Complex division, imag part

Percentage Accurate: 62.4% → 62.4%

Time: 1.5s

Alternatives: 6

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 62.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.3%

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

Alternative 2: 40.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.3%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 39.1%

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

Alternative 3: 17.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.3%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 19.9%

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

Alternative 4: 6.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.2 \cdot 10^{+103}:\\ \;\;\;\;a \cdot d\\ \mathbf{else}:\\ \;\;\;\;c \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (if (<= c -4.2e+103) (* a d) (* c c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -4.2e+103) {
		tmp = a * d;
	} else {
		tmp = c * c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-4.2d+103)) then
        tmp = a * d
    else
        tmp = c * c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -4.2e+103) {
		tmp = a * d;
	} else {
		tmp = c * c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -4.2e+103:
		tmp = a * d
	else:
		tmp = c * c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -4.2e+103)
		tmp = Float64(a * d);
	else
		tmp = Float64(c * c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -4.2e+103)
		tmp = a * d;
	else
		tmp = c * c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -4.2e+103], N[(a * d), $MachinePrecision], N[(c * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -4.2000000000000003e103

   1. Initial program 43.8%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 11.2%

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

   if -4.2000000000000003e103 < c

   1. Initial program 63.0%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 4.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 3.0%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 5.3%

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

Alternative 5: 5.4% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.3%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 3.8%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 4.6%

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

9. Taylor expanded in b around inf

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

11. Applied rewrites 6.0%

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

Alternative 6: 3.8% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.3%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 3.8%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 4.6%

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

Developer Target 1: 99.3% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (< (fabs d) (fabs c))
   (/ (- b (* a (/ d c))) (+ c (* d (/ d c))))
   (/ (+ (- a) (* b (/ 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 = (b - (a * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (-a + (b * (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 = (b - (a * (d / c))) / (c + (d * (d / c)))
    else
        tmp = (-a + (b * (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 = (b - (a * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (-a + (b * (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 = (b - (a * (d / c))) / (c + (d * (d / c)))
	else:
		tmp = (-a + (b * (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(b - Float64(a * Float64(d / c))) / Float64(c + Float64(d * Float64(d / c))));
	else
		tmp = Float64(Float64(Float64(-a) + Float64(b * 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 = (b - (a * (d / c))) / (c + (d * (d / c)));
	else
		tmp = (-a + (b * (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[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c + N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-a) + N[(b * 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, imag part"
  :precision binary64
  :pre (TRUE)

  :alt
  (! :herbie-platform default (if (< (fabs d) (fabs c)) (/ (- b (* a (/ d c))) (+ c (* d (/ d c)))) (/ (+ (- a) (* b (/ c d))) (+ d (* c (/ c d))))))

  (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))