Complex division, imag part

Percentage Accurate: 61.2% → 82.2%

Time: 10.3s

Alternatives: 9

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: 61.2% 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: 82.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.55 \cdot 10^{+74}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;d \leq -2.6 \cdot 10^{-67}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{-121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, b, a \cdot \frac{d}{c}\right)}{-c}\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, c, d \cdot \left(-a\right)\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -3.55e+74)
   (/ (- (* b (/ c d)) a) d)
   (if (<= d -2.6e-67)
     (/ (- (* b c) (* d a)) (+ (* c c) (* d d)))
     (if (<= d 2e-121)
       (/ (fma -1.0 b (* a (/ d c))) (- c))
       (if (<= d 2.4e+35)
         (/ (fma b c (* d (- a))) (fma d d (* c c)))
         (/ (- (* c (/ b d)) a) d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -3.55e+74) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (d <= -2.6e-67) {
		tmp = ((b * c) - (d * a)) / ((c * c) + (d * d));
	} else if (d <= 2e-121) {
		tmp = fma(-1.0, b, (a * (d / c))) / -c;
	} else if (d <= 2.4e+35) {
		tmp = fma(b, c, (d * -a)) / fma(d, d, (c * c));
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -3.55e+74)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	elseif (d <= -2.6e-67)
		tmp = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 2e-121)
		tmp = Float64(fma(-1.0, b, Float64(a * Float64(d / c))) / Float64(-c));
	elseif (d <= 2.4e+35)
		tmp = Float64(fma(b, c, Float64(d * Float64(-a))) / fma(d, d, Float64(c * c)));
	else
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -3.55e+74], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -2.6e-67], N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2e-121], N[(N[(-1.0 * b + N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-c)), $MachinePrecision], If[LessEqual[d, 2.4e+35], N[(N[(b * c + N[(d * (-a)), $MachinePrecision]), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -3.55000000000000001e74

   1. Initial program 40.3%

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

      1. fmm-def 40.3%

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

      2. distribute-rgt-neg-out 40.3%

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

      3. +-commutative 40.3%

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

      4. fma-define 40.3%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in d around -inf 82.4%

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

      1. mul-1-neg 82.4%

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

      2. mul-1-neg 82.4%

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

      3. associate-/l* 93.5%

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

   7. Simplified 93.5%

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

   8. Taylor expanded in a around 0 79.9%

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

      1. mul-1-neg 79.9%

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

      2. distribute-frac-neg2 79.9%

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

      3. +-commutative 79.9%

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

      4. distribute-frac-neg2 79.9%

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

      5. sub-neg 79.9%

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

      6. unpow2 79.9%

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

      7. associate-/l/ 82.4%

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

      8. div-sub 82.4%

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

      9. associate-*r/ 93.5%

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

   10. Simplified 93.5%

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

   if -3.55000000000000001e74 < d < -2.5999999999999999e-67

   1. Initial program 86.4%

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

   if -2.5999999999999999e-67 < d < 2e-121

   1. Initial program 76.8%

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

      1. fmm-def 76.8%

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

      2. distribute-rgt-neg-out 76.8%

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

      3. +-commutative 76.8%

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

      4. fma-define 76.8%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in c around -inf 97.8%

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

      1. mul-1-neg 97.8%

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

      2. fma-define 97.8%

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

      3. associate-/l* 97.8%

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

   7. Simplified 97.8%

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

   if 2e-121 < d < 2.40000000000000015e35

   1. Initial program 86.4%

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

      1. fmm-def 86.4%

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

      2. distribute-rgt-neg-out 86.4%

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

      3. +-commutative 86.4%

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

      4. fma-define 86.4%

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

   3. Simplified 86.4%

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

   if 2.40000000000000015e35 < d

   1. Initial program 48.8%

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

      1. fmm-def 48.8%

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

      2. distribute-rgt-neg-out 48.8%

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

      3. +-commutative 48.8%

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

      4. fma-define 48.8%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in d around -inf 88.9%

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

      1. mul-1-neg 88.9%

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

      2. distribute-neg-frac2 88.9%

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

      3. mul-1-neg 88.9%

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

      4. unsub-neg 88.9%

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

      5. *-commutative 88.9%

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

      6. associate-/l* 92.0%

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

   7. Simplified 92.0%

      \[\leadsto \color{blue}{\frac{a - c \cdot \frac{b}{d}}{-d}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.55 \cdot 10^{+74}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;d \leq -2.6 \cdot 10^{-67}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{-121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, b, a \cdot \frac{d}{c}\right)}{-c}\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, c, d \cdot \left(-a\right)\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -2.4 \cdot 10^{+72}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;d \leq -4.8 \cdot 10^{-65}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{-120}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, b, a \cdot \frac{d}{c}\right)}{-c}\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{+35}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* d a)) (+ (* c c) (* d d)))))
   (if (<= d -2.4e+72)
     (/ (- (* b (/ c d)) a) d)
     (if (<= d -4.8e-65)
       t_0
       (if (<= d 2.1e-120)
         (/ (fma -1.0 b (* a (/ d c))) (- c))
         (if (<= d 2.4e+35) t_0 (/ (- (* c (/ b d)) a) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -2.4e+72) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (d <= -4.8e-65) {
		tmp = t_0;
	} else if (d <= 2.1e-120) {
		tmp = fma(-1.0, b, (a * (d / c))) / -c;
	} else if (d <= 2.4e+35) {
		tmp = t_0;
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -2.4e+72)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	elseif (d <= -4.8e-65)
		tmp = t_0;
	elseif (d <= 2.1e-120)
		tmp = Float64(fma(-1.0, b, Float64(a * Float64(d / c))) / Float64(-c));
	elseif (d <= 2.4e+35)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.4e+72], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -4.8e-65], t$95$0, If[LessEqual[d, 2.1e-120], N[(N[(-1.0 * b + N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-c)), $MachinePrecision], If[LessEqual[d, 2.4e+35], t$95$0, N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.4000000000000001e72

   1. Initial program 40.3%

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

      1. fmm-def 40.3%

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

      2. distribute-rgt-neg-out 40.3%

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

      3. +-commutative 40.3%

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

      4. fma-define 40.3%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in d around -inf 82.4%

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

      1. mul-1-neg 82.4%

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

      2. mul-1-neg 82.4%

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

      3. associate-/l* 93.5%

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

   7. Simplified 93.5%

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

   8. Taylor expanded in a around 0 79.9%

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

      1. mul-1-neg 79.9%

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

      2. distribute-frac-neg2 79.9%

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

      3. +-commutative 79.9%

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

      4. distribute-frac-neg2 79.9%

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

      5. sub-neg 79.9%

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

      6. unpow2 79.9%

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

      7. associate-/l/ 82.4%

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

      8. div-sub 82.4%

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

      9. associate-*r/ 93.5%

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

   10. Simplified 93.5%

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

   if -2.4000000000000001e72 < d < -4.8000000000000003e-65 or 2.1e-120 < d < 2.40000000000000015e35

   1. Initial program 86.4%

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

   if -4.8000000000000003e-65 < d < 2.1e-120

   1. Initial program 76.8%

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

      1. fmm-def 76.8%

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

      2. distribute-rgt-neg-out 76.8%

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

      3. +-commutative 76.8%

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

      4. fma-define 76.8%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in c around -inf 97.8%

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

      1. mul-1-neg 97.8%

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

      2. fma-define 97.8%

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

      3. associate-/l* 97.8%

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

   7. Simplified 97.8%

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

   if 2.40000000000000015e35 < d

   1. Initial program 48.8%

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

      1. fmm-def 48.8%

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

      2. distribute-rgt-neg-out 48.8%

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

      3. +-commutative 48.8%

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

      4. fma-define 48.8%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in d around -inf 88.9%

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

      1. mul-1-neg 88.9%

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

      2. distribute-neg-frac2 88.9%

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

      3. mul-1-neg 88.9%

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

      4. unsub-neg 88.9%

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

      5. *-commutative 88.9%

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

      6. associate-/l* 92.0%

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

   7. Simplified 92.0%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.4 \cdot 10^{+72}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;d \leq -4.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{-120}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, b, a \cdot \frac{d}{c}\right)}{-c}\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{+35}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -6.8 \cdot 10^{+72}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;d \leq -2.5 \cdot 10^{-67}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-119}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{+35}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* d a)) (+ (* c c) (* d d)))))
   (if (<= d -6.8e+72)
     (/ (- (* b (/ c d)) a) d)
     (if (<= d -2.5e-67)
       t_0
       (if (<= d 4.5e-119)
         (/ (- b (/ (* d a) c)) c)
         (if (<= d 2.4e+35) t_0 (/ (- (* c (/ b d)) a) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -6.8e+72) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (d <= -2.5e-67) {
		tmp = t_0;
	} else if (d <= 4.5e-119) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 2.4e+35) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = ((b * c) - (d * a)) / ((c * c) + (d * d))
    if (d <= (-6.8d+72)) then
        tmp = ((b * (c / d)) - a) / d
    else if (d <= (-2.5d-67)) then
        tmp = t_0
    else if (d <= 4.5d-119) then
        tmp = (b - ((d * a) / c)) / c
    else if (d <= 2.4d+35) then
        tmp = t_0
    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 t_0 = ((b * c) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -6.8e+72) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (d <= -2.5e-67) {
		tmp = t_0;
	} else if (d <= 4.5e-119) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 2.4e+35) {
		tmp = t_0;
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * c) - (d * a)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -6.8e+72:
		tmp = ((b * (c / d)) - a) / d
	elif d <= -2.5e-67:
		tmp = t_0
	elif d <= 4.5e-119:
		tmp = (b - ((d * a) / c)) / c
	elif d <= 2.4e+35:
		tmp = t_0
	else:
		tmp = ((c * (b / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -6.8e+72)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	elseif (d <= -2.5e-67)
		tmp = t_0;
	elseif (d <= 4.5e-119)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (d <= 2.4e+35)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b * c) - (d * a)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (d <= -6.8e+72)
		tmp = ((b * (c / d)) - a) / d;
	elseif (d <= -2.5e-67)
		tmp = t_0;
	elseif (d <= 4.5e-119)
		tmp = (b - ((d * a) / c)) / c;
	elseif (d <= 2.4e+35)
		tmp = t_0;
	else
		tmp = ((c * (b / d)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -6.8e+72], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -2.5e-67], t$95$0, If[LessEqual[d, 4.5e-119], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2.4e+35], t$95$0, N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -6.7999999999999997e72

   1. Initial program 40.3%

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

      1. fmm-def 40.3%

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

      2. distribute-rgt-neg-out 40.3%

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

      3. +-commutative 40.3%

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

      4. fma-define 40.3%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in d around -inf 82.4%

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

      1. mul-1-neg 82.4%

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

      2. mul-1-neg 82.4%

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

      3. associate-/l* 93.5%

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

   7. Simplified 93.5%

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

   8. Taylor expanded in a around 0 79.9%

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

      1. mul-1-neg 79.9%

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

      2. distribute-frac-neg2 79.9%

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

      3. +-commutative 79.9%

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

      4. distribute-frac-neg2 79.9%

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

      5. sub-neg 79.9%

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

      6. unpow2 79.9%

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

      7. associate-/l/ 82.4%

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

      8. div-sub 82.4%

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

      9. associate-*r/ 93.5%

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

   10. Simplified 93.5%

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

   if -6.7999999999999997e72 < d < -2.4999999999999999e-67 or 4.5000000000000003e-119 < d < 2.40000000000000015e35

   1. Initial program 86.4%

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

   if -2.4999999999999999e-67 < d < 4.5000000000000003e-119

   1. Initial program 76.8%

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

      1. fmm-def 76.8%

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

      2. distribute-rgt-neg-out 76.8%

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

      3. +-commutative 76.8%

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

      4. fma-define 76.8%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in c around inf 97.8%

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

      1. mul-1-neg 97.8%

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

      2. unsub-neg 97.8%

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

      3. *-commutative 97.8%

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

   7. Simplified 97.8%

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

   if 2.40000000000000015e35 < d

   1. Initial program 48.8%

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

      1. fmm-def 48.8%

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

      2. distribute-rgt-neg-out 48.8%

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

      3. +-commutative 48.8%

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

      4. fma-define 48.8%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in d around -inf 88.9%

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

      1. mul-1-neg 88.9%

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

      2. distribute-neg-frac2 88.9%

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

      3. mul-1-neg 88.9%

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

      4. unsub-neg 88.9%

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

      5. *-commutative 88.9%

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

      6. associate-/l* 92.0%

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

   7. Simplified 92.0%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.8 \cdot 10^{+72}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;d \leq -2.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-119}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{+35}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -8.5 \cdot 10^{-59} \lor \neg \left(d \leq 2.1 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -8.5e-59) (not (<= d 2.1e-12)))
   (/ (- (* b (/ c d)) a) d)
   (/ (- b (/ (* d a) c)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -8.5e-59) || !(d <= 2.1e-12)) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - ((d * a) / 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 ((d <= (-8.5d-59)) .or. (.not. (d <= 2.1d-12))) then
        tmp = ((b * (c / d)) - a) / d
    else
        tmp = (b - ((d * a) / c)) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -8.5e-59) || !(d <= 2.1e-12)) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - ((d * a) / c)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -8.5e-59) or not (d <= 2.1e-12):
		tmp = ((b * (c / d)) - a) / d
	else:
		tmp = (b - ((d * a) / c)) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -8.5e-59) || !(d <= 2.1e-12))
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -8.5e-59) || ~((d <= 2.1e-12)))
		tmp = ((b * (c / d)) - a) / d;
	else
		tmp = (b - ((d * a) / c)) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -8.5e-59], N[Not[LessEqual[d, 2.1e-12]], $MachinePrecision]], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -8.49999999999999933e-59 or 2.09999999999999994e-12 < d

   1. Initial program 56.0%

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

      1. fmm-def 56.0%

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

      2. distribute-rgt-neg-out 56.0%

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

      3. +-commutative 56.0%

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

      4. fma-define 56.1%

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

   3. Simplified 56.1%

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

   5. Taylor expanded in d around -inf 80.2%

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

      1. mul-1-neg 80.2%

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

      2. mul-1-neg 80.2%

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

      3. associate-/l* 84.4%

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

   7. Simplified 84.4%

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

   8. Taylor expanded in a around 0 79.4%

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

      1. mul-1-neg 79.4%

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

      2. distribute-frac-neg2 79.4%

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

      3. +-commutative 79.4%

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

      4. distribute-frac-neg2 79.4%

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

      5. sub-neg 79.4%

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

      6. unpow2 79.4%

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

      7. associate-/l/ 80.2%

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

      8. div-sub 80.2%

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

      9. associate-*r/ 84.4%

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

   10. Simplified 84.4%

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

   if -8.49999999999999933e-59 < d < 2.09999999999999994e-12

   1. Initial program 78.8%

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

      1. fmm-def 78.8%

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

      2. distribute-rgt-neg-out 78.8%

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

      3. +-commutative 78.8%

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

      4. fma-define 78.8%

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

   3. Simplified 78.8%

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

   5. Taylor expanded in c around inf 90.8%

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

      1. mul-1-neg 90.8%

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

      2. unsub-neg 90.8%

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

      3. *-commutative 90.8%

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

   7. Simplified 90.8%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.5 \cdot 10^{-59} \lor \neg \left(d \leq 2.1 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.5 \cdot 10^{+25} \lor \neg \left(d \leq 5.9 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -3.5e+25) (not (<= d 5.9e-8)))
   (/ a (- d))
   (/ (- b (/ (* d a) c)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -3.5e+25) || !(d <= 5.9e-8)) {
		tmp = a / -d;
	} else {
		tmp = (b - ((d * a) / 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 ((d <= (-3.5d+25)) .or. (.not. (d <= 5.9d-8))) then
        tmp = a / -d
    else
        tmp = (b - ((d * a) / c)) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -3.5e+25) || !(d <= 5.9e-8)) {
		tmp = a / -d;
	} else {
		tmp = (b - ((d * a) / c)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -3.5e+25) or not (d <= 5.9e-8):
		tmp = a / -d
	else:
		tmp = (b - ((d * a) / c)) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -3.5e+25) || !(d <= 5.9e-8))
		tmp = Float64(a / Float64(-d));
	else
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -3.5e+25) || ~((d <= 5.9e-8)))
		tmp = a / -d;
	else
		tmp = (b - ((d * a) / c)) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -3.5e+25], N[Not[LessEqual[d, 5.9e-8]], $MachinePrecision]], N[(a / (-d)), $MachinePrecision], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -3.49999999999999999e25 or 5.8999999999999999e-8 < d

   1. Initial program 49.3%

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

      1. fmm-def 49.3%

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

      2. distribute-rgt-neg-out 49.3%

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

      3. +-commutative 49.3%

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

      4. fma-define 49.3%

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

   3. Simplified 49.3%

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

   5. Taylor expanded in c around 0 77.0%

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

      1. associate-*r/ 77.0%

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

      2. neg-mul-1 77.0%

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

   7. Simplified 77.0%

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

   if -3.49999999999999999e25 < d < 5.8999999999999999e-8

   1. Initial program 81.3%

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

      1. fmm-def 81.3%

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

      2. distribute-rgt-neg-out 81.3%

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

      3. +-commutative 81.3%

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

      4. fma-define 81.3%

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

   3. Simplified 81.3%

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

   5. Taylor expanded in c around inf 83.0%

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

      1. mul-1-neg 83.0%

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

      2. unsub-neg 83.0%

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

      3. *-commutative 83.0%

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

   7. Simplified 83.0%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.5 \cdot 10^{+25} \lor \neg \left(d \leq 5.9 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{-59}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.55e-59)
   (/ (- (* b (/ c d)) a) d)
   (if (<= d 3.8e-11) (/ (- b (/ (* d a) c)) c) (/ (- (* c (/ b d)) a) d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.55e-59) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (d <= 3.8e-11) {
		tmp = (b - ((d * a) / c)) / 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 (d <= (-1.55d-59)) then
        tmp = ((b * (c / d)) - a) / d
    else if (d <= 3.8d-11) then
        tmp = (b - ((d * a) / c)) / 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 (d <= -1.55e-59) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (d <= 3.8e-11) {
		tmp = (b - ((d * a) / c)) / c;
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -1.55e-59:
		tmp = ((b * (c / d)) - a) / d
	elif d <= 3.8e-11:
		tmp = (b - ((d * a) / c)) / c
	else:
		tmp = ((c * (b / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.55e-59)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	elseif (d <= 3.8e-11)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / 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 (d <= -1.55e-59)
		tmp = ((b * (c / d)) - a) / d;
	elseif (d <= 3.8e-11)
		tmp = (b - ((d * a) / c)) / c;
	else
		tmp = ((c * (b / d)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.55e-59], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 3.8e-11], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.55e-59

   1. Initial program 58.6%

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

      1. fmm-def 58.6%

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

      2. distribute-rgt-neg-out 58.6%

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

      3. +-commutative 58.6%

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

      4. fma-define 58.6%

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

   3. Simplified 58.6%

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

   5. Taylor expanded in d around -inf 73.7%

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

      1. mul-1-neg 73.7%

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

      2. mul-1-neg 73.7%

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

      3. associate-/l* 80.4%

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

   7. Simplified 80.4%

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

   8. Taylor expanded in a around 0 72.1%

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

      1. mul-1-neg 72.1%

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

      2. distribute-frac-neg2 72.1%

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

      3. +-commutative 72.1%

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

      4. distribute-frac-neg2 72.1%

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

      5. sub-neg 72.1%

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

      6. unpow2 72.1%

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

      7. associate-/l/ 73.7%

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

      8. div-sub 73.7%

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

      9. associate-*r/ 80.4%

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

   10. Simplified 80.4%

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

   if -1.55e-59 < d < 3.7999999999999998e-11

   1. Initial program 78.8%

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

      1. fmm-def 78.8%

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

      2. distribute-rgt-neg-out 78.8%

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

      3. +-commutative 78.8%

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

      4. fma-define 78.8%

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

   3. Simplified 78.8%

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

   5. Taylor expanded in c around inf 90.8%

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

      1. mul-1-neg 90.8%

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

      2. unsub-neg 90.8%

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

      3. *-commutative 90.8%

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

   7. Simplified 90.8%

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

   if 3.7999999999999998e-11 < d

   1. Initial program 53.3%

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

      1. fmm-def 53.3%

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

      2. distribute-rgt-neg-out 53.3%

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

      3. +-commutative 53.3%

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

      4. fma-define 53.4%

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

   3. Simplified 53.4%

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

   5. Taylor expanded in d around -inf 87.1%

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

      1. mul-1-neg 87.1%

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

      2. distribute-neg-frac2 87.1%

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

      3. mul-1-neg 87.1%

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

      4. unsub-neg 87.1%

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

      5. *-commutative 87.1%

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

      6. associate-/l* 89.9%

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

   7. Simplified 89.9%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{-59}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -8.5 \cdot 10^{-68} \lor \neg \left(d \leq 5 \cdot 10^{-64}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -8.5e-68) (not (<= d 5e-64))) (/ a (- d)) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -8.5e-68) || !(d <= 5e-64)) {
		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 <= (-8.5d-68)) .or. (.not. (d <= 5d-64))) 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 <= -8.5e-68) || !(d <= 5e-64)) {
		tmp = a / -d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -8.5e-68) or not (d <= 5e-64):
		tmp = a / -d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -8.5e-68) || !(d <= 5e-64))
		tmp = Float64(a / Float64(-d));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -8.5e-68) || ~((d <= 5e-64)))
		tmp = a / -d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -8.5e-68], N[Not[LessEqual[d, 5e-64]], $MachinePrecision]], N[(a / (-d)), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -8.50000000000000026e-68 or 5.00000000000000033e-64 < d

   1. Initial program 58.4%

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

      1. fmm-def 58.4%

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

      2. distribute-rgt-neg-out 58.4%

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

      3. +-commutative 58.4%

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

      4. fma-define 58.4%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in c around 0 68.4%

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

      1. associate-*r/ 68.4%

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

      2. neg-mul-1 68.4%

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

   7. Simplified 68.4%

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

   if -8.50000000000000026e-68 < d < 5.00000000000000033e-64

   1. Initial program 78.2%

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

      1. fmm-def 78.2%

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

      2. distribute-rgt-neg-out 78.2%

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

      3. +-commutative 78.2%

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

      4. fma-define 78.2%

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

   3. Simplified 78.2%

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

   5. Taylor expanded in c around inf 75.5%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.5 \cdot 10^{-68} \lor \neg \left(d \leq 5 \cdot 10^{-64}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 46.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.4 \cdot 10^{+176} \lor \neg \left(d \leq 1.15 \cdot 10^{+190}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -4.4e+176) (not (<= d 1.15e+190))) (/ a d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4.4e+176) || !(d <= 1.15e+190)) {
		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 <= (-4.4d+176)) .or. (.not. (d <= 1.15d+190))) 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 <= -4.4e+176) || !(d <= 1.15e+190)) {
		tmp = a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -4.4e+176) or not (d <= 1.15e+190):
		tmp = a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -4.4e+176) || !(d <= 1.15e+190))
		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 <= -4.4e+176) || ~((d <= 1.15e+190)))
		tmp = a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -4.4e+176], N[Not[LessEqual[d, 1.15e+190]], $MachinePrecision]], N[(a / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.40000000000000015e176 or 1.15e190 < d

   1. Initial program 37.8%

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

      1. fmm-def 37.8%

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

      2. distribute-rgt-neg-out 37.8%

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

      3. +-commutative 37.8%

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

      4. fma-define 37.8%

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

   3. Simplified 37.8%

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

   5. Taylor expanded in c around 0 92.3%

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

      1. associate-*r/ 92.3%

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

      2. neg-mul-1 92.3%

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

   7. Simplified 92.3%

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

      1. div-inv 92.2%

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

      2. add-sqr-sqrt 49.9%

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

      3. sqrt-unprod 57.4%

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

      4. sqr-neg 57.4%

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

      5. sqrt-unprod 19.1%

         \[\leadsto \color{blue}{\left(\sqrt{a} \cdot \sqrt{a}\right)} \cdot \frac{1}{d} \]

      6. add-sqr-sqrt 39.0%

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

   9. Applied egg-rr 39.0%

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

       1. associate-*r/ 39.0%

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

       2. *-rgt-identity 39.0%

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

   11. Simplified 39.0%

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

   if -4.40000000000000015e176 < d < 1.15e190

   1. Initial program 73.1%

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

      1. fmm-def 73.1%

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

      2. distribute-rgt-neg-out 73.1%

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

      3. +-commutative 73.1%

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

      4. fma-define 73.1%

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

   3. Simplified 73.1%

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

   5. Taylor expanded in c around inf 48.7%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.4 \cdot 10^{+176} \lor \neg \left(d \leq 1.15 \cdot 10^{+190}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 10.5% 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 66.2%

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

   1. fmm-def 66.2%

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

   2. distribute-rgt-neg-out 66.2%

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

   3. +-commutative 66.2%

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

   4. fma-define 66.2%

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

3. Simplified 66.2%

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

5. Taylor expanded in c around 0 47.6%

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

   1. associate-*r/ 47.6%

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

   2. neg-mul-1 47.6%

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

7. Simplified 47.6%

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

   1. div-inv 47.5%

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

   2. add-sqr-sqrt 25.4%

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

   3. sqrt-unprod 26.3%

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

   4. sqr-neg 26.3%

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

   5. sqrt-unprod 6.0%

      \[\leadsto \color{blue}{\left(\sqrt{a} \cdot \sqrt{a}\right)} \cdot \frac{1}{d} \]

   6. add-sqr-sqrt 11.2%

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

9. Applied egg-rr 11.2%

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

    1. associate-*r/ 11.2%

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

    2. *-rgt-identity 11.2%

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

11. Simplified 11.2%

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

Developer Target 1: 99.2% 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 2024152 
(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))))