + d \cdot b\right)\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{+74}:\\ \;\;\;\;\frac{a + \left(d \cdot b\right) \cdot \frac{1}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a}{d} \cdot c}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{d} + \frac{\frac{a}{d} \cdot c}{d}\\ \mathbf{if}\;d \leq -6 \cdot 10^{+36}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -3.7 \cdot 10^{-112}:\\ \;\;\;\;\frac{a \cdot c + d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 7.7 \cdot 10^{+74}:\\ \;\;\;\;\frac{a + \frac{1}{\frac{c}{d \cdot b}}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (/ b d) (/ (* (/ a d) c) d))))
   (if (<= d -6e+36)
     t_0
     (if (<= d -3.7e-112)
       (/ (+ (* a c) (* d b)) (+ (* c c) (* d d)))
       (if (<= d 7.7e+74) (/ (+ a (/ 1.0 (/ c (* d b)))) c) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b / d) + (((a / d) * c) / d);
	double tmp;
	if (d <= -6e+36) {
		tmp = t_0;
	} else if (d <= -3.7e-112) {
		tmp = ((a * c) + (d * b)) / ((c * c) + (d * d));
	} else if (d <= 7.7e+74) {
		tmp = (a + (1.0 / (c / (d * b)))) / 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) + (((a / d) * c) / d)
    if (d <= (-6d+36)) then
        tmp = t_0
    else if (d <= (-3.7d-112)) then
        tmp = ((a * c) + (d * b)) / ((c * c) + (d * d))
    else if (d <= 7.7d+74) then
        tmp = (a + (1.0d0 / (c / (d * b)))) / 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) + (((a / d) * c) / d);
	double tmp;
	if (d <= -6e+36) {
		tmp = t_0;
	} else if (d <= -3.7e-112) {
		tmp = ((a * c) + (d * b)) / ((c * c) + (d * d));
	} else if (d <= 7.7e+74) {
		tmp = (a + (1.0 / (c / (d * b)))) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b / d) + (((a / d) * c) / d)
	tmp = 0
	if d <= -6e+36:
		tmp = t_0
	elif d <= -3.7e-112:
		tmp = ((a * c) + (d * b)) / ((c * c) + (d * d))
	elif d <= 7.7e+74:
		tmp = (a + (1.0 / (c / (d * b)))) / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b / d) + Float64(Float64(Float64(a / d) * c) / d))
	tmp = 0.0
	if (d <= -6e+36)
		tmp = t_0;
	elseif (d <= -3.7e-112)
		tmp = Float64(Float64(Float64(a * c) + Float64(d * b)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 7.7e+74)
		tmp = Float64(Float64(a + Float64(1.0 / Float64(c / Float64(d * b)))) / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b / d) + (((a / d) * c) / d);
	tmp = 0.0;
	if (d <= -6e+36)
		tmp = t_0;
	elseif (d <= -3.7e-112)
		tmp = ((a * c) + (d * b)) / ((c * c) + (d * d));
	elseif (d <= 7.7e+74)
		tmp = (a + (1.0 / (c / (d * b)))) / 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[(N[(a / d), $MachinePrecision] * c), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -6e+36], t$95$0, If[LessEqual[d, -3.7e-112], N[(N[(N[(a * c), $MachinePrecision] + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.7e+74], N[(N[(a + N[(1.0 / N[(c / N[(d * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -6e36 or 7.70000000000000042e74 < d

   1. Initial program 38.8%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{a \cdot c}{{d}^{2}}} \]
   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. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. /-lowering-/.f64 N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 80.1%

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

   5. Simplified 80.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 82.5%

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

   7. Applied egg-rr 82.5%

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

   if -6e36 < d < -3.6999999999999998e-112

   1. Initial program 94.3%

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

   if -3.6999999999999998e-112 < d < 7.70000000000000042e74

   1. Initial program 66.7%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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 86.4%

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

   5. Simplified 86.4%

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

      1. clear-num N/A

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 86.4%

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

   7. Applied egg-rr 86.4%

      \[\leadsto \frac{a + \color{blue}{\frac{1}{\frac{c}{d \cdot b}}}}{c} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6 \cdot 10^{+36}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a}{d} \cdot c}{d}\\ \mathbf{elif}\;d \leq -3.7 \cdot 10^{-112}:\\ \;\;\;\;\frac{a \cdot c + d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 7.7 \cdot 10^{+74}:\\ \;\;\;\;\frac{a + \frac{1}{\frac{c}{d \cdot b}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a}{d} \cdot c}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{d} + \frac{\frac{a}{d} \cdot c}{d}\\ \mathbf{if}\;d \leq -11500000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 7.7 \cdot 10^{+74}:\\ \;\;\;\;\frac{a + \frac{1}{\frac{c}{d \cdot b}}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (/ b d) (/ (* (/ a d) c) d))))
   (if (<= d -11500000.0)
     t_0
     (if (<= d 7.7e+74) (/ (+ a (/ 1.0 (/ c (* d b)))) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b / d) + (((a / d) * c) / d);
	double tmp;
	if (d <= -11500000.0) {
		tmp = t_0;
	} else if (d <= 7.7e+74) {
		tmp = (a + (1.0 / (c / (d * b)))) / 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) + (((a / d) * c) / d)
    if (d <= (-11500000.0d0)) then
        tmp = t_0
    else if (d <= 7.7d+74) then
        tmp = (a + (1.0d0 / (c / (d * b)))) / 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) + (((a / d) * c) / d);
	double tmp;
	if (d <= -11500000.0) {
		tmp = t_0;
	} else if (d <= 7.7e+74) {
		tmp = (a + (1.0 / (c / (d * b)))) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b / d) + (((a / d) * c) / d)
	tmp = 0
	if d <= -11500000.0:
		tmp = t_0
	elif d <= 7.7e+74:
		tmp = (a + (1.0 / (c / (d * b)))) / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b / d) + Float64(Float64(Float64(a / d) * c) / d))
	tmp = 0.0
	if (d <= -11500000.0)
		tmp = t_0;
	elseif (d <= 7.7e+74)
		tmp = Float64(Float64(a + Float64(1.0 / Float64(c / Float64(d * b)))) / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b / d) + (((a / d) * c) / d);
	tmp = 0.0;
	if (d <= -11500000.0)
		tmp = t_0;
	elseif (d <= 7.7e+74)
		tmp = (a + (1.0 / (c / (d * b)))) / 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[(N[(a / d), $MachinePrecision] * c), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -11500000.0], t$95$0, If[LessEqual[d, 7.7e+74], N[(N[(a + N[(1.0 / N[(c / N[(d * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.15e7 or 7.70000000000000042e74 < d

   1. Initial program 42.4%

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

   3. Taylor expanded in c around 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. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. /-lowering-/.f64 N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 79.7%

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

   5. Simplified 79.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 81.9%

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

   7. Applied egg-rr 81.9%

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

   if -1.15e7 < d < 7.70000000000000042e74

   1. Initial program 72.5%

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

   3. Taylor expanded in c around inf

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

      1. /-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 81.0%

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

   5. Simplified 81.0%

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

      1. clear-num N/A

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 81.0%

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

   7. Applied egg-rr 81.0%

      \[\leadsto \frac{a + \color{blue}{\frac{1}{\frac{c}{d \cdot b}}}}{c} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 78.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{d} + \frac{\frac{a}{d} \cdot c}{d}\\ \mathbf{if}\;d \leq -5200000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 7.8 \cdot 10^{+74}:\\ \;\;\;\;\frac{a + \left(d \cdot b\right) \cdot \frac{1}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (/ b d) (/ (* (/ a d) c) d))))
   (if (<= d -5200000.0)
     t_0
     (if (<= d 7.8e+74) (/ (+ a (* (* d b) (/ 1.0 c))) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b / d) + (((a / d) * c) / d);
	double tmp;
	if (d <= -5200000.0) {
		tmp = t_0;
	} else if (d <= 7.8e+74) {
		tmp = (a + ((d * b) * (1.0 / 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) + (((a / d) * c) / d)
    if (d <= (-5200000.0d0)) then
        tmp = t_0
    else if (d <= 7.8d+74) then
        tmp = (a + ((d * b) * (1.0d0 / 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) + (((a / d) * c) / d);
	double tmp;
	if (d <= -5200000.0) {
		tmp = t_0;
	} else if (d <= 7.8e+74) {
		tmp = (a + ((d * b) * (1.0 / c))) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b / d) + (((a / d) * c) / d)
	tmp = 0
	if d <= -5200000.0:
		tmp = t_0
	elif d <= 7.8e+74:
		tmp = (a + ((d * b) * (1.0 / c))) / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b / d) + Float64(Float64(Float64(a / d) * c) / d))
	tmp = 0.0
	if (d <= -5200000.0)
		tmp = t_0;
	elseif (d <= 7.8e+74)
		tmp = Float64(Float64(a + Float64(Float64(d * b) * Float64(1.0 / c))) / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b / d) + (((a / d) * c) / d);
	tmp = 0.0;
	if (d <= -5200000.0)
		tmp = t_0;
	elseif (d <= 7.8e+74)
		tmp = (a + ((d * b) * (1.0 / 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[(N[(a / d), $MachinePrecision] * c), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5200000.0], t$95$0, If[LessEqual[d, 7.8e+74], N[(N[(a + N[(N[(d * b), $MachinePrecision] * N[(1.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -5.2e6 or 7.80000000000000015e74 < d

   1. Initial program 42.4%

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

   3. Taylor expanded in c around 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. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. /-lowering-/.f64 N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 79.7%

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

   5. Simplified 79.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 81.9%

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

   7. Applied egg-rr 81.9%

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

   if -5.2e6 < d < 7.80000000000000015e74

   1. Initial program 72.5%

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

   3. Taylor expanded in c around inf

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

      1. /-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 81.0%

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

   5. Simplified 81.0%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 81.0%

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

   7. Applied egg-rr 81.0%

      \[\leadsto \frac{a + \color{blue}{\frac{1}{c} \cdot \left(d \cdot b\right)}}{c} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5200000:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a}{d} \cdot c}{d}\\ \mathbf{elif}\;d \leq 7.8 \cdot 10^{+74}:\\ \;\;\;\;\frac{a + \left(d \cdot b\right) \cdot \frac{1}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a}{d} \cdot c}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{d} + \frac{\frac{a}{d} \cdot c}{d}\\ \mathbf{if}\;d \leq -5600000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 7.7 \cdot 10^{+74}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (/ b d) (/ (* (/ a d) c) d))))
   (if (<= d -5600000.0)
     t_0
     (if (<= d 7.7e+74) (/ (+ a (/ (* d b) c)) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b / d) + (((a / d) * c) / d);
	double tmp;
	if (d <= -5600000.0) {
		tmp = t_0;
	} else if (d <= 7.7e+74) {
		tmp = (a + ((d * b) / 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) + (((a / d) * c) / d)
    if (d <= (-5600000.0d0)) then
        tmp = t_0
    else if (d <= 7.7d+74) then
        tmp = (a + ((d * b) / 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) + (((a / d) * c) / d);
	double tmp;
	if (d <= -5600000.0) {
		tmp = t_0;
	} else if (d <= 7.7e+74) {
		tmp = (a + ((d * b) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b / d) + (((a / d) * c) / d)
	tmp = 0
	if d <= -5600000.0:
		tmp = t_0
	elif d <= 7.7e+74:
		tmp = (a + ((d * b) / c)) / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b / d) + Float64(Float64(Float64(a / d) * c) / d))
	tmp = 0.0
	if (d <= -5600000.0)
		tmp = t_0;
	elseif (d <= 7.7e+74)
		tmp = Float64(Float64(a + Float64(Float64(d * b) / c)) / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b / d) + (((a / d) * c) / d);
	tmp = 0.0;
	if (d <= -5600000.0)
		tmp = t_0;
	elseif (d <= 7.7e+74)
		tmp = (a + ((d * b) / 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[(N[(a / d), $MachinePrecision] * c), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5600000.0], t$95$0, If[LessEqual[d, 7.7e+74], N[(N[(a + N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -5.6e6 or 7.70000000000000042e74 < d

   1. Initial program 42.4%

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

   3. Taylor expanded in c around 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. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. /-lowering-/.f64 N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 79.7%

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

   5. Simplified 79.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 81.9%

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

   7. Applied egg-rr 81.9%

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

   if -5.6e6 < d < 7.70000000000000042e74

   1. Initial program 72.5%

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

   3. Taylor expanded in c around inf

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

      1. /-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 81.0%

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

   5. Simplified 81.0%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5600000:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a}{d} \cdot c}{d}\\ \mathbf{elif}\;d \leq 7.7 \cdot 10^{+74}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a}{d} \cdot c}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -7000000:\\ \;\;\;\;\frac{b + \frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;d \leq 7.7 \cdot 10^{+74}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -7000000.0)
   (/ (+ b (/ (* a c) d)) d)
   (if (<= d 7.7e+74) (/ (+ a (/ (* d b) c)) c) (/ b d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -7000000.0) {
		tmp = (b + ((a * c) / d)) / d;
	} else if (d <= 7.7e+74) {
		tmp = (a + ((d * b) / 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 <= (-7000000.0d0)) then
        tmp = (b + ((a * c) / d)) / d
    else if (d <= 7.7d+74) then
        tmp = (a + ((d * b) / 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 <= -7000000.0) {
		tmp = (b + ((a * c) / d)) / d;
	} else if (d <= 7.7e+74) {
		tmp = (a + ((d * b) / c)) / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -7000000.0:
		tmp = (b + ((a * c) / d)) / d
	elif d <= 7.7e+74:
		tmp = (a + ((d * b) / c)) / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -7000000.0)
		tmp = Float64(Float64(b + Float64(Float64(a * c) / d)) / d);
	elseif (d <= 7.7e+74)
		tmp = Float64(Float64(a + Float64(Float64(d * b) / 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 <= -7000000.0)
		tmp = (b + ((a * c) / d)) / d;
	elseif (d <= 7.7e+74)
		tmp = (a + ((d * b) / c)) / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -7000000.0], N[(N[(b + N[(N[(a * c), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 7.7e+74], N[(N[(a + N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -7e6

   1. Initial program 46.2%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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. /-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 82.9%

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

   5. Simplified 82.9%

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

   if -7e6 < d < 7.70000000000000042e74

   1. Initial program 72.5%

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

   3. Taylor expanded in c around inf

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

      1. /-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 81.0%

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

   5. Simplified 81.0%

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

   if 7.70000000000000042e74 < d

   1. Initial program 36.0%

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

   3. Taylor expanded in c around 0

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

      1. /-lowering-/.f64 74.5%

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

   5. Simplified 74.5%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7000000:\\ \;\;\;\;\frac{b + \frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;d \leq 7.7 \cdot 10^{+74}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.45 \cdot 10^{+41}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 7.7 \cdot 10^{+74}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.45e+41)
   (/ b d)
   (if (<= d 7.7e+74) (/ (+ a (/ (* d b) c)) c) (/ b d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.45e+41) {
		tmp = b / d;
	} else if (d <= 7.7e+74) {
		tmp = (a + ((d * b) / 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 <= (-1.45d+41)) then
        tmp = b / d
    else if (d <= 7.7d+74) then
        tmp = (a + ((d * b) / 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 <= -1.45e+41) {
		tmp = b / d;
	} else if (d <= 7.7e+74) {
		tmp = (a + ((d * b) / c)) / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -1.45e+41:
		tmp = b / d
	elif d <= 7.7e+74:
		tmp = (a + ((d * b) / c)) / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.45e+41)
		tmp = Float64(b / d);
	elseif (d <= 7.7e+74)
		tmp = Float64(Float64(a + Float64(Float64(d * b) / 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 <= -1.45e+41)
		tmp = b / d;
	elseif (d <= 7.7e+74)
		tmp = (a + ((d * b) / c)) / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.45e+41], N[(b / d), $MachinePrecision], If[LessEqual[d, 7.7e+74], N[(N[(a + N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.44999999999999994e41 or 7.70000000000000042e74 < d

   1. Initial program 39.2%

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

   3. Taylor expanded in c around 0

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

      1. /-lowering-/.f64 76.2%

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

   5. Simplified 76.2%

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

   if -1.44999999999999994e41 < d < 7.70000000000000042e74

   1. Initial program 73.3%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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 78.7%

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

   5. Simplified 78.7%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.45 \cdot 10^{+41}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 7.7 \cdot 10^{+74}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -45000000000000:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 10^{+78}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -45000000000000.0) (/ b d) (if (<= d 1e+78) (/ a c) (/ b d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -45000000000000.0) {
		tmp = b / d;
	} else if (d <= 1e+78) {
		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 <= (-45000000000000.0d0)) then
        tmp = b / d
    else if (d <= 1d+78) 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 <= -45000000000000.0) {
		tmp = b / d;
	} else if (d <= 1e+78) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -45000000000000.0:
		tmp = b / d
	elif d <= 1e+78:
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -45000000000000.0)
		tmp = Float64(b / d);
	elseif (d <= 1e+78)
		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 <= -45000000000000.0)
		tmp = b / d;
	elseif (d <= 1e+78)
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -45000000000000.0], N[(b / d), $MachinePrecision], If[LessEqual[d, 1e+78], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -4.5e13 or 1.00000000000000001e78 < d

   1. Initial program 41.9%

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

   3. Taylor expanded in c around 0

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

      1. /-lowering-/.f64 74.4%

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

   5. Simplified 74.4%

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

   if -4.5e13 < d < 1.00000000000000001e78

   1. Initial program 72.7%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 66.0%

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

   5. Simplified 66.0%

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

Alternative 9: 42.7% 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 58.4%

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

3. Taylor expanded in c around inf

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

   1. /-lowering-/.f64 44.7%

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

5. Simplified 44.7%

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

Developer Target 1: 99.5% 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 2024158 
(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))))