Complex division, real part

Percentage Accurate: 62.2% → 82.9%

Time: 10.8s

Alternatives: 8

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b + c \cdot \frac{a}{d}}{d}\\ \mathbf{if}\;d \leq -4 \cdot 10^{+80}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -8.8 \cdot 10^{-112}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-164}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{+25}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ b (* c (/ a d))) d)))
   (if (<= d -4e+80)
     t_0
     (if (<= d -8.8e-112)
       (/ (fma a c (* d b)) (fma c c (* d d)))
       (if (<= d 3.5e-164)
         (/ (+ a (/ (* d b) c)) c)
         (if (<= d 2.8e+25)
           (/ (+ (* d b) (* c a)) (+ (* d d) (* c c)))
           t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b + (c * (a / d))) / d;
	double tmp;
	if (d <= -4e+80) {
		tmp = t_0;
	} else if (d <= -8.8e-112) {
		tmp = fma(a, c, (d * b)) / fma(c, c, (d * d));
	} else if (d <= 3.5e-164) {
		tmp = (a + ((d * b) / c)) / c;
	} else if (d <= 2.8e+25) {
		tmp = ((d * b) + (c * a)) / ((d * d) + (c * c));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b + Float64(c * Float64(a / d))) / d)
	tmp = 0.0
	if (d <= -4e+80)
		tmp = t_0;
	elseif (d <= -8.8e-112)
		tmp = Float64(fma(a, c, Float64(d * b)) / fma(c, c, Float64(d * d)));
	elseif (d <= 3.5e-164)
		tmp = Float64(Float64(a + Float64(Float64(d * b) / c)) / c);
	elseif (d <= 2.8e+25)
		tmp = Float64(Float64(Float64(d * b) + Float64(c * a)) / Float64(Float64(d * d) + Float64(c * c)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -4e+80], t$95$0, If[LessEqual[d, -8.8e-112], N[(N[(a * c + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.5e-164], N[(N[(a + N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2.8e+25], N[(N[(N[(d * b), $MachinePrecision] + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -4e80 or 2.8000000000000002e25 < d

   1. Initial program 40.9%

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

      1. fma-define 40.9%

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

      2. fma-define 40.9%

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

   3. Simplified 40.9%

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

   5. Taylor expanded in d around inf 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-/l* 83.1%

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

   7. Applied egg-rr 83.1%

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

   if -4e80 < d < -8.80000000000000085e-112

   1. Initial program 94.3%

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

      1. fma-define 94.3%

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

      2. fma-define 94.3%

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

   3. Simplified 94.3%

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

   if -8.80000000000000085e-112 < d < 3.5e-164

   1. Initial program 59.4%

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

      1. fma-define 59.4%

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

      2. fma-define 59.4%

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

   3. Simplified 59.4%

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

   5. Taylor expanded in c around inf 92.7%

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

      1. *-commutative 92.7%

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

   7. Simplified 92.7%

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

   if 3.5e-164 < d < 2.8000000000000002e25

   1. Initial program 76.9%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{+80}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \mathbf{elif}\;d \leq -8.8 \cdot 10^{-112}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-164}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{+25}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ t_1 := \frac{b + c \cdot \frac{a}{d}}{d}\\ \mathbf{if}\;d \leq -1.8 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -2.55 \cdot 10^{-112}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-164}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{+25}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* d b) (* c a)) (+ (* d d) (* c c))))
        (t_1 (/ (+ b (* c (/ a d))) d)))
   (if (<= d -1.8e+80)
     t_1
     (if (<= d -2.55e-112)
       t_0
       (if (<= d 3.5e-164)
         (/ (+ a (/ (* d b) c)) c)
         (if (<= d 2.8e+25) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((d * b) + (c * a)) / ((d * d) + (c * c));
	double t_1 = (b + (c * (a / d))) / d;
	double tmp;
	if (d <= -1.8e+80) {
		tmp = t_1;
	} else if (d <= -2.55e-112) {
		tmp = t_0;
	} else if (d <= 3.5e-164) {
		tmp = (a + ((d * b) / c)) / c;
	} else if (d <= 2.8e+25) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = ((d * b) + (c * a)) / ((d * d) + (c * c))
    t_1 = (b + (c * (a / d))) / d
    if (d <= (-1.8d+80)) then
        tmp = t_1
    else if (d <= (-2.55d-112)) then
        tmp = t_0
    else if (d <= 3.5d-164) then
        tmp = (a + ((d * b) / c)) / c
    else if (d <= 2.8d+25) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((d * b) + (c * a)) / ((d * d) + (c * c));
	double t_1 = (b + (c * (a / d))) / d;
	double tmp;
	if (d <= -1.8e+80) {
		tmp = t_1;
	} else if (d <= -2.55e-112) {
		tmp = t_0;
	} else if (d <= 3.5e-164) {
		tmp = (a + ((d * b) / c)) / c;
	} else if (d <= 2.8e+25) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((d * b) + (c * a)) / ((d * d) + (c * c))
	t_1 = (b + (c * (a / d))) / d
	tmp = 0
	if d <= -1.8e+80:
		tmp = t_1
	elif d <= -2.55e-112:
		tmp = t_0
	elif d <= 3.5e-164:
		tmp = (a + ((d * b) / c)) / c
	elif d <= 2.8e+25:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(d * b) + Float64(c * a)) / Float64(Float64(d * d) + Float64(c * c)))
	t_1 = Float64(Float64(b + Float64(c * Float64(a / d))) / d)
	tmp = 0.0
	if (d <= -1.8e+80)
		tmp = t_1;
	elseif (d <= -2.55e-112)
		tmp = t_0;
	elseif (d <= 3.5e-164)
		tmp = Float64(Float64(a + Float64(Float64(d * b) / c)) / c);
	elseif (d <= 2.8e+25)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((d * b) + (c * a)) / ((d * d) + (c * c));
	t_1 = (b + (c * (a / d))) / d;
	tmp = 0.0;
	if (d <= -1.8e+80)
		tmp = t_1;
	elseif (d <= -2.55e-112)
		tmp = t_0;
	elseif (d <= 3.5e-164)
		tmp = (a + ((d * b) / c)) / c;
	elseif (d <= 2.8e+25)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(d * b), $MachinePrecision] + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -1.8e+80], t$95$1, If[LessEqual[d, -2.55e-112], t$95$0, If[LessEqual[d, 3.5e-164], N[(N[(a + N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2.8e+25], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.79999999999999997e80 or 2.8000000000000002e25 < d

   1. Initial program 40.9%

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

      1. fma-define 40.9%

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

      2. fma-define 40.9%

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

   3. Simplified 40.9%

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

   5. Taylor expanded in d around inf 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-/l* 83.1%

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

   7. Applied egg-rr 83.1%

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

   if -1.79999999999999997e80 < d < -2.55000000000000007e-112 or 3.5e-164 < d < 2.8000000000000002e25

   1. Initial program 84.1%

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

   if -2.55000000000000007e-112 < d < 3.5e-164

   1. Initial program 59.4%

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

      1. fma-define 59.4%

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

      2. fma-define 59.4%

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

   3. Simplified 59.4%

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

   5. Taylor expanded in c around inf 92.7%

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

      1. *-commutative 92.7%

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

   7. Simplified 92.7%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.8 \cdot 10^{+80}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \mathbf{elif}\;d \leq -2.55 \cdot 10^{-112}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-164}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{+25}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -7.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{a + \frac{d}{\frac{c}{b}}}{c}\\ \mathbf{elif}\;c \leq 4 \cdot 10^{+36}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{1}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -7.5e+72)
   (/ (+ a (/ d (/ c b))) c)
   (if (<= c 4e+36)
     (/ (+ b (* a (/ c d))) d)
     (/ (+ a (* b (/ 1.0 (/ c d)))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -7.5e+72) {
		tmp = (a + (d / (c / b))) / c;
	} else if (c <= 4e+36) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = (a + (b * (1.0 / (c / d)))) / 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 <= (-7.5d+72)) then
        tmp = (a + (d / (c / b))) / c
    else if (c <= 4d+36) then
        tmp = (b + (a * (c / d))) / d
    else
        tmp = (a + (b * (1.0d0 / (c / d)))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -7.5e+72) {
		tmp = (a + (d / (c / b))) / c;
	} else if (c <= 4e+36) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = (a + (b * (1.0 / (c / d)))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -7.5e+72:
		tmp = (a + (d / (c / b))) / c
	elif c <= 4e+36:
		tmp = (b + (a * (c / d))) / d
	else:
		tmp = (a + (b * (1.0 / (c / d)))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -7.5e+72)
		tmp = Float64(Float64(a + Float64(d / Float64(c / b))) / c);
	elseif (c <= 4e+36)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	else
		tmp = Float64(Float64(a + Float64(b * Float64(1.0 / Float64(c / d)))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -7.5e+72)
		tmp = (a + (d / (c / b))) / c;
	elseif (c <= 4e+36)
		tmp = (b + (a * (c / d))) / d;
	else
		tmp = (a + (b * (1.0 / (c / d)))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -7.5e+72], N[(N[(a + N[(d / N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 4e+36], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(N[(a + N[(b * N[(1.0 / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -7.50000000000000027e72

   1. Initial program 42.6%

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

      1. fma-define 42.6%

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

      2. fma-define 42.6%

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

   3. Simplified 42.6%

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

   5. Taylor expanded in c around inf 76.0%

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

      1. *-commutative 76.0%

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

   7. Simplified 76.0%

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

      1. associate-/l* 83.4%

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

      2. *-commutative 83.4%

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

   9. Applied egg-rr 83.4%

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

       1. *-commutative 83.4%

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

       2. clear-num 84.6%

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

       3. un-div-inv 84.6%

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

   11. Applied egg-rr 84.6%

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

   if -7.50000000000000027e72 < c < 4.00000000000000017e36

   1. Initial program 69.9%

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

      1. fma-define 69.9%

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

      2. fma-define 69.9%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in d around inf 78.9%

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

      1. associate-/l* 79.1%

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

   7. Simplified 79.1%

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

   if 4.00000000000000017e36 < c

   1. Initial program 55.5%

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

      1. fma-define 55.5%

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

      2. fma-define 55.5%

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

   3. Simplified 55.5%

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

   5. Taylor expanded in c around inf 78.7%

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

      1. *-commutative 78.7%

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

   7. Simplified 78.7%

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

      1. associate-/l* 82.5%

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

      2. *-commutative 82.5%

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

   9. Applied egg-rr 82.5%

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

       1. *-commutative 82.5%

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

       2. associate-/l* 78.7%

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

       3. clear-num 78.7%

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

       4. associate-/r* 82.6%

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

   11. Applied egg-rr 82.6%

       \[\leadsto \frac{a + \color{blue}{\frac{1}{\frac{\frac{c}{d}}{b}}}}{c} \]
   12. Step-by-step derivation

       1. associate-/r/ 82.6%

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

   13. Simplified 82.6%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{a + \frac{d}{\frac{c}{b}}}{c}\\ \mathbf{elif}\;c \leq 4 \cdot 10^{+36}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{1}{\frac{c}{d}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.2 \cdot 10^{+72} \lor \neg \left(c \leq 10^{+31}\right):\\ \;\;\;\;\frac{a + \frac{d}{\frac{c}{b}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -4.2e+72) (not (<= c 1e+31)))
   (/ (+ a (/ d (/ c b))) c)
   (/ (+ b (* a (/ c d))) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4.2e+72) || !(c <= 1e+31)) {
		tmp = (a + (d / (c / b))) / c;
	} else {
		tmp = (b + (a * (c / d))) / 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 ((c <= (-4.2d+72)) .or. (.not. (c <= 1d+31))) then
        tmp = (a + (d / (c / b))) / c
    else
        tmp = (b + (a * (c / d))) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4.2e+72) || !(c <= 1e+31)) {
		tmp = (a + (d / (c / b))) / c;
	} else {
		tmp = (b + (a * (c / d))) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -4.2e+72) or not (c <= 1e+31):
		tmp = (a + (d / (c / b))) / c
	else:
		tmp = (b + (a * (c / d))) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -4.2e+72) || !(c <= 1e+31))
		tmp = Float64(Float64(a + Float64(d / Float64(c / b))) / c);
	else
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -4.2e+72) || ~((c <= 1e+31)))
		tmp = (a + (d / (c / b))) / c;
	else
		tmp = (b + (a * (c / d))) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -4.2e+72], N[Not[LessEqual[c, 1e+31]], $MachinePrecision]], N[(N[(a + N[(d / N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -4.2000000000000003e72 or 9.9999999999999996e30 < c

   1. Initial program 49.0%

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

      1. fma-define 49.0%

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

      2. fma-define 49.0%

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

   3. Simplified 49.0%

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

   5. Taylor expanded in c around inf 77.4%

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

      1. *-commutative 77.4%

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

   7. Simplified 77.4%

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

      1. associate-/l* 83.0%

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

      2. *-commutative 83.0%

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

   9. Applied egg-rr 83.0%

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

       1. *-commutative 83.0%

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

       2. clear-num 83.6%

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

       3. un-div-inv 83.6%

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

   11. Applied egg-rr 83.6%

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

   if -4.2000000000000003e72 < c < 9.9999999999999996e30

   1. Initial program 69.9%

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

      1. fma-define 69.9%

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

      2. fma-define 69.9%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in d around inf 78.9%

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

      1. associate-/l* 79.1%

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

   7. Simplified 79.1%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.2 \cdot 10^{+72} \lor \neg \left(c \leq 10^{+31}\right):\\ \;\;\;\;\frac{a + \frac{d}{\frac{c}{b}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -14000000000000 \lor \neg \left(d \leq 8.2 \cdot 10^{+82}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -14000000000000.0) (not (<= d 8.2e+82)))
   (/ b d)
   (/ (+ a (/ (* d b) c)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -14000000000000.0) || !(d <= 8.2e+82)) {
		tmp = b / d;
	} else {
		tmp = (a + ((d * b) / 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 ((d <= (-14000000000000.0d0)) .or. (.not. (d <= 8.2d+82))) then
        tmp = b / d
    else
        tmp = (a + ((d * b) / c)) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -14000000000000.0) || !(d <= 8.2e+82)) {
		tmp = b / d;
	} else {
		tmp = (a + ((d * b) / c)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -14000000000000.0) or not (d <= 8.2e+82):
		tmp = b / d
	else:
		tmp = (a + ((d * b) / c)) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -14000000000000.0) || !(d <= 8.2e+82))
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(a + Float64(Float64(d * b) / c)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -14000000000000.0) || ~((d <= 8.2e+82)))
		tmp = b / d;
	else
		tmp = (a + ((d * b) / c)) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -14000000000000.0], N[Not[LessEqual[d, 8.2e+82]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(N[(a + N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.4e13 or 8.1999999999999999e82 < d

   1. Initial program 50.2%

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

      1. fma-define 50.2%

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

      2. fma-define 50.2%

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

   3. Simplified 50.2%

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

   5. Taylor expanded in c around 0 73.0%

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

   if -1.4e13 < d < 8.1999999999999999e82

   1. Initial program 68.2%

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

      1. fma-define 68.2%

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

      2. fma-define 68.2%

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

   3. Simplified 68.2%

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

   5. Taylor expanded in c around inf 74.7%

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

      1. *-commutative 74.7%

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

   7. Simplified 74.7%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -14000000000000 \lor \neg \left(d \leq 8.2 \cdot 10^{+82}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.7 \cdot 10^{+14} \lor \neg \left(d \leq 9.5 \cdot 10^{+84}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -2.7e+14) (not (<= d 9.5e+84)))
   (/ b d)
   (/ (+ a (* b (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.7e+14) || !(d <= 9.5e+84)) {
		tmp = b / 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 ((d <= (-2.7d+14)) .or. (.not. (d <= 9.5d+84))) then
        tmp = b / 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 ((d <= -2.7e+14) || !(d <= 9.5e+84)) {
		tmp = b / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -2.7e+14) or not (d <= 9.5e+84):
		tmp = b / d
	else:
		tmp = (a + (b * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -2.7e+14) || !(d <= 9.5e+84))
		tmp = Float64(b / 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 ((d <= -2.7e+14) || ~((d <= 9.5e+84)))
		tmp = b / d;
	else
		tmp = (a + (b * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -2.7e+14], N[Not[LessEqual[d, 9.5e+84]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.7e14 or 9.49999999999999979e84 < d

   1. Initial program 50.2%

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

      1. fma-define 50.2%

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

      2. fma-define 50.2%

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

   3. Simplified 50.2%

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

   5. Taylor expanded in c around 0 73.0%

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

   if -2.7e14 < d < 9.49999999999999979e84

   1. Initial program 68.2%

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

      1. fma-define 68.2%

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

      2. fma-define 68.2%

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

   3. Simplified 68.2%

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

   5. Taylor expanded in c around inf 74.7%

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

      1. *-commutative 74.7%

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

   7. Simplified 74.7%

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

   8. Taylor expanded in d around 0 74.7%

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

      1. associate-*r/ 74.2%

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

   10. Simplified 74.2%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.7 \cdot 10^{+14} \lor \neg \left(d \leq 9.5 \cdot 10^{+84}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.5 \cdot 10^{+73} \lor \neg \left(c \leq 1.25 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -4.5e+73) (not (<= c 1.25e+31))) (/ a c) (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4.5e+73) || !(c <= 1.25e+31)) {
		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 ((c <= (-4.5d+73)) .or. (.not. (c <= 1.25d+31))) 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 ((c <= -4.5e+73) || !(c <= 1.25e+31)) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -4.5e+73) or not (c <= 1.25e+31):
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -4.5e+73) || !(c <= 1.25e+31))
		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 ((c <= -4.5e+73) || ~((c <= 1.25e+31)))
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -4.5e+73], N[Not[LessEqual[c, 1.25e+31]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -4.49999999999999985e73 or 1.25000000000000007e31 < c

   1. Initial program 49.0%

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

      1. fma-define 49.0%

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

      2. fma-define 49.0%

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

   3. Simplified 49.0%

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

   5. Taylor expanded in c around inf 68.0%

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

   if -4.49999999999999985e73 < c < 1.25000000000000007e31

   1. Initial program 69.9%

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

      1. fma-define 69.9%

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

      2. fma-define 69.9%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in c around 0 65.2%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.5 \cdot 10^{+73} \lor \neg \left(c \leq 1.25 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.0% 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 61.5%

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

   1. fma-define 61.5%

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

   2. fma-define 61.5%

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

3. Simplified 61.5%

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

5. Taylor expanded in c around inf 40.2%

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

Developer Target 1: 99.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|d\right| < \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (fabs(d) < fabs(c)) {
		tmp = (a + (b * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (abs(d) < abs(c)) then
        tmp = (a + (b * (d / c))) / (c + (d * (d / c)))
    else
        tmp = (b + (a * (c / d))) / (d + (c * (c / d)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (Math.abs(d) < Math.abs(c)) {
		tmp = (a + (b * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if math.fabs(d) < math.fabs(c):
		tmp = (a + (b * (d / c))) / (c + (d * (d / c)))
	else:
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (abs(d) < abs(c))
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / Float64(c + Float64(d * Float64(d / c))));
	else
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / Float64(d + Float64(c * Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (abs(d) < abs(c))
		tmp = (a + (b * (d / c))) / (c + (d * (d / c)));
	else
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Reproduce

herbie shell --seed 2024139 
(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))))