Complex division, imag part

Percentage Accurate: 60.8% → 79.2%

Time: 11.2s

Alternatives: 8

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: 60.8% 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: 79.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.2 \cdot 10^{+90}:\\ \;\;\;\;\frac{b - \frac{d}{c} \cdot a}{c}\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{-122}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{+83}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, c, d \cdot \left(-a\right)\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3.2e+90)
   (/ (- b (* (/ d c) a)) c)
   (if (<= c 1.5e-122)
     (/ (- (* b (/ c d)) a) d)
     (if (<= c 7.5e+83)
       (/ (fma b c (* d (- a))) (fma d d (* c c)))
       (/ (- b (/ a (/ c d))) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.2e+90) {
		tmp = (b - ((d / c) * a)) / c;
	} else if (c <= 1.5e-122) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (c <= 7.5e+83) {
		tmp = fma(b, c, (d * -a)) / fma(d, d, (c * c));
	} else {
		tmp = (b - (a / (c / d))) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3.2e+90)
		tmp = Float64(Float64(b - Float64(Float64(d / c) * a)) / c);
	elseif (c <= 1.5e-122)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	elseif (c <= 7.5e+83)
		tmp = Float64(fma(b, c, Float64(d * Float64(-a))) / fma(d, d, Float64(c * c)));
	else
		tmp = Float64(Float64(b - Float64(a / Float64(c / d))) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3.2e+90], N[(N[(b - N[(N[(d / c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 1.5e-122], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 7.5e+83], N[(N[(b * c + N[(d * (-a)), $MachinePrecision]), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -3.19999999999999998e90

   1. Initial program 39.4%

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

      1. fmm-def 39.4%

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

      2. distribute-rgt-neg-out 39.4%

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

      3. +-commutative 39.4%

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

      4. fma-define 39.4%

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

   3. Simplified 39.4%

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

   5. Taylor expanded in c around inf 87.6%

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

      1. mul-1-neg 87.6%

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

      2. unsub-neg 87.6%

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

      3. *-commutative 87.6%

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

   7. Simplified 87.6%

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

      1. *-commutative 87.6%

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

      2. associate-*r/ 90.2%

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

      3. *-commutative 90.2%

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

   9. Applied egg-rr 90.2%

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

   if -3.19999999999999998e90 < c < 1.50000000000000002e-122

   1. Initial program 69.5%

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

      1. fmm-def 69.5%

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

      2. distribute-rgt-neg-out 69.5%

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

      3. +-commutative 69.5%

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

      4. fma-define 69.5%

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

   3. Simplified 69.5%

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

   5. Taylor expanded in d around inf 80.7%

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

      1. +-commutative 80.7%

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

      2. mul-1-neg 80.7%

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

      3. unsub-neg 80.7%

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

      4. associate-*r/ 81.6%

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

   7. Applied egg-rr 81.6%

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

   if 1.50000000000000002e-122 < c < 7.49999999999999989e83

   1. Initial program 81.4%

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

      1. fmm-def 81.5%

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

      2. distribute-rgt-neg-out 81.5%

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

      3. +-commutative 81.5%

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

      4. fma-define 81.5%

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

   3. Simplified 81.5%

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

   if 7.49999999999999989e83 < c

   1. Initial program 34.0%

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

      1. fmm-def 34.0%

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

      2. distribute-rgt-neg-out 34.0%

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

      3. +-commutative 34.0%

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

      4. fma-define 34.0%

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

   3. Simplified 34.0%

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

   5. Taylor expanded in c around inf 72.7%

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

      1. mul-1-neg 72.7%

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

      2. unsub-neg 72.7%

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

      3. *-commutative 72.7%

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

   7. Simplified 72.7%

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

   8. Taylor expanded in b around 0 61.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c}} \]
   9. Step-by-step derivation

      1. +-commutative 61.8%

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

      2. mul-1-neg 61.8%

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

      3. unpow2 61.8%

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

      4. associate-/l/ 72.7%

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

      5. sub-neg 72.7%

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

      6. div-sub 72.7%

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

      7. associate-*l/ 78.2%

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

      8. associate-/r/ 78.2%

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

   10. Simplified 78.2%

       \[\leadsto \color{blue}{\frac{b - \frac{a}{\frac{c}{d}}}{c}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.2 \cdot 10^{+90}:\\ \;\;\;\;\frac{b - \frac{d}{c} \cdot a}{c}\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{-122}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{+83}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, c, d \cdot \left(-a\right)\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.4 \cdot 10^{+90}:\\ \;\;\;\;\frac{b - \frac{d}{c} \cdot a}{c}\\ \mathbf{elif}\;c \leq 2.3 \cdot 10^{-121}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 5.7 \cdot 10^{+83}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.4e+90)
   (/ (- b (* (/ d c) a)) c)
   (if (<= c 2.3e-121)
     (/ (- (* b (/ c d)) a) d)
     (if (<= c 5.7e+83)
       (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))
       (/ (- b (/ a (/ c d))) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.4e+90) {
		tmp = (b - ((d / c) * a)) / c;
	} else if (c <= 2.3e-121) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (c <= 5.7e+83) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} else {
		tmp = (b - (a / (c / d))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-2.4d+90)) then
        tmp = (b - ((d / c) * a)) / c
    else if (c <= 2.3d-121) then
        tmp = ((b * (c / d)) - a) / d
    else if (c <= 5.7d+83) then
        tmp = ((c * b) - (d * a)) / ((c * c) + (d * d))
    else
        tmp = (b - (a / (c / d))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.4e+90) {
		tmp = (b - ((d / c) * a)) / c;
	} else if (c <= 2.3e-121) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (c <= 5.7e+83) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} else {
		tmp = (b - (a / (c / d))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -2.4e+90:
		tmp = (b - ((d / c) * a)) / c
	elif c <= 2.3e-121:
		tmp = ((b * (c / d)) - a) / d
	elif c <= 5.7e+83:
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d))
	else:
		tmp = (b - (a / (c / d))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.4e+90)
		tmp = Float64(Float64(b - Float64(Float64(d / c) * a)) / c);
	elseif (c <= 2.3e-121)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	elseif (c <= 5.7e+83)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = Float64(Float64(b - Float64(a / Float64(c / d))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -2.4e+90)
		tmp = (b - ((d / c) * a)) / c;
	elseif (c <= 2.3e-121)
		tmp = ((b * (c / d)) - a) / d;
	elseif (c <= 5.7e+83)
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	else
		tmp = (b - (a / (c / d))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.4e+90], N[(N[(b - N[(N[(d / c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 2.3e-121], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 5.7e+83], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.4000000000000001e90

   1. Initial program 39.4%

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

      1. fmm-def 39.4%

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

      2. distribute-rgt-neg-out 39.4%

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

      3. +-commutative 39.4%

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

      4. fma-define 39.4%

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

   3. Simplified 39.4%

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

   5. Taylor expanded in c around inf 87.6%

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

      1. mul-1-neg 87.6%

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

      2. unsub-neg 87.6%

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

      3. *-commutative 87.6%

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

   7. Simplified 87.6%

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

      1. *-commutative 87.6%

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

      2. associate-*r/ 90.2%

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

      3. *-commutative 90.2%

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

   9. Applied egg-rr 90.2%

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

   if -2.4000000000000001e90 < c < 2.30000000000000012e-121

   1. Initial program 69.5%

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

      1. fmm-def 69.5%

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

      2. distribute-rgt-neg-out 69.5%

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

      3. +-commutative 69.5%

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

      4. fma-define 69.5%

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

   3. Simplified 69.5%

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

   5. Taylor expanded in d around inf 80.7%

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

      1. +-commutative 80.7%

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

      2. mul-1-neg 80.7%

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

      3. unsub-neg 80.7%

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

      4. associate-*r/ 81.6%

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

   7. Applied egg-rr 81.6%

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

   if 2.30000000000000012e-121 < c < 5.69999999999999949e83

   1. Initial program 81.4%

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

   if 5.69999999999999949e83 < c

   1. Initial program 34.0%

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

      1. fmm-def 34.0%

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

      2. distribute-rgt-neg-out 34.0%

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

      3. +-commutative 34.0%

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

      4. fma-define 34.0%

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

   3. Simplified 34.0%

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

   5. Taylor expanded in c around inf 72.7%

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

      1. mul-1-neg 72.7%

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

      2. unsub-neg 72.7%

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

      3. *-commutative 72.7%

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

   7. Simplified 72.7%

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

   8. Taylor expanded in b around 0 61.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c}} \]
   9. Step-by-step derivation

      1. +-commutative 61.8%

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

      2. mul-1-neg 61.8%

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

      3. unpow2 61.8%

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

      4. associate-/l/ 72.7%

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

      5. sub-neg 72.7%

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

      6. div-sub 72.7%

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

      7. associate-*l/ 78.2%

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

      8. associate-/r/ 78.2%

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

   10. Simplified 78.2%

       \[\leadsto \color{blue}{\frac{b - \frac{a}{\frac{c}{d}}}{c}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.4 \cdot 10^{+90}:\\ \;\;\;\;\frac{b - \frac{d}{c} \cdot a}{c}\\ \mathbf{elif}\;c \leq 2.3 \cdot 10^{-121}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 5.7 \cdot 10^{+83}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{+39} \lor \neg \left(d \leq 3.9 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -5e+39) (not (<= d 3.9e+15)))
   (/ (- a) d)
   (/ (- b (/ a (/ c d))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5e+39) || !(d <= 3.9e+15)) {
		tmp = -a / d;
	} else {
		tmp = (b - (a / (c / d))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-5d+39)) .or. (.not. (d <= 3.9d+15))) then
        tmp = -a / d
    else
        tmp = (b - (a / (c / d))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5e+39) || !(d <= 3.9e+15)) {
		tmp = -a / d;
	} else {
		tmp = (b - (a / (c / d))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -5e+39) or not (d <= 3.9e+15):
		tmp = -a / d
	else:
		tmp = (b - (a / (c / d))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -5e+39) || !(d <= 3.9e+15))
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(Float64(b - Float64(a / Float64(c / d))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -5e+39) || ~((d <= 3.9e+15)))
		tmp = -a / d;
	else
		tmp = (b - (a / (c / d))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -5e+39], N[Not[LessEqual[d, 3.9e+15]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(N[(b - N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -5.00000000000000015e39 or 3.9e15 < d

   1. Initial program 46.9%

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

      1. fmm-def 46.9%

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

      2. distribute-rgt-neg-out 46.9%

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

      3. +-commutative 46.9%

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

      4. fma-define 46.9%

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

   3. Simplified 46.9%

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

   5. Taylor expanded in c around 0 71.4%

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

      1. associate-*r/ 71.4%

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

      2. neg-mul-1 71.4%

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

   7. Simplified 71.4%

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

   if -5.00000000000000015e39 < d < 3.9e15

   1. Initial program 71.9%

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

      1. fmm-def 71.9%

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

      2. distribute-rgt-neg-out 71.9%

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

      3. +-commutative 71.9%

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

      4. fma-define 71.9%

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

   3. Simplified 71.9%

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

   5. Taylor expanded in c around inf 77.3%

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

      1. mul-1-neg 77.3%

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

      2. unsub-neg 77.3%

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

      3. *-commutative 77.3%

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

   7. Simplified 77.3%

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

   8. Taylor expanded in b around 0 70.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c}} \]
   9. Step-by-step derivation

      1. +-commutative 70.9%

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

      2. mul-1-neg 70.9%

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

      3. unpow2 70.9%

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

      4. associate-/l/ 77.2%

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

      5. sub-neg 77.2%

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

      6. div-sub 77.3%

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

      7. associate-*l/ 75.2%

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

      8. associate-/r/ 77.3%

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

   10. Simplified 77.3%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{+39} \lor \neg \left(d \leq 3.9 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -7.8 \cdot 10^{+39} \lor \neg \left(d \leq 2.15 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d}{c} \cdot a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -7.8e+39) (not (<= d 2.15e+15)))
   (/ (- a) d)
   (/ (- b (* (/ d c) a)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -7.8e+39) || !(d <= 2.15e+15)) {
		tmp = -a / d;
	} else {
		tmp = (b - ((d / c) * 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 ((d <= (-7.8d+39)) .or. (.not. (d <= 2.15d+15))) then
        tmp = -a / d
    else
        tmp = (b - ((d / c) * a)) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -7.8e+39) || !(d <= 2.15e+15)) {
		tmp = -a / d;
	} else {
		tmp = (b - ((d / c) * a)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -7.8e+39) or not (d <= 2.15e+15):
		tmp = -a / d
	else:
		tmp = (b - ((d / c) * a)) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -7.8e+39) || !(d <= 2.15e+15))
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(Float64(b - Float64(Float64(d / c) * a)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -7.8e+39) || ~((d <= 2.15e+15)))
		tmp = -a / d;
	else
		tmp = (b - ((d / c) * a)) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -7.8e+39], N[Not[LessEqual[d, 2.15e+15]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(N[(b - N[(N[(d / c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -7.8000000000000002e39 or 2.15e15 < d

   1. Initial program 46.9%

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

      1. fmm-def 46.9%

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

      2. distribute-rgt-neg-out 46.9%

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

      3. +-commutative 46.9%

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

      4. fma-define 46.9%

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

   3. Simplified 46.9%

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

   5. Taylor expanded in c around 0 71.4%

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

      1. associate-*r/ 71.4%

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

      2. neg-mul-1 71.4%

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

   7. Simplified 71.4%

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

   if -7.8000000000000002e39 < d < 2.15e15

   1. Initial program 71.9%

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

      1. fmm-def 71.9%

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

      2. distribute-rgt-neg-out 71.9%

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

      3. +-commutative 71.9%

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

      4. fma-define 71.9%

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

   3. Simplified 71.9%

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

   5. Taylor expanded in c around inf 77.3%

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

      1. mul-1-neg 77.3%

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

      2. unsub-neg 77.3%

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

      3. *-commutative 77.3%

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

   7. Simplified 77.3%

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

      1. *-commutative 77.3%

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

      2. associate-*r/ 77.3%

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

      3. *-commutative 77.3%

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

   9. Applied egg-rr 77.3%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.8 \cdot 10^{+39} \lor \neg \left(d \leq 2.15 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d}{c} \cdot a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.4 \cdot 10^{+90}:\\ \;\;\;\;\frac{b - \frac{d}{c} \cdot a}{c}\\ \mathbf{elif}\;c \leq 5 \cdot 10^{+65}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3.4e+90)
   (/ (- b (* (/ d c) a)) c)
   (if (<= c 5e+65) (/ (- (* b (/ c d)) a) d) (/ (- b (/ a (/ c d))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.4e+90) {
		tmp = (b - ((d / c) * a)) / c;
	} else if (c <= 5e+65) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - (a / (c / d))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-3.4d+90)) then
        tmp = (b - ((d / c) * a)) / c
    else if (c <= 5d+65) then
        tmp = ((b * (c / d)) - a) / d
    else
        tmp = (b - (a / (c / d))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.4e+90) {
		tmp = (b - ((d / c) * a)) / c;
	} else if (c <= 5e+65) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - (a / (c / d))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -3.4e+90:
		tmp = (b - ((d / c) * a)) / c
	elif c <= 5e+65:
		tmp = ((b * (c / d)) - a) / d
	else:
		tmp = (b - (a / (c / d))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3.4e+90)
		tmp = Float64(Float64(b - Float64(Float64(d / c) * a)) / c);
	elseif (c <= 5e+65)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(a / Float64(c / d))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -3.4e+90)
		tmp = (b - ((d / c) * a)) / c;
	elseif (c <= 5e+65)
		tmp = ((b * (c / d)) - a) / d;
	else
		tmp = (b - (a / (c / d))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3.4e+90], N[(N[(b - N[(N[(d / c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 5e+65], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.40000000000000018e90

   1. Initial program 39.4%

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

      1. fmm-def 39.4%

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

      2. distribute-rgt-neg-out 39.4%

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

      3. +-commutative 39.4%

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

      4. fma-define 39.4%

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

   3. Simplified 39.4%

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

   5. Taylor expanded in c around inf 87.6%

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

      1. mul-1-neg 87.6%

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

      2. unsub-neg 87.6%

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

      3. *-commutative 87.6%

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

   7. Simplified 87.6%

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

      1. *-commutative 87.6%

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

      2. associate-*r/ 90.2%

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

      3. *-commutative 90.2%

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

   9. Applied egg-rr 90.2%

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

   if -3.40000000000000018e90 < c < 4.99999999999999973e65

   1. Initial program 72.0%

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

      1. fmm-def 72.0%

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

      2. distribute-rgt-neg-out 72.0%

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

      3. +-commutative 72.0%

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

      4. fma-define 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in d around inf 77.0%

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

      1. +-commutative 77.0%

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

      2. mul-1-neg 77.0%

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

      3. unsub-neg 77.0%

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

      4. associate-*r/ 78.4%

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

   7. Applied egg-rr 78.4%

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

   if 4.99999999999999973e65 < c

   1. Initial program 37.3%

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

      1. fmm-def 37.3%

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

      2. distribute-rgt-neg-out 37.3%

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

      3. +-commutative 37.3%

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

      4. fma-define 37.3%

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

   3. Simplified 37.3%

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

   5. Taylor expanded in c around inf 72.9%

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

      1. mul-1-neg 72.9%

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

      2. unsub-neg 72.9%

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

      3. *-commutative 72.9%

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

   7. Simplified 72.9%

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

   8. Taylor expanded in b around 0 62.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c}} \]
   9. Step-by-step derivation

      1. +-commutative 62.8%

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

      2. mul-1-neg 62.8%

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

      3. unpow2 62.8%

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

      4. associate-/l/ 72.9%

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

      5. sub-neg 72.9%

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

      6. div-sub 72.9%

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

      7. associate-*l/ 78.0%

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

      8. associate-/r/ 78.0%

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

   10. Simplified 78.0%

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

Alternative 6: 64.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.8 \cdot 10^{+36} \lor \neg \left(d \leq 2.15 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -3.8e+36) (not (<= d 2.15e+15))) (/ (- a) d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -3.8e+36) || !(d <= 2.15e+15)) {
		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 <= (-3.8d+36)) .or. (.not. (d <= 2.15d+15))) 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 <= -3.8e+36) || !(d <= 2.15e+15)) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -3.8e+36) or not (d <= 2.15e+15):
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -3.8e+36) || !(d <= 2.15e+15))
		tmp = Float64(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 <= -3.8e+36) || ~((d <= 2.15e+15)))
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -3.8e+36], N[Not[LessEqual[d, 2.15e+15]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -3.80000000000000025e36 or 2.15e15 < d

   1. Initial program 47.7%

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

      1. fmm-def 47.7%

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

      2. distribute-rgt-neg-out 47.7%

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

      3. +-commutative 47.7%

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

      4. fma-define 47.7%

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

   3. Simplified 47.7%

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

   5. Taylor expanded in c around 0 70.3%

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

      1. associate-*r/ 70.3%

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

      2. neg-mul-1 70.3%

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

   7. Simplified 70.3%

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

   if -3.80000000000000025e36 < d < 2.15e15

   1. Initial program 71.5%

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

      1. fmm-def 71.5%

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

      2. distribute-rgt-neg-out 71.5%

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

      3. +-commutative 71.5%

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

      4. fma-define 71.5%

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

   3. Simplified 71.5%

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

   5. Taylor expanded in c around inf 59.9%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.8 \cdot 10^{+36} \lor \neg \left(d \leq 2.15 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 7.2 \cdot 10^{+200}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (if (<= d 7.2e+200) (/ b c) (/ a d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= 7.2e+200) {
		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 (d <= 7.2d+200) 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 (d <= 7.2e+200) {
		tmp = b / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= 7.2e+200:
		tmp = b / c
	else:
		tmp = a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= 7.2e+200)
		tmp = Float64(b / c);
	else
		tmp = Float64(a / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= 7.2e+200)
		tmp = b / c;
	else
		tmp = a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, 7.2e+200], N[(b / c), $MachinePrecision], N[(a / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 7.1999999999999995e200

   1. Initial program 64.1%

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

      1. fmm-def 64.1%

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

      2. distribute-rgt-neg-out 64.1%

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

      3. +-commutative 64.1%

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

      4. fma-define 64.1%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in c around inf 43.0%

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

   if 7.1999999999999995e200 < d

   1. Initial program 24.8%

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

      1. fmm-def 24.8%

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

      2. distribute-rgt-neg-out 24.8%

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

      3. +-commutative 24.8%

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

      4. fma-define 24.8%

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

   3. Simplified 24.8%

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

   5. Taylor expanded in c around 0 89.6%

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

      1. associate-*r/ 89.6%

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

      2. neg-mul-1 89.6%

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

   7. Simplified 89.6%

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

      1. add-sqr-sqrt 55.7%

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

      2. sqrt-unprod 57.2%

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

      3. sqr-neg 57.2%

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

      4. sqrt-unprod 12.1%

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

      5. add-sqr-sqrt 25.4%

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

      6. div-inv 25.4%

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

   9. Applied egg-rr 25.4%

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

       1. associate-*r/ 25.4%

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

       2. *-rgt-identity 25.4%

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

   11. Simplified 25.4%

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

Alternative 8: 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 60.0%

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

   1. fmm-def 60.0%

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

   2. distribute-rgt-neg-out 60.0%

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

   3. +-commutative 60.0%

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

   4. fma-define 60.0%

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

3. Simplified 60.0%

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

5. Taylor expanded in c around 0 44.5%

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

   1. associate-*r/ 44.5%

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

   2. neg-mul-1 44.5%

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

7. Simplified 44.5%

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

   1. add-sqr-sqrt 24.6%

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

   2. sqrt-unprod 23.9%

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

   3. sqr-neg 23.9%

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

   4. sqrt-unprod 4.1%

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

   5. add-sqr-sqrt 8.5%

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

   6. div-inv 8.5%

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

9. Applied egg-rr 8.5%

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

    1. associate-*r/ 8.5%

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

    2. *-rgt-identity 8.5%

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

11. Simplified 8.5%

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

Developer Target 1: 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 2024157 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :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))))