Complex division, real part

Percentage Accurate: 62.4% → 79.9%

Time: 8.4s

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: 62.4% 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: 79.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b + \frac{a}{d} \cdot c}{d}\\ \mathbf{if}\;d \leq -1.6 \cdot 10^{+97}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-165}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+92}:\\ \;\;\;\;\frac{a \cdot c + d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ b (* (/ a d) c)) d)))
   (if (<= d -1.6e+97)
     t_0
     (if (<= d 3.2e-165)
       (/ (+ a (* b (/ d c))) c)
       (if (<= d 1.7e+92) (/ (+ (* a c) (* d b)) (+ (* c c) (* d d))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b + ((a / d) * c)) / d;
	double tmp;
	if (d <= -1.6e+97) {
		tmp = t_0;
	} else if (d <= 3.2e-165) {
		tmp = (a + (b * (d / c))) / c;
	} else if (d <= 1.7e+92) {
		tmp = ((a * c) + (d * b)) / ((c * c) + (d * d));
	} 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 + ((a / d) * c)) / d
    if (d <= (-1.6d+97)) then
        tmp = t_0
    else if (d <= 3.2d-165) then
        tmp = (a + (b * (d / c))) / c
    else if (d <= 1.7d+92) then
        tmp = ((a * c) + (d * b)) / ((c * c) + (d * d))
    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 + ((a / d) * c)) / d;
	double tmp;
	if (d <= -1.6e+97) {
		tmp = t_0;
	} else if (d <= 3.2e-165) {
		tmp = (a + (b * (d / c))) / c;
	} else if (d <= 1.7e+92) {
		tmp = ((a * c) + (d * b)) / ((c * c) + (d * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b + ((a / d) * c)) / d
	tmp = 0
	if d <= -1.6e+97:
		tmp = t_0
	elif d <= 3.2e-165:
		tmp = (a + (b * (d / c))) / c
	elif d <= 1.7e+92:
		tmp = ((a * c) + (d * b)) / ((c * c) + (d * d))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b + Float64(Float64(a / d) * c)) / d)
	tmp = 0.0
	if (d <= -1.6e+97)
		tmp = t_0;
	elseif (d <= 3.2e-165)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	elseif (d <= 1.7e+92)
		tmp = Float64(Float64(Float64(a * c) + Float64(d * b)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b + ((a / d) * c)) / d;
	tmp = 0.0;
	if (d <= -1.6e+97)
		tmp = t_0;
	elseif (d <= 3.2e-165)
		tmp = (a + (b * (d / c))) / c;
	elseif (d <= 1.7e+92)
		tmp = ((a * c) + (d * b)) / ((c * c) + (d * d));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b + N[(N[(a / d), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -1.6e+97], t$95$0, If[LessEqual[d, 3.2e-165], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.7e+92], N[(N[(N[(a * c), $MachinePrecision] + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.60000000000000008e97 or 1.6999999999999999e92 < d

   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 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 80.7%

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

   5. Simplified 80.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 85.3%

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

   7. Applied egg-rr 85.3%

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

   if -1.60000000000000008e97 < d < 3.20000000000000013e-165

   1. Initial program 75.6%

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

   3. Taylor expanded in c around inf

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

      1. /-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 83.3%

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

   5. Simplified 83.3%

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   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 84.2%

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

   7. Applied egg-rr 84.2%

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

   if 3.20000000000000013e-165 < d < 1.6999999999999999e92

   1. Initial program 88.7%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{+97}:\\ \;\;\;\;\frac{b + \frac{a}{d} \cdot c}{d}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-165}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+92}:\\ \;\;\;\;\frac{a \cdot c + d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{d} \cdot c}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{b + \frac{a \cdot c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5.8e-19)
   (/ (+ a (* d (/ b c))) c)
   (if (<= c 7.2e-7) (/ (+ b (/ (* a c) d)) d) (/ (+ a (* b (/ d c))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.8e-19) {
		tmp = (a + (d * (b / c))) / c;
	} else if (c <= 7.2e-7) {
		tmp = (b + ((a * c) / d)) / d;
	} 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) :: tmp
    if (c <= (-5.8d-19)) then
        tmp = (a + (d * (b / c))) / c
    else if (c <= 7.2d-7) then
        tmp = (b + ((a * c) / d)) / d
    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 tmp;
	if (c <= -5.8e-19) {
		tmp = (a + (d * (b / c))) / c;
	} else if (c <= 7.2e-7) {
		tmp = (b + ((a * c) / d)) / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -5.8e-19:
		tmp = (a + (d * (b / c))) / c
	elif c <= 7.2e-7:
		tmp = (b + ((a * c) / d)) / d
	else:
		tmp = (a + (b * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -5.8e-19)
		tmp = Float64(Float64(a + Float64(d * Float64(b / c))) / c);
	elseif (c <= 7.2e-7)
		tmp = Float64(Float64(b + Float64(Float64(a * c) / d)) / d);
	else
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -5.8e-19)
		tmp = (a + (d * (b / c))) / c;
	elseif (c <= 7.2e-7)
		tmp = (b + ((a * c) / d)) / d;
	else
		tmp = (a + (b * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -5.8e-19], N[(N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 7.2e-7], N[(N[(b + N[(N[(a * c), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -5.8e-19

   1. Initial program 62.0%

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

   3. Taylor expanded in c around inf

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

      1. /-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 71.4%

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

   5. Simplified 71.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 71.6%

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

   7. Applied egg-rr 71.6%

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

   if -5.8e-19 < c < 7.19999999999999989e-7

   1. Initial program 75.7%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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 85.0%

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

   5. Simplified 85.0%

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

   if 7.19999999999999989e-7 < c

   1. Initial program 47.6%

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

   3. Taylor expanded in c around inf

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

      1. /-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 79.4%

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

   5. Simplified 79.4%

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   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 82.6%

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

   7. Applied egg-rr 82.6%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{b + \frac{a \cdot c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b + \frac{a}{d} \cdot c}{d}\\ \mathbf{if}\;d \leq -1.6 \cdot 10^{+97}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 3.45 \cdot 10^{-25}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ b (* (/ a d) c)) d)))
   (if (<= d -1.6e+97)
     t_0
     (if (<= d 3.45e-25) (/ (+ a (* b (/ d c))) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b + ((a / d) * c)) / d;
	double tmp;
	if (d <= -1.6e+97) {
		tmp = t_0;
	} else if (d <= 3.45e-25) {
		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 + ((a / d) * c)) / d
    if (d <= (-1.6d+97)) then
        tmp = t_0
    else if (d <= 3.45d-25) 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 + ((a / d) * c)) / d;
	double tmp;
	if (d <= -1.6e+97) {
		tmp = t_0;
	} else if (d <= 3.45e-25) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b + ((a / d) * c)) / d
	tmp = 0
	if d <= -1.6e+97:
		tmp = t_0
	elif d <= 3.45e-25:
		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(a / d) * c)) / d)
	tmp = 0.0
	if (d <= -1.6e+97)
		tmp = t_0;
	elseif (d <= 3.45e-25)
		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 + ((a / d) * c)) / d;
	tmp = 0.0;
	if (d <= -1.6e+97)
		tmp = t_0;
	elseif (d <= 3.45e-25)
		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[(a / d), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -1.6e+97], t$95$0, If[LessEqual[d, 3.45e-25], 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 < -1.60000000000000008e97 or 3.44999999999999987e-25 < d

   1. Initial program 48.1%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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 77.8%

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

   5. Simplified 77.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 81.5%

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

   7. Applied egg-rr 81.5%

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

   if -1.60000000000000008e97 < d < 3.44999999999999987e-25

   1. Initial program 78.7%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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 80.2%

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

   5. Simplified 80.2%

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   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 80.8%

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

   7. Applied egg-rr 80.8%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{+97}:\\ \;\;\;\;\frac{b + \frac{a}{d} \cdot c}{d}\\ \mathbf{elif}\;d \leq 3.45 \cdot 10^{-25}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{d} \cdot c}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{+97}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 9 \cdot 10^{-9}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.9e+97)
   (/ b d)
   (if (<= d 9e-9) (/ (+ a (* b (/ d c))) c) (/ b d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.9e+97) {
		tmp = b / d;
	} else if (d <= 9e-9) {
		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 <= (-1.9d+97)) then
        tmp = b / d
    else if (d <= 9d-9) 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 <= -1.9e+97) {
		tmp = b / d;
	} else if (d <= 9e-9) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -1.9e+97:
		tmp = b / d
	elif d <= 9e-9:
		tmp = (a + (b * (d / c))) / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.9e+97)
		tmp = Float64(b / d);
	elseif (d <= 9e-9)
		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 <= -1.9e+97)
		tmp = b / d;
	elseif (d <= 9e-9)
		tmp = (a + (b * (d / c))) / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.9e+97], N[(b / d), $MachinePrecision], If[LessEqual[d, 9e-9], 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 < -1.90000000000000018e97 or 8.99999999999999953e-9 < d

   1. Initial program 47.1%

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

   3. Taylor expanded in c around 0

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

      1. /-lowering-/.f64 69.8%

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

   5. Simplified 69.8%

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

   if -1.90000000000000018e97 < d < 8.99999999999999953e-9

   1. Initial program 78.9%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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 79.3%

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

   5. Simplified 79.3%

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   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 80.0%

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

   7. Applied egg-rr 80.0%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{+97}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 9 \cdot 10^{-9}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{+97}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 9.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.6e+97)
   (/ b d)
   (if (<= d 9.6e-9) (/ (+ a (* d (/ b c))) c) (/ b d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.6e+97) {
		tmp = b / d;
	} else if (d <= 9.6e-9) {
		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 <= (-1.6d+97)) then
        tmp = b / d
    else if (d <= 9.6d-9) 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 <= -1.6e+97) {
		tmp = b / d;
	} else if (d <= 9.6e-9) {
		tmp = (a + (d * (b / c))) / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -1.6e+97:
		tmp = b / d
	elif d <= 9.6e-9:
		tmp = (a + (d * (b / c))) / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.6e+97)
		tmp = Float64(b / d);
	elseif (d <= 9.6e-9)
		tmp = Float64(Float64(a + Float64(d * Float64(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 <= -1.6e+97)
		tmp = b / d;
	elseif (d <= 9.6e-9)
		tmp = (a + (d * (b / c))) / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.6e+97], N[(b / d), $MachinePrecision], If[LessEqual[d, 9.6e-9], N[(N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.60000000000000008e97 or 9.5999999999999999e-9 < d

   1. Initial program 47.1%

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

   3. Taylor expanded in c around 0

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

      1. /-lowering-/.f64 69.8%

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

   5. Simplified 69.8%

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

   if -1.60000000000000008e97 < d < 9.5999999999999999e-9

   1. Initial program 78.9%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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 79.3%

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

   5. Simplified 79.3%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 78.7%

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

   7. Applied egg-rr 78.7%

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

Alternative 6: 64.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.55 \cdot 10^{-33}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 1.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.55e-33) (/ a c) (if (<= c 1.1e-7) (/ b d) (/ a c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.55e-33) {
		tmp = a / c;
	} else if (c <= 1.1e-7) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-1.55d-33)) then
        tmp = a / c
    else if (c <= 1.1d-7) then
        tmp = b / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.55e-33) {
		tmp = a / c;
	} else if (c <= 1.1e-7) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.55e-33:
		tmp = a / c
	elif c <= 1.1e-7:
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.55e-33)
		tmp = Float64(a / c);
	elseif (c <= 1.1e-7)
		tmp = Float64(b / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.55e-33)
		tmp = a / c;
	elseif (c <= 1.1e-7)
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.55e-33], N[(a / c), $MachinePrecision], If[LessEqual[c, 1.1e-7], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -1.54999999999999998e-33 or 1.1000000000000001e-7 < c

   1. Initial program 54.3%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 68.0%

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

   5. Simplified 68.0%

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

   if -1.54999999999999998e-33 < c < 1.1000000000000001e-7

   1. Initial program 76.1%

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

   3. Taylor expanded in c around 0

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

      1. /-lowering-/.f64 71.7%

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

   5. Simplified 71.7%

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

Alternative 7: 42.3% 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 65.4%

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

3. Taylor expanded in c around inf

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

   1. /-lowering-/.f64 42.2%

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

5. Simplified 42.2%

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

Developer Target 1: 99.4% 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 2024163 
(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))))