Complex division, real part

Percentage Accurate: 61.9% → 61.9%

Time: 1.3s

Alternatives: 5

Speedup: 1.0×

Specification

\[\mathsf{TRUE}\left(\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.9% 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: 61.9% 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]

Derivation

1. Initial program 54.2%

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

Alternative 2: 49.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot c + d \cdot d\\ t_1 := \frac{b \cdot d}{t\_0}\\ \mathbf{if}\;b \leq -1.9 \cdot 10^{-56}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 0.00055:\\ \;\;\;\;\frac{a \cdot c}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (* c c) (* d d))) (t_1 (/ (* b d) t_0)))
   (if (<= b -1.9e-56) t_1 (if (<= b 0.00055) (/ (* a c) t_0) t_1))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c * c) + (d * d);
	double t_1 = (b * d) / t_0;
	double tmp;
	if (b <= -1.9e-56) {
		tmp = t_1;
	} else if (b <= 0.00055) {
		tmp = (a * c) / 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 = (c * c) + (d * d)
    t_1 = (b * d) / t_0
    if (b <= (-1.9d-56)) then
        tmp = t_1
    else if (b <= 0.00055d0) then
        tmp = (a * c) / 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 = (c * c) + (d * d);
	double t_1 = (b * d) / t_0;
	double tmp;
	if (b <= -1.9e-56) {
		tmp = t_1;
	} else if (b <= 0.00055) {
		tmp = (a * c) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (c * c) + (d * d)
	t_1 = (b * d) / t_0
	tmp = 0
	if b <= -1.9e-56:
		tmp = t_1
	elif b <= 0.00055:
		tmp = (a * c) / t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c * c) + Float64(d * d))
	t_1 = Float64(Float64(b * d) / t_0)
	tmp = 0.0
	if (b <= -1.9e-56)
		tmp = t_1;
	elseif (b <= 0.00055)
		tmp = Float64(Float64(a * c) / t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (c * c) + (d * d);
	t_1 = (b * d) / t_0;
	tmp = 0.0;
	if (b <= -1.9e-56)
		tmp = t_1;
	elseif (b <= 0.00055)
		tmp = (a * c) / t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * d), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[b, -1.9e-56], t$95$1, If[LessEqual[b, 0.00055], N[(N[(a * c), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.9000000000000001e-56 or 5.50000000000000033e-4 < b

   1. Initial program 46.4%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 39.1%

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

   if -1.9000000000000001e-56 < b < 5.50000000000000033e-4

   1. Initial program 63.5%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 57.2%

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

Alternative 3: 40.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 35.7%

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

Alternative 4: 5.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ c \cdot c \end{array} \]

FPCore

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

C

double code(double a, double b, double c, double d) {
	return c * 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 = c * c
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 3.8%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 4.9%

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

9. Taylor expanded in a around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(-1 \cdot \frac{c}{{c}^{2} + {d}^{2}} + -1 \cdot \frac{b \cdot d}{a \cdot \left({c}^{2} + {d}^{2}\right)}\right)\right)} \]

11. Applied rewrites 5.0%

    \[\leadsto \color{blue}{c \cdot c} \]
12. Add Preprocessing

Alternative 5: 5.3% accurate, 6.3× speedup

Math

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

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 54.2%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 3.8%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 4.9%

   \[\leadsto a \cdot \color{blue}{c} \]
9. Add Preprocessing

Developer Target 1: 99.5% accurate, 0.6× 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 2024321 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64
  :pre (TRUE)

  :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))))