Complex division, real part

Percentage Accurate: 62.2% → 82.9%

Time: 8.6s

Alternatives: 6

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: 62.2% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + d \cdot b}{c \cdot c + d \cdot d}\\ t_1 := \frac{b + \frac{a}{d} \cdot c}{d}\\ \mathbf{if}\;d \leq -4 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -8.8 \cdot 10^{-112}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-164}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{+25}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* d b)) (+ (* c c) (* d d))))
        (t_1 (/ (+ b (* (/ a d) c)) d)))
   (if (<= d -4e+80)
     t_1
     (if (<= d -8.8e-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 = ((a * c) + (d * b)) / ((c * c) + (d * d));
	double t_1 = (b + ((a / d) * c)) / d;
	double tmp;
	if (d <= -4e+80) {
		tmp = t_1;
	} else if (d <= -8.8e-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 = ((a * c) + (d * b)) / ((c * c) + (d * d))
    t_1 = (b + ((a / d) * c)) / d
    if (d <= (-4d+80)) then
        tmp = t_1
    else if (d <= (-8.8d-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 = ((a * c) + (d * b)) / ((c * c) + (d * d));
	double t_1 = (b + ((a / d) * c)) / d;
	double tmp;
	if (d <= -4e+80) {
		tmp = t_1;
	} else if (d <= -8.8e-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 = ((a * c) + (d * b)) / ((c * c) + (d * d))
	t_1 = (b + ((a / d) * c)) / d
	tmp = 0
	if d <= -4e+80:
		tmp = t_1
	elif d <= -8.8e-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(a * c) + Float64(d * b)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(b + Float64(Float64(a / d) * c)) / d)
	tmp = 0.0
	if (d <= -4e+80)
		tmp = t_1;
	elseif (d <= -8.8e-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 = ((a * c) + (d * b)) / ((c * c) + (d * d));
	t_1 = (b + ((a / d) * c)) / d;
	tmp = 0.0;
	if (d <= -4e+80)
		tmp = t_1;
	elseif (d <= -8.8e-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[(a * c), $MachinePrecision] + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b + N[(N[(a / d), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -4e+80], t$95$1, If[LessEqual[d, -8.8e-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 < -4e80 or 2.8000000000000002e25 < d

   1. Initial program 40.9%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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 83.1%

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

   7. Applied egg-rr 83.1%

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

   if -4e80 < d < -8.80000000000000085e-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 -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. 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 92.7%

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

   5. Simplified 92.7%

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

4. Final simplification 86.3%

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

Alternative 2: 78.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b + \frac{a}{d} \cdot c}{d}\\ \mathbf{if}\;d \leq -270000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 3.9 \cdot 10^{+87}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ b (* (/ a d) c)) d)))
   (if (<= d -270000000000.0)
     t_0
     (if (<= d 3.9e+87) (/ (+ a (/ (* d b) 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 <= -270000000000.0) {
		tmp = t_0;
	} else if (d <= 3.9e+87) {
		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 + ((a / d) * c)) / d
    if (d <= (-270000000000.0d0)) then
        tmp = t_0
    else if (d <= 3.9d+87) 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 + ((a / d) * c)) / d;
	double tmp;
	if (d <= -270000000000.0) {
		tmp = t_0;
	} else if (d <= 3.9e+87) {
		tmp = (a + ((d * b) / 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 <= -270000000000.0:
		tmp = t_0
	elif d <= 3.9e+87:
		tmp = (a + ((d * b) / 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 <= -270000000000.0)
		tmp = t_0;
	elseif (d <= 3.9e+87)
		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 + ((a / d) * c)) / d;
	tmp = 0.0;
	if (d <= -270000000000.0)
		tmp = t_0;
	elseif (d <= 3.9e+87)
		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 + N[(N[(a / d), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -270000000000.0], t$95$0, If[LessEqual[d, 3.9e+87], 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 < -2.7e11 or 3.9000000000000002e87 < d

   1. Initial program 50.2%

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

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

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

   7. Applied egg-rr 85.0%

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

   if -2.7e11 < d < 3.9000000000000002e87

   1. Initial program 68.2%

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

   3. Taylor expanded in c around inf

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

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

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

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

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

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

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

      4. *-lowering-*.f64 74.7%

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

   5. Simplified 74.7%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -270000000000:\\ \;\;\;\;\frac{b + \frac{a}{d} \cdot c}{d}\\ \mathbf{elif}\;d \leq 3.9 \cdot 10^{+87}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{d} \cdot c}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{if}\;c \leq -4.2 \cdot 10^{+72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{+31}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ a (/ (* d b) c)) c)))
   (if (<= c -4.2e+72) t_0 (if (<= c 1.9e+31) (/ (+ b (* a (/ c d))) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (a + ((d * b) / c)) / c;
	double tmp;
	if (c <= -4.2e+72) {
		tmp = t_0;
	} else if (c <= 1.9e+31) {
		tmp = (b + (a * (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 = (a + ((d * b) / c)) / c
    if (c <= (-4.2d+72)) then
        tmp = t_0
    else if (c <= 1.9d+31) then
        tmp = (b + (a * (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 = (a + ((d * b) / c)) / c;
	double tmp;
	if (c <= -4.2e+72) {
		tmp = t_0;
	} else if (c <= 1.9e+31) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (a + ((d * b) / c)) / c
	tmp = 0
	if c <= -4.2e+72:
		tmp = t_0
	elif c <= 1.9e+31:
		tmp = (b + (a * (c / d))) / d
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(a + Float64(Float64(d * b) / c)) / c)
	tmp = 0.0
	if (c <= -4.2e+72)
		tmp = t_0;
	elseif (c <= 1.9e+31)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (a + ((d * b) / c)) / c;
	tmp = 0.0;
	if (c <= -4.2e+72)
		tmp = t_0;
	elseif (c <= 1.9e+31)
		tmp = (b + (a * (c / d))) / d;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a + N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -4.2e+72], t$95$0, If[LessEqual[c, 1.9e+31], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -4.2000000000000003e72 or 1.9000000000000001e31 < c

   1. Initial program 49.0%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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 77.4%

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

   5. Simplified 77.4%

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

   if -4.2000000000000003e72 < c < 1.9000000000000001e31

   1. Initial program 69.9%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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 78.9%

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

   5. Simplified 78.9%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 79.1%

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

   7. Applied egg-rr 79.1%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.2 \cdot 10^{+72}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{+31}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -7 \cdot 10^{+16}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 2.35 \cdot 10^{+82}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -7e+16)
   (/ b d)
   (if (<= d 2.35e+82) (/ (+ a (/ (* d b) c)) c) (/ b d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -7e+16) {
		tmp = b / d;
	} else if (d <= 2.35e+82) {
		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 <= (-7d+16)) then
        tmp = b / d
    else if (d <= 2.35d+82) 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 <= -7e+16) {
		tmp = b / d;
	} else if (d <= 2.35e+82) {
		tmp = (a + ((d * b) / c)) / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -7e+16:
		tmp = b / d
	elif d <= 2.35e+82:
		tmp = (a + ((d * b) / c)) / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -7e+16)
		tmp = Float64(b / d);
	elseif (d <= 2.35e+82)
		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 <= -7e+16)
		tmp = b / d;
	elseif (d <= 2.35e+82)
		tmp = (a + ((d * b) / c)) / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -7e+16], N[(b / d), $MachinePrecision], If[LessEqual[d, 2.35e+82], 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 < -7e16 or 2.35e82 < d

   1. Initial program 50.2%

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

   3. Taylor expanded in c around 0

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

      1. /-lowering-/.f64 73.0%

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

   5. Simplified 73.0%

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

   if -7e16 < d < 2.35e82

   1. Initial program 68.2%

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

   3. Taylor expanded in c around inf

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

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

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

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

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

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

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

      4. *-lowering-*.f64 74.7%

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

   5. Simplified 74.7%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7 \cdot 10^{+16}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 2.35 \cdot 10^{+82}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.3 \cdot 10^{+72}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 3.2 \cdot 10^{+33}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -4.3e+72) (/ a c) (if (<= c 3.2e+33) (/ b d) (/ a c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -4.3e+72) {
		tmp = a / c;
	} else if (c <= 3.2e+33) {
		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 <= (-4.3d+72)) then
        tmp = a / c
    else if (c <= 3.2d+33) 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 <= -4.3e+72) {
		tmp = a / c;
	} else if (c <= 3.2e+33) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -4.3e+72:
		tmp = a / c
	elif c <= 3.2e+33:
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -4.3e+72)
		tmp = Float64(a / c);
	elseif (c <= 3.2e+33)
		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 <= -4.3e+72)
		tmp = a / c;
	elseif (c <= 3.2e+33)
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -4.3e+72], N[(a / c), $MachinePrecision], If[LessEqual[c, 3.2e+33], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -4.3000000000000001e72 or 3.20000000000000017e33 < c

   1. Initial program 49.0%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 68.0%

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

   5. Simplified 68.0%

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

   if -4.3000000000000001e72 < c < 3.20000000000000017e33

   1. Initial program 69.9%

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

   3. Taylor expanded in c around 0

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

      1. /-lowering-/.f64 65.2%

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

   5. Simplified 65.2%

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

Alternative 6: 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. Add Preprocessing

3. Taylor expanded in c around inf

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

   1. /-lowering-/.f64 40.2%

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

5. Simplified 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))))