Complex division, imag part

Percentage Accurate: 62.0% → 80.5%

Time: 12.6s

Alternatives: 10

Speedup: 1.1×

Specification

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: 62.0% 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: 80.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.1 \cdot 10^{+139}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;c \leq -7.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + d \cdot \left(a \cdot \frac{-1}{c}\right)}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.1e+139)
   (/ (- b (/ a (/ c d))) c)
   (if (<= c -7.2e-54)
     (/ (- (* c b) (* a d)) (+ (* c c) (* d d)))
     (if (<= c 7.2e+15)
       (/ (- (/ b (/ d c)) a) d)
       (/ (+ b (* d (* a (/ -1.0 c)))) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.1e+139) {
		tmp = (b - (a / (c / d))) / c;
	} else if (c <= -7.2e-54) {
		tmp = ((c * b) - (a * d)) / ((c * c) + (d * d));
	} else if (c <= 7.2e+15) {
		tmp = ((b / (d / c)) - a) / d;
	} else {
		tmp = (b + (d * (a * (-1.0 / 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 <= (-2.1d+139)) then
        tmp = (b - (a / (c / d))) / c
    else if (c <= (-7.2d-54)) then
        tmp = ((c * b) - (a * d)) / ((c * c) + (d * d))
    else if (c <= 7.2d+15) then
        tmp = ((b / (d / c)) - a) / d
    else
        tmp = (b + (d * (a * ((-1.0d0) / 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 <= -2.1e+139) {
		tmp = (b - (a / (c / d))) / c;
	} else if (c <= -7.2e-54) {
		tmp = ((c * b) - (a * d)) / ((c * c) + (d * d));
	} else if (c <= 7.2e+15) {
		tmp = ((b / (d / c)) - a) / d;
	} else {
		tmp = (b + (d * (a * (-1.0 / c)))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -2.1e+139:
		tmp = (b - (a / (c / d))) / c
	elif c <= -7.2e-54:
		tmp = ((c * b) - (a * d)) / ((c * c) + (d * d))
	elif c <= 7.2e+15:
		tmp = ((b / (d / c)) - a) / d
	else:
		tmp = (b + (d * (a * (-1.0 / c)))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.1e+139)
		tmp = Float64(Float64(b - Float64(a / Float64(c / d))) / c);
	elseif (c <= -7.2e-54)
		tmp = Float64(Float64(Float64(c * b) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 7.2e+15)
		tmp = Float64(Float64(Float64(b / Float64(d / c)) - a) / d);
	else
		tmp = Float64(Float64(b + Float64(d * Float64(a * Float64(-1.0 / c)))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -2.1e+139)
		tmp = (b - (a / (c / d))) / c;
	elseif (c <= -7.2e-54)
		tmp = ((c * b) - (a * d)) / ((c * c) + (d * d));
	elseif (c <= 7.2e+15)
		tmp = ((b / (d / c)) - a) / d;
	else
		tmp = (b + (d * (a * (-1.0 / c)))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.1e+139], N[(N[(b - N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -7.2e-54], N[(N[(N[(c * b), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 7.2e+15], N[(N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b + N[(d * N[(a * N[(-1.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.0999999999999999e139

   1. Initial program 31.1%

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

   3. Taylor expanded in c around inf 89.0%

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

      1. mul-1-neg 89.0%

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

      2. unsub-neg 89.0%

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

      3. associate-/l* 95.6%

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

   5. Simplified 95.6%

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

      1. clear-num 95.6%

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

      2. un-div-inv 95.6%

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

   7. Applied egg-rr 95.6%

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

   if -2.0999999999999999e139 < c < -7.19999999999999953e-54

   1. Initial program 79.6%

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

   if -7.19999999999999953e-54 < c < 7.2e15

   1. Initial program 70.6%

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

   3. Taylor expanded in d around -inf 86.0%

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

      1. mul-1-neg 86.0%

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

      2. distribute-neg-frac2 86.0%

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

      3. mul-1-neg 86.0%

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

      4. unsub-neg 86.0%

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

      5. associate-/l* 86.0%

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

   5. Simplified 86.0%

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

      1. clear-num 86.0%

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

      2. un-div-inv 86.1%

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

   7. Applied egg-rr 86.1%

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

   if 7.2e15 < c

   1. Initial program 43.6%

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

   3. Taylor expanded in c around inf 78.9%

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

      1. associate-*r/ 86.1%

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

      2. *-commutative 86.1%

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

      3. div-inv 86.1%

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

      4. associate-*l* 86.2%

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

   5. Applied egg-rr 86.2%

      \[\leadsto \frac{b + -1 \cdot \color{blue}{\left(d \cdot \left(\frac{1}{c} \cdot a\right)\right)}}{c} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.1 \cdot 10^{+139}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;c \leq -7.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + d \cdot \left(a \cdot \frac{-1}{c}\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -41000000:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;c \leq 4.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + d \cdot \left(a \cdot \frac{-1}{c}\right)}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -41000000.0)
   (/ (- b (/ a (/ c d))) c)
   (if (<= c 4.4e+15)
     (/ (- (/ b (/ d c)) a) d)
     (/ (+ b (* d (* a (/ -1.0 c)))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -41000000.0) {
		tmp = (b - (a / (c / d))) / c;
	} else if (c <= 4.4e+15) {
		tmp = ((b / (d / c)) - a) / d;
	} else {
		tmp = (b + (d * (a * (-1.0 / 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 <= (-41000000.0d0)) then
        tmp = (b - (a / (c / d))) / c
    else if (c <= 4.4d+15) then
        tmp = ((b / (d / c)) - a) / d
    else
        tmp = (b + (d * (a * ((-1.0d0) / 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 <= -41000000.0) {
		tmp = (b - (a / (c / d))) / c;
	} else if (c <= 4.4e+15) {
		tmp = ((b / (d / c)) - a) / d;
	} else {
		tmp = (b + (d * (a * (-1.0 / c)))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -41000000.0:
		tmp = (b - (a / (c / d))) / c
	elif c <= 4.4e+15:
		tmp = ((b / (d / c)) - a) / d
	else:
		tmp = (b + (d * (a * (-1.0 / c)))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -41000000.0)
		tmp = Float64(Float64(b - Float64(a / Float64(c / d))) / c);
	elseif (c <= 4.4e+15)
		tmp = Float64(Float64(Float64(b / Float64(d / c)) - a) / d);
	else
		tmp = Float64(Float64(b + Float64(d * Float64(a * Float64(-1.0 / c)))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -41000000.0)
		tmp = (b - (a / (c / d))) / c;
	elseif (c <= 4.4e+15)
		tmp = ((b / (d / c)) - a) / d;
	else
		tmp = (b + (d * (a * (-1.0 / c)))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -41000000.0], N[(N[(b - N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 4.4e+15], N[(N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b + N[(d * N[(a * N[(-1.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -4.1e7

   1. Initial program 50.5%

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

   3. Taylor expanded in c around inf 78.8%

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

      1. mul-1-neg 78.8%

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

      2. unsub-neg 78.8%

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

      3. associate-/l* 82.6%

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

   5. Simplified 82.6%

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

      1. clear-num 82.5%

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

      2. un-div-inv 82.6%

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

   7. Applied egg-rr 82.6%

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

   if -4.1e7 < c < 4.4e15

   1. Initial program 72.2%

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

   3. Taylor expanded in d around -inf 84.1%

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

      1. mul-1-neg 84.1%

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

      2. distribute-neg-frac2 84.1%

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

      3. mul-1-neg 84.1%

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

      4. unsub-neg 84.1%

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

      5. associate-/l* 84.2%

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

   5. Simplified 84.2%

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

      1. clear-num 84.2%

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

      2. un-div-inv 84.2%

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

   7. Applied egg-rr 84.2%

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

   if 4.4e15 < c

   1. Initial program 43.6%

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

   3. Taylor expanded in c around inf 78.9%

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

      1. associate-*r/ 86.1%

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

      2. *-commutative 86.1%

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

      3. div-inv 86.1%

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

      4. associate-*l* 86.2%

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

   5. Applied egg-rr 86.2%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -41000000:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;c \leq 4.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + d \cdot \left(a \cdot \frac{-1}{c}\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -11600 \lor \neg \left(c \leq 2.9 \cdot 10^{+17}\right):\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -11600.0) (not (<= c 2.9e+17)))
   (/ (- b (/ a (/ c d))) c)
   (/ (- (/ b (/ d c)) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -11600.0) || !(c <= 2.9e+17)) {
		tmp = (b - (a / (c / d))) / c;
	} else {
		tmp = ((b / (d / c)) - a) / 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 <= (-11600.0d0)) .or. (.not. (c <= 2.9d+17))) then
        tmp = (b - (a / (c / d))) / c
    else
        tmp = ((b / (d / c)) - a) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -11600.0) || !(c <= 2.9e+17)) {
		tmp = (b - (a / (c / d))) / c;
	} else {
		tmp = ((b / (d / c)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -11600.0) or not (c <= 2.9e+17):
		tmp = (b - (a / (c / d))) / c
	else:
		tmp = ((b / (d / c)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -11600.0) || !(c <= 2.9e+17))
		tmp = Float64(Float64(b - Float64(a / Float64(c / d))) / c);
	else
		tmp = Float64(Float64(Float64(b / Float64(d / c)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -11600.0) || ~((c <= 2.9e+17)))
		tmp = (b - (a / (c / d))) / c;
	else
		tmp = ((b / (d / c)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -11600.0], N[Not[LessEqual[c, 2.9e+17]], $MachinePrecision]], N[(N[(b - N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -11600 or 2.9e17 < c

   1. Initial program 47.7%

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

   3. Taylor expanded in c around inf 78.9%

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

      1. mul-1-neg 78.9%

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

      2. unsub-neg 78.9%

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

      3. associate-/l* 84.0%

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

   5. Simplified 84.0%

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

      1. clear-num 84.0%

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

      2. un-div-inv 84.1%

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

   7. Applied egg-rr 84.1%

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

   if -11600 < c < 2.9e17

   1. Initial program 72.2%

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

   3. Taylor expanded in d around -inf 84.1%

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

      1. mul-1-neg 84.1%

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

      2. distribute-neg-frac2 84.1%

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

      3. mul-1-neg 84.1%

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

      4. unsub-neg 84.1%

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

      5. associate-/l* 84.2%

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

   5. Simplified 84.2%

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

      1. clear-num 84.2%

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

      2. un-div-inv 84.2%

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

   7. Applied egg-rr 84.2%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -11600 \lor \neg \left(c \leq 2.9 \cdot 10^{+17}\right):\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -112000000 \lor \neg \left(c \leq 3.5 \cdot 10^{+17}\right):\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -112000000.0) (not (<= c 3.5e+17)))
   (/ (- b (/ a (/ c d))) c)
   (/ (- (* b (/ c d)) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -112000000.0) || !(c <= 3.5e+17)) {
		tmp = (b - (a / (c / d))) / c;
	} else {
		tmp = ((b * (c / d)) - a) / 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 <= (-112000000.0d0)) .or. (.not. (c <= 3.5d+17))) then
        tmp = (b - (a / (c / d))) / c
    else
        tmp = ((b * (c / d)) - a) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -112000000.0) || !(c <= 3.5e+17)) {
		tmp = (b - (a / (c / d))) / c;
	} else {
		tmp = ((b * (c / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -112000000.0) or not (c <= 3.5e+17):
		tmp = (b - (a / (c / d))) / c
	else:
		tmp = ((b * (c / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -112000000.0) || !(c <= 3.5e+17))
		tmp = Float64(Float64(b - Float64(a / Float64(c / d))) / c);
	else
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -112000000.0) || ~((c <= 3.5e+17)))
		tmp = (b - (a / (c / d))) / c;
	else
		tmp = ((b * (c / d)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -112000000.0], N[Not[LessEqual[c, 3.5e+17]], $MachinePrecision]], N[(N[(b - N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.12e8 or 3.5e17 < c

   1. Initial program 47.7%

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

   3. Taylor expanded in c around inf 78.9%

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

      1. mul-1-neg 78.9%

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

      2. unsub-neg 78.9%

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

      3. associate-/l* 84.0%

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

   5. Simplified 84.0%

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

      1. clear-num 84.0%

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

      2. un-div-inv 84.1%

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

   7. Applied egg-rr 84.1%

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

   if -1.12e8 < c < 3.5e17

   1. Initial program 72.2%

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

   3. Taylor expanded in d around -inf 84.1%

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

      1. mul-1-neg 84.1%

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

      2. distribute-neg-frac2 84.1%

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

      3. mul-1-neg 84.1%

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

      4. unsub-neg 84.1%

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

      5. associate-/l* 84.2%

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

   5. Simplified 84.2%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -112000000 \lor \neg \left(c \leq 3.5 \cdot 10^{+17}\right):\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -135000 \lor \neg \left(c \leq 2.5 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -135000.0) (not (<= c 2.5e+16)))
   (/ (- b (/ a (/ c d))) c)
   (/ (- (* c (/ b d)) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -135000.0) || !(c <= 2.5e+16)) {
		tmp = (b - (a / (c / d))) / c;
	} else {
		tmp = ((c * (b / d)) - a) / 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 <= (-135000.0d0)) .or. (.not. (c <= 2.5d+16))) then
        tmp = (b - (a / (c / d))) / c
    else
        tmp = ((c * (b / d)) - a) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -135000.0) || !(c <= 2.5e+16)) {
		tmp = (b - (a / (c / d))) / c;
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -135000.0) or not (c <= 2.5e+16):
		tmp = (b - (a / (c / d))) / c
	else:
		tmp = ((c * (b / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -135000.0) || !(c <= 2.5e+16))
		tmp = Float64(Float64(b - Float64(a / Float64(c / d))) / c);
	else
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -135000.0) || ~((c <= 2.5e+16)))
		tmp = (b - (a / (c / d))) / c;
	else
		tmp = ((c * (b / d)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -135000.0], N[Not[LessEqual[c, 2.5e+16]], $MachinePrecision]], N[(N[(b - N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -135000 or 2.5e16 < c

   1. Initial program 47.7%

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

   3. Taylor expanded in c around inf 78.9%

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

      1. mul-1-neg 78.9%

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

      2. unsub-neg 78.9%

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

      3. associate-/l* 84.0%

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

   5. Simplified 84.0%

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

      1. clear-num 84.0%

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

      2. un-div-inv 84.1%

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

   7. Applied egg-rr 84.1%

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

   if -135000 < c < 2.5e16

   1. Initial program 72.2%

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

   3. Taylor expanded in d around -inf 84.1%

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

      1. mul-1-neg 84.1%

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

      2. distribute-neg-frac2 84.1%

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

      3. mul-1-neg 84.1%

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

      4. unsub-neg 84.1%

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

      5. associate-/l* 84.2%

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

   5. Simplified 84.2%

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

   6. Taylor expanded in d around inf 84.1%

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

      1. neg-mul-1 84.1%

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

      2. associate-*r/ 84.2%

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

      3. +-commutative 84.2%

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

      4. unsub-neg 84.2%

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

      5. associate-*r/ 84.1%

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

      6. *-commutative 84.1%

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

      7. associate-/l* 82.6%

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

   8. Simplified 82.6%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -135000 \lor \neg \left(c \leq 2.5 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -9 \cdot 10^{-73} \lor \neg \left(c \leq 6500\right):\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -9e-73) (not (<= c 6500.0)))
   (/ (- b (/ a (/ c d))) c)
   (/ a (- d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -9e-73) || !(c <= 6500.0)) {
		tmp = (b - (a / (c / d))) / c;
	} else {
		tmp = a / -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 <= (-9d-73)) .or. (.not. (c <= 6500.0d0))) then
        tmp = (b - (a / (c / d))) / c
    else
        tmp = a / -d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -9e-73) || !(c <= 6500.0)) {
		tmp = (b - (a / (c / d))) / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -9e-73) or not (c <= 6500.0):
		tmp = (b - (a / (c / d))) / c
	else:
		tmp = a / -d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -9e-73) || !(c <= 6500.0))
		tmp = Float64(Float64(b - Float64(a / Float64(c / d))) / c);
	else
		tmp = Float64(a / Float64(-d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -9e-73) || ~((c <= 6500.0)))
		tmp = (b - (a / (c / d))) / c;
	else
		tmp = a / -d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -9e-73], N[Not[LessEqual[c, 6500.0]], $MachinePrecision]], N[(N[(b - N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(a / (-d)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -9e-73 or 6500 < c

   1. Initial program 51.7%

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

   3. Taylor expanded in c around inf 75.8%

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

      1. mul-1-neg 75.8%

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

      2. unsub-neg 75.8%

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

      3. associate-/l* 80.5%

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

   5. Simplified 80.5%

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

      1. clear-num 80.4%

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

      2. un-div-inv 80.5%

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

   7. Applied egg-rr 80.5%

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

   if -9e-73 < c < 6500

   1. Initial program 70.2%

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

   3. Taylor expanded in c around 0 72.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   4. Step-by-step derivation

      1. associate-*r/ 72.2%

         \[\leadsto \color{blue}{\frac{-1 \cdot a}{d}} \]

      2. neg-mul-1 72.2%

         \[\leadsto \frac{\color{blue}{-a}}{d} \]

   5. Simplified 72.2%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -9 \cdot 10^{-73} \lor \neg \left(c \leq 6500\right):\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.35 \cdot 10^{-72} \lor \neg \left(c \leq 2600\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.35e-72) (not (<= c 2600.0)))
   (/ (- b (* a (/ d c))) c)
   (/ a (- d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.35e-72) || !(c <= 2600.0)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = a / -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 <= (-1.35d-72)) .or. (.not. (c <= 2600.0d0))) then
        tmp = (b - (a * (d / c))) / c
    else
        tmp = a / -d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.35e-72) || !(c <= 2600.0)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.35e-72) or not (c <= 2600.0):
		tmp = (b - (a * (d / c))) / c
	else:
		tmp = a / -d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.35e-72) || !(c <= 2600.0))
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	else
		tmp = Float64(a / Float64(-d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.35e-72) || ~((c <= 2600.0)))
		tmp = (b - (a * (d / c))) / c;
	else
		tmp = a / -d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.35e-72], N[Not[LessEqual[c, 2600.0]], $MachinePrecision]], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(a / (-d)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.35e-72 or 2600 < c

   1. Initial program 51.7%

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

   3. Taylor expanded in c around inf 75.8%

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

      1. mul-1-neg 75.8%

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

      2. unsub-neg 75.8%

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

      3. associate-/l* 80.5%

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

   5. Simplified 80.5%

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

   if -1.35e-72 < c < 2600

   1. Initial program 70.2%

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

   3. Taylor expanded in c around 0 72.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   4. Step-by-step derivation

      1. associate-*r/ 72.2%

         \[\leadsto \color{blue}{\frac{-1 \cdot a}{d}} \]

      2. neg-mul-1 72.2%

         \[\leadsto \frac{\color{blue}{-a}}{d} \]

   5. Simplified 72.2%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.35 \cdot 10^{-72} \lor \neg \left(c \leq 2600\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.06 \cdot 10^{-94} \lor \neg \left(c \leq 480000\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.06e-94) (not (<= c 480000.0))) (/ b c) (/ a (- d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.06e-94) || !(c <= 480000.0)) {
		tmp = b / c;
	} else {
		tmp = a / -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 <= (-1.06d-94)) .or. (.not. (c <= 480000.0d0))) then
        tmp = b / c
    else
        tmp = a / -d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.06e-94) || !(c <= 480000.0)) {
		tmp = b / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.06e-94) or not (c <= 480000.0):
		tmp = b / c
	else:
		tmp = a / -d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.06e-94) || !(c <= 480000.0))
		tmp = Float64(b / c);
	else
		tmp = Float64(a / Float64(-d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.06e-94) || ~((c <= 480000.0)))
		tmp = b / c;
	else
		tmp = a / -d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.06e-94], N[Not[LessEqual[c, 480000.0]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[(a / (-d)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.06e-94 or 4.8e5 < c

   1. Initial program 52.6%

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

   3. Taylor expanded in c around inf 67.3%

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

   if -1.06e-94 < c < 4.8e5

   1. Initial program 69.7%

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

   3. Taylor expanded in c around 0 74.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   4. Step-by-step derivation

      1. associate-*r/ 74.5%

         \[\leadsto \color{blue}{\frac{-1 \cdot a}{d}} \]

      2. neg-mul-1 74.5%

         \[\leadsto \frac{\color{blue}{-a}}{d} \]

   5. Simplified 74.5%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.06 \cdot 10^{-94} \lor \neg \left(c \leq 480000\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -8.8 \cdot 10^{+178} \lor \neg \left(d \leq 4.7 \cdot 10^{+120}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -8.8e+178) (not (<= d 4.7e+120))) (/ a d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -8.8e+178) || !(d <= 4.7e+120)) {
		tmp = a / d;
	} else {
		tmp = b / 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 <= (-8.8d+178)) .or. (.not. (d <= 4.7d+120))) then
        tmp = a / d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -8.8e+178) || !(d <= 4.7e+120)) {
		tmp = a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -8.8e+178) or not (d <= 4.7e+120):
		tmp = a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -8.8e+178) || !(d <= 4.7e+120))
		tmp = Float64(a / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -8.8e+178) || ~((d <= 4.7e+120)))
		tmp = a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -8.8e+178], N[Not[LessEqual[d, 4.7e+120]], $MachinePrecision]], N[(a / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -8.79999999999999989e178 or 4.69999999999999993e120 < d

   1. Initial program 43.2%

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

   3. Taylor expanded in c around 0 87.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   4. Step-by-step derivation

      1. associate-*r/ 87.3%

         \[\leadsto \color{blue}{\frac{-1 \cdot a}{d}} \]

      2. neg-mul-1 87.3%

         \[\leadsto \frac{\color{blue}{-a}}{d} \]

   5. Simplified 87.3%

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

      1. neg-sub0 87.3%

         \[\leadsto \frac{\color{blue}{0 - a}}{d} \]

      2. sub-neg 87.3%

         \[\leadsto \frac{\color{blue}{0 + \left(-a\right)}}{d} \]

      3. add-sqr-sqrt 46.0%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{-a} \cdot \sqrt{-a}}}{d} \]

      4. sqrt-unprod 52.3%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}}{d} \]

      5. sqr-neg 52.3%

         \[\leadsto \frac{0 + \sqrt{\color{blue}{a \cdot a}}}{d} \]

      6. sqrt-unprod 14.9%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{a} \cdot \sqrt{a}}}{d} \]

      7. add-sqr-sqrt 43.9%

         \[\leadsto \frac{0 + \color{blue}{a}}{d} \]

   7. Applied egg-rr 43.9%

      \[\leadsto \frac{\color{blue}{0 + a}}{d} \]
   8. Step-by-step derivation

      1. +-lft-identity 43.9%

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

   9. Simplified 43.9%

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

   if -8.79999999999999989e178 < d < 4.69999999999999993e120

   1. Initial program 65.6%

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

   3. Taylor expanded in c around inf 55.3%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.8 \cdot 10^{+178} \lor \neg \left(d \leq 4.7 \cdot 10^{+120}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 11.2% accurate, 5.0× speedup

Math

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

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 59.6%

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

3. Taylor expanded in c around 0 42.5%

   \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
4. Step-by-step derivation

   1. associate-*r/ 42.5%

      \[\leadsto \color{blue}{\frac{-1 \cdot a}{d}} \]

   2. neg-mul-1 42.5%

      \[\leadsto \frac{\color{blue}{-a}}{d} \]

5. Simplified 42.5%

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

   1. neg-sub0 42.5%

      \[\leadsto \frac{\color{blue}{0 - a}}{d} \]

   2. sub-neg 42.5%

      \[\leadsto \frac{\color{blue}{0 + \left(-a\right)}}{d} \]

   3. add-sqr-sqrt 20.8%

      \[\leadsto \frac{0 + \color{blue}{\sqrt{-a} \cdot \sqrt{-a}}}{d} \]

   4. sqrt-unprod 23.1%

      \[\leadsto \frac{0 + \color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}}{d} \]

   5. sqr-neg 23.1%

      \[\leadsto \frac{0 + \sqrt{\color{blue}{a \cdot a}}}{d} \]

   6. sqrt-unprod 5.9%

      \[\leadsto \frac{0 + \color{blue}{\sqrt{a} \cdot \sqrt{a}}}{d} \]

   7. add-sqr-sqrt 14.4%

      \[\leadsto \frac{0 + \color{blue}{a}}{d} \]

7. Applied egg-rr 14.4%

   \[\leadsto \frac{\color{blue}{0 + a}}{d} \]
8. Step-by-step derivation

   1. +-lft-identity 14.4%

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

9. Simplified 14.4%

   \[\leadsto \frac{\color{blue}{a}}{d} \]
10. Add Preprocessing

Developer target: 99.4% accurate, 0.1× 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 2024110 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :alt
  (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))))