Complex division, imag part

Percentage Accurate: 61.4% → 84.5%

Time: 10.2s

Alternatives: 11

Speedup: 1.5×

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: 61.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{if}\;d \leq -1.15 \cdot 10^{+87}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -7.5 \cdot 10^{-153}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-161}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+128}:\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{d}{t\_0}, \frac{b \cdot c}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))) (t_1 (/ (fma c (/ b d) (- a)) d)))
   (if (<= d -1.15e+87)
     t_1
     (if (<= d -7.5e-153)
       (/ (- (* b c) (* d a)) (fma c c (* d d)))
       (if (<= d 8e-161)
         (/ (- b (* a (/ d c))) c)
         (if (<= d 5.5e+128) (fma (- a) (/ d t_0) (/ (* b c) t_0)) t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double t_1 = fma(c, (b / d), -a) / d;
	double tmp;
	if (d <= -1.15e+87) {
		tmp = t_1;
	} else if (d <= -7.5e-153) {
		tmp = ((b * c) - (d * a)) / fma(c, c, (d * d));
	} else if (d <= 8e-161) {
		tmp = (b - (a * (d / c))) / c;
	} else if (d <= 5.5e+128) {
		tmp = fma(-a, (d / t_0), ((b * c) / t_0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = Float64(fma(c, Float64(b / d), Float64(-a)) / d)
	tmp = 0.0
	if (d <= -1.15e+87)
		tmp = t_1;
	elseif (d <= -7.5e-153)
		tmp = Float64(Float64(Float64(b * c) - Float64(d * a)) / fma(c, c, Float64(d * d)));
	elseif (d <= 8e-161)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (d <= 5.5e+128)
		tmp = fma(Float64(-a), Float64(d / t_0), Float64(Float64(b * c) / t_0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -1.15e+87], t$95$1, If[LessEqual[d, -7.5e-153], N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8e-161], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 5.5e+128], N[((-a) * N[(d / t$95$0), $MachinePrecision] + N[(N[(b * c), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.1500000000000001e87 or 5.4999999999999998e128 < d

   1. Initial program 33.8%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 85.1

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

   5. Applied rewrites 85.1%

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

   if -1.1500000000000001e87 < d < -7.5e-153

   1. Initial program 85.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 86.0

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

   4. Applied rewrites 86.0%

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

   if -7.5e-153 < d < 8.00000000000000022e-161

   1. Initial program 75.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 95.7

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

   5. Applied rewrites 95.7%

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

   7. Applied rewrites 98.5%

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

   if 8.00000000000000022e-161 < d < 5.4999999999999998e128

   1. Initial program 78.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 81.4

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

   4. Applied rewrites 81.4%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.15 \cdot 10^{+87}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq -7.5 \cdot 10^{-153}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-161}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+128}:\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{b \cdot c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - d \cdot a}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ t_1 := \frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{if}\;d \leq -1.15 \cdot 10^{+87}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -7.5 \cdot 10^{-153}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.18 \cdot 10^{-148}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{+96}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* d a)) (fma c c (* d d))))
        (t_1 (/ (fma c (/ b d) (- a)) d)))
   (if (<= d -1.15e+87)
     t_1
     (if (<= d -7.5e-153)
       t_0
       (if (<= d 1.18e-148)
         (/ (- b (* a (/ d c))) c)
         (if (<= d 1.6e+96) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (d * a)) / fma(c, c, (d * d));
	double t_1 = fma(c, (b / d), -a) / d;
	double tmp;
	if (d <= -1.15e+87) {
		tmp = t_1;
	} else if (d <= -7.5e-153) {
		tmp = t_0;
	} else if (d <= 1.18e-148) {
		tmp = (b - (a * (d / c))) / c;
	} else if (d <= 1.6e+96) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(d * a)) / fma(c, c, Float64(d * d)))
	t_1 = Float64(fma(c, Float64(b / d), Float64(-a)) / d)
	tmp = 0.0
	if (d <= -1.15e+87)
		tmp = t_1;
	elseif (d <= -7.5e-153)
		tmp = t_0;
	elseif (d <= 1.18e-148)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (d <= 1.6e+96)
		tmp = t_0;
	else
		tmp = t_1;
	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[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -1.15e+87], t$95$1, If[LessEqual[d, -7.5e-153], t$95$0, If[LessEqual[d, 1.18e-148], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.6e+96], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.1500000000000001e87 or 1.60000000000000003e96 < d

   1. Initial program 39.1%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 82.9

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

   5. Applied rewrites 82.9%

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

   if -1.1500000000000001e87 < d < -7.5e-153 or 1.18e-148 < d < 1.60000000000000003e96

   1. Initial program 83.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 83.4

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

   4. Applied rewrites 83.4%

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

   if -7.5e-153 < d < 1.18e-148

   1. Initial program 76.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 95.9

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

   5. Applied rewrites 95.9%

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

   7. Applied rewrites 98.5%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.15 \cdot 10^{+87}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq -7.5 \cdot 10^{-153}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq 1.18 \cdot 10^{-148}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{+96}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot c - d \cdot a\\ \mathbf{if}\;c \leq -1.12 \cdot 10^{+74}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -5 \cdot 10^{-102}:\\ \;\;\;\;\frac{b \cdot c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;c \leq 6 \cdot 10^{-118}:\\ \;\;\;\;\frac{t\_0}{d \cdot d}\\ \mathbf{elif}\;c \leq 4.8 \cdot 10^{+142}:\\ \;\;\;\;\frac{t\_0}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* b c) (* d a))))
   (if (<= c -1.12e+74)
     (/ b c)
     (if (<= c -5e-102)
       (/ (* b c) (fma d d (* c c)))
       (if (<= c 6e-118)
         (/ t_0 (* d d))
         (if (<= c 4.8e+142) (/ t_0 (* c c)) (/ b c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b * c) - (d * a);
	double tmp;
	if (c <= -1.12e+74) {
		tmp = b / c;
	} else if (c <= -5e-102) {
		tmp = (b * c) / fma(d, d, (c * c));
	} else if (c <= 6e-118) {
		tmp = t_0 / (d * d);
	} else if (c <= 4.8e+142) {
		tmp = t_0 / (c * c);
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b * c) - Float64(d * a))
	tmp = 0.0
	if (c <= -1.12e+74)
		tmp = Float64(b / c);
	elseif (c <= -5e-102)
		tmp = Float64(Float64(b * c) / fma(d, d, Float64(c * c)));
	elseif (c <= 6e-118)
		tmp = Float64(t_0 / Float64(d * d));
	elseif (c <= 4.8e+142)
		tmp = Float64(t_0 / Float64(c * c));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.12e+74], N[(b / c), $MachinePrecision], If[LessEqual[c, -5e-102], N[(N[(b * c), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 6e-118], N[(t$95$0 / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 4.8e+142], N[(t$95$0 / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(b / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.12000000000000003e74 or 4.7999999999999998e142 < c

   1. Initial program 40.9%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b}{c}} \]
   4. Step-by-step derivation

      1. lower-/.f64 75.3

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

   5. Applied rewrites 75.3%

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

   if -1.12000000000000003e74 < c < -5.00000000000000026e-102

   1. Initial program 74.2%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \frac{c \cdot b}{\color{blue}{\mathsf{fma}\left(d, d, {c}^{2}\right)}} \]

      7. unpow2 N/A

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

      8. lower-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

   if -5.00000000000000026e-102 < c < 6.00000000000000035e-118

   1. Initial program 82.3%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 79.0

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

   5. Applied rewrites 79.0%

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

   if 6.00000000000000035e-118 < c < 4.7999999999999998e142

   1. Initial program 78.2%

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

   3. Taylor expanded in c around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 54.5

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

   5. Applied rewrites 54.5%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.12 \cdot 10^{+74}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -5 \cdot 10^{-102}:\\ \;\;\;\;\frac{b \cdot c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;c \leq 6 \cdot 10^{-118}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d}\\ \mathbf{elif}\;c \leq 4.8 \cdot 10^{+142}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -1.35 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{-90}:\\ \;\;\;\;\frac{b \cdot c}{t\_0}\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{-88}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{+137}:\\ \;\;\;\;\left(-a\right) \cdot \frac{d}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))) (t_1 (/ (- a) d)))
   (if (<= d -1.35e+23)
     t_1
     (if (<= d -1.35e-90)
       (/ (* b c) t_0)
       (if (<= d 6.8e-88)
         (/ (- (* b c) (* d a)) (* c c))
         (if (<= d 3.6e+137) (* (- a) (/ d t_0)) t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double t_1 = -a / d;
	double tmp;
	if (d <= -1.35e+23) {
		tmp = t_1;
	} else if (d <= -1.35e-90) {
		tmp = (b * c) / t_0;
	} else if (d <= 6.8e-88) {
		tmp = ((b * c) - (d * a)) / (c * c);
	} else if (d <= 3.6e+137) {
		tmp = -a * (d / t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (d <= -1.35e+23)
		tmp = t_1;
	elseif (d <= -1.35e-90)
		tmp = Float64(Float64(b * c) / t_0);
	elseif (d <= 6.8e-88)
		tmp = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(c * c));
	elseif (d <= 3.6e+137)
		tmp = Float64(Float64(-a) * Float64(d / t_0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-a) / d), $MachinePrecision]}, If[LessEqual[d, -1.35e+23], t$95$1, If[LessEqual[d, -1.35e-90], N[(N[(b * c), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[d, 6.8e-88], N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.6e+137], N[((-a) * N[(d / t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.3499999999999999e23 or 3.6e137 < d

   1. Initial program 37.0%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{a}{d}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{a}{\mathsf{neg}\left(d\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{a}{\color{blue}{\mathsf{neg}\left(d\right)}} \]

      6. lower-neg.f64 71.2

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

   5. Applied rewrites 71.2%

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

   if -1.3499999999999999e23 < d < -1.34999999999999998e-90

   1. Initial program 89.0%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \frac{c \cdot b}{\color{blue}{\mathsf{fma}\left(d, d, {c}^{2}\right)}} \]

      7. unpow2 N/A

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

      8. lower-*.f64 64.0

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

   5. Applied rewrites 64.0%

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

   if -1.34999999999999998e-90 < d < 6.79999999999999949e-88

   1. Initial program 80.1%

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

   3. Taylor expanded in c around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 73.5

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

   5. Applied rewrites 73.5%

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

   if 6.79999999999999949e-88 < d < 3.6e137

   1. Initial program 73.1%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \frac{d \cdot a}{\mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(d, d, {c}^{2}\right)}\right)} \]

      10. unpow2 N/A

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

      11. lower-*.f64 55.8

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

   5. Applied rewrites 55.8%

      \[\leadsto \color{blue}{\frac{d \cdot a}{-\mathsf{fma}\left(d, d, c \cdot c\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 59.3%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.35 \cdot 10^{+23}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{-90}:\\ \;\;\;\;\frac{b \cdot c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{-88}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{+137}:\\ \;\;\;\;\left(-a\right) \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -1.35 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -5.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{b \cdot c}{t\_0}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-157}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{+137}:\\ \;\;\;\;\left(-a\right) \cdot \frac{d}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))) (t_1 (/ (- a) d)))
   (if (<= d -1.35e+23)
     t_1
     (if (<= d -5.5e-106)
       (/ (* b c) t_0)
       (if (<= d 1.45e-157)
         (/ b c)
         (if (<= d 3.6e+137) (* (- a) (/ d t_0)) t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double t_1 = -a / d;
	double tmp;
	if (d <= -1.35e+23) {
		tmp = t_1;
	} else if (d <= -5.5e-106) {
		tmp = (b * c) / t_0;
	} else if (d <= 1.45e-157) {
		tmp = b / c;
	} else if (d <= 3.6e+137) {
		tmp = -a * (d / t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (d <= -1.35e+23)
		tmp = t_1;
	elseif (d <= -5.5e-106)
		tmp = Float64(Float64(b * c) / t_0);
	elseif (d <= 1.45e-157)
		tmp = Float64(b / c);
	elseif (d <= 3.6e+137)
		tmp = Float64(Float64(-a) * Float64(d / t_0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-a) / d), $MachinePrecision]}, If[LessEqual[d, -1.35e+23], t$95$1, If[LessEqual[d, -5.5e-106], N[(N[(b * c), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[d, 1.45e-157], N[(b / c), $MachinePrecision], If[LessEqual[d, 3.6e+137], N[((-a) * N[(d / t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.3499999999999999e23 or 3.6e137 < d

   1. Initial program 37.0%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{a}{d}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{a}{\mathsf{neg}\left(d\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{a}{\color{blue}{\mathsf{neg}\left(d\right)}} \]

      6. lower-neg.f64 71.2

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

   5. Applied rewrites 71.2%

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

   if -1.3499999999999999e23 < d < -5.5000000000000001e-106

   1. Initial program 90.1%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \frac{c \cdot b}{\color{blue}{\mathsf{fma}\left(d, d, {c}^{2}\right)}} \]

      7. unpow2 N/A

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

      8. lower-*.f64 64.4

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

   5. Applied rewrites 64.4%

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

   if -5.5000000000000001e-106 < d < 1.44999999999999994e-157

   1. Initial program 77.7%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b}{c}} \]
   4. Step-by-step derivation

      1. lower-/.f64 72.2

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

   5. Applied rewrites 72.2%

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

   if 1.44999999999999994e-157 < d < 3.6e137

   1. Initial program 76.6%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \frac{d \cdot a}{\mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(d, d, {c}^{2}\right)}\right)} \]

      10. unpow2 N/A

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

      11. lower-*.f64 56.8

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

   5. Applied rewrites 56.8%

      \[\leadsto \color{blue}{\frac{d \cdot a}{-\mathsf{fma}\left(d, d, c \cdot c\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 59.4%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.35 \cdot 10^{+23}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -5.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{b \cdot c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-157}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{+137}:\\ \;\;\;\;\left(-a\right) \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{if}\;c \leq -1.12 \cdot 10^{+74}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -6.5 \cdot 10^{-153}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{-131}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 5 \cdot 10^{+38}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (* b c) (fma d d (* c c)))))
   (if (<= c -1.12e+74)
     (/ b c)
     (if (<= c -6.5e-153)
       t_0
       (if (<= c 1.65e-131) (/ (- a) d) (if (<= c 5e+38) t_0 (/ b c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b * c) / fma(d, d, (c * c));
	double tmp;
	if (c <= -1.12e+74) {
		tmp = b / c;
	} else if (c <= -6.5e-153) {
		tmp = t_0;
	} else if (c <= 1.65e-131) {
		tmp = -a / d;
	} else if (c <= 5e+38) {
		tmp = t_0;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b * c) / fma(d, d, Float64(c * c)))
	tmp = 0.0
	if (c <= -1.12e+74)
		tmp = Float64(b / c);
	elseif (c <= -6.5e-153)
		tmp = t_0;
	elseif (c <= 1.65e-131)
		tmp = Float64(Float64(-a) / d);
	elseif (c <= 5e+38)
		tmp = t_0;
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * c), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.12e+74], N[(b / c), $MachinePrecision], If[LessEqual[c, -6.5e-153], t$95$0, If[LessEqual[c, 1.65e-131], N[((-a) / d), $MachinePrecision], If[LessEqual[c, 5e+38], t$95$0, N[(b / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.12000000000000003e74 or 4.9999999999999997e38 < c

   1. Initial program 47.2%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b}{c}} \]
   4. Step-by-step derivation

      1. lower-/.f64 70.1

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

   5. Applied rewrites 70.1%

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

   if -1.12000000000000003e74 < c < -6.50000000000000032e-153 or 1.6500000000000001e-131 < c < 4.9999999999999997e38

   1. Initial program 80.9%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \frac{c \cdot b}{\color{blue}{\mathsf{fma}\left(d, d, {c}^{2}\right)}} \]

      7. unpow2 N/A

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

      8. lower-*.f64 60.9

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

   5. Applied rewrites 60.9%

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

   if -6.50000000000000032e-153 < c < 1.6500000000000001e-131

   1. Initial program 79.7%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{a}{d}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{a}{\mathsf{neg}\left(d\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{a}{\color{blue}{\mathsf{neg}\left(d\right)}} \]

      6. lower-neg.f64 70.8

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

   5. Applied rewrites 70.8%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.12 \cdot 10^{+74}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -6.5 \cdot 10^{-153}:\\ \;\;\;\;\frac{b \cdot c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{-131}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 5 \cdot 10^{+38}:\\ \;\;\;\;\frac{b \cdot c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -1.18 \cdot 10^{+86}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -1.42 \cdot 10^{-90}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{+32}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -1.18e+86)
     t_0
     (if (<= d -1.42e-90)
       (/ (- (* b c) (* d a)) (* d d))
       (if (<= d 2.8e+32) (/ (- b (* a (/ d c))) c) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -1.18e+86) {
		tmp = t_0;
	} else if (d <= -1.42e-90) {
		tmp = ((b * c) - (d * a)) / (d * d);
	} else if (d <= 2.8e+32) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = t_0;
	}
	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 = -a / d
    if (d <= (-1.18d+86)) then
        tmp = t_0
    else if (d <= (-1.42d-90)) then
        tmp = ((b * c) - (d * a)) / (d * d)
    else if (d <= 2.8d+32) then
        tmp = (b - (a * (d / c))) / c
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -1.18e+86) {
		tmp = t_0;
	} else if (d <= -1.42e-90) {
		tmp = ((b * c) - (d * a)) / (d * d);
	} else if (d <= 2.8e+32) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = -a / d
	tmp = 0
	if d <= -1.18e+86:
		tmp = t_0
	elif d <= -1.42e-90:
		tmp = ((b * c) - (d * a)) / (d * d)
	elif d <= 2.8e+32:
		tmp = (b - (a * (d / c))) / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (d <= -1.18e+86)
		tmp = t_0;
	elseif (d <= -1.42e-90)
		tmp = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(d * d));
	elseif (d <= 2.8e+32)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = -a / d;
	tmp = 0.0;
	if (d <= -1.18e+86)
		tmp = t_0;
	elseif (d <= -1.42e-90)
		tmp = ((b * c) - (d * a)) / (d * d);
	elseif (d <= 2.8e+32)
		tmp = (b - (a * (d / c))) / c;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, If[LessEqual[d, -1.18e+86], t$95$0, If[LessEqual[d, -1.42e-90], N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.8e+32], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.18e86 or 2.8e32 < d

   1. Initial program 46.2%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{a}{d}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{a}{\mathsf{neg}\left(d\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{a}{\color{blue}{\mathsf{neg}\left(d\right)}} \]

      6. lower-neg.f64 69.6

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

   5. Applied rewrites 69.6%

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

   if -1.18e86 < d < -1.41999999999999998e-90

   1. Initial program 83.1%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.7

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

   5. Applied rewrites 61.7%

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

   if -1.41999999999999998e-90 < d < 2.8e32

   1. Initial program 78.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 84.0

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

   5. Applied rewrites 84.0%

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

   7. Applied rewrites 85.4%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.18 \cdot 10^{+86}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -1.42 \cdot 10^{-90}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{+32}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{if}\;d \leq -7.8 \cdot 10^{-88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{+53}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma c (/ b d) (- a)) d)))
   (if (<= d -7.8e-88)
     t_0
     (if (<= d 1.35e+53) (/ (- b (* a (/ d c))) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(c, (b / d), -a) / d;
	double tmp;
	if (d <= -7.8e-88) {
		tmp = t_0;
	} else if (d <= 1.35e+53) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(c, Float64(b / d), Float64(-a)) / d)
	tmp = 0.0
	if (d <= -7.8e-88)
		tmp = t_0;
	elseif (d <= 1.35e+53)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -7.8e-88], t$95$0, If[LessEqual[d, 1.35e+53], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -7.79999999999999985e-88 or 1.3500000000000001e53 < d

   1. Initial program 54.3%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 77.1

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

   5. Applied rewrites 77.1%

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

   if -7.79999999999999985e-88 < d < 1.3500000000000001e53

   1. Initial program 78.9%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

   7. Applied rewrites 83.3%

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

Alternative 9: 77.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{if}\;c \leq -2.2 \cdot 10^{+39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 2.1 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (* a (/ d c))) c)))
   (if (<= c -2.2e+39) t_0 (if (<= c 2.1e+36) (/ (- (/ (* b c) d) a) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - (a * (d / c))) / c;
	double tmp;
	if (c <= -2.2e+39) {
		tmp = t_0;
	} else if (c <= 2.1e+36) {
		tmp = (((b * c) / d) - a) / d;
	} else {
		tmp = t_0;
	}
	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 - (a * (d / c))) / c
    if (c <= (-2.2d+39)) then
        tmp = t_0
    else if (c <= 2.1d+36) then
        tmp = (((b * c) / d) - a) / d
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b - (a * (d / c))) / c;
	double tmp;
	if (c <= -2.2e+39) {
		tmp = t_0;
	} else if (c <= 2.1e+36) {
		tmp = (((b * c) / d) - a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b - (a * (d / c))) / c
	tmp = 0
	if c <= -2.2e+39:
		tmp = t_0
	elif c <= 2.1e+36:
		tmp = (((b * c) / d) - a) / d
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(a * Float64(d / c))) / c)
	tmp = 0.0
	if (c <= -2.2e+39)
		tmp = t_0;
	elseif (c <= 2.1e+36)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b - (a * (d / c))) / c;
	tmp = 0.0;
	if (c <= -2.2e+39)
		tmp = t_0;
	elseif (c <= 2.1e+36)
		tmp = (((b * c) / d) - a) / d;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -2.2e+39], t$95$0, If[LessEqual[c, 2.1e+36], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -2.2000000000000001e39 or 2.10000000000000004e36 < c

   1. Initial program 49.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 78.7

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

   5. Applied rewrites 78.7%

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

   7. Applied rewrites 83.2%

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

   if -2.2000000000000001e39 < c < 2.10000000000000004e36

   1. Initial program 79.9%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + c \cdot \left(\frac{b}{{d}^{2}} + \frac{a \cdot c}{{d}^{3}}\right)} \]

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 76.5%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.2 \cdot 10^{+39}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 2.1 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1850000000000:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 3.9 \cdot 10^{+29}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1850000000000.0) (/ b c) (if (<= c 3.9e+29) (/ (- a) d) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1850000000000.0) {
		tmp = b / c;
	} else if (c <= 3.9e+29) {
		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 (c <= (-1850000000000.0d0)) then
        tmp = b / c
    else if (c <= 3.9d+29) 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 (c <= -1850000000000.0) {
		tmp = b / c;
	} else if (c <= 3.9e+29) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1850000000000.0:
		tmp = b / c
	elif c <= 3.9e+29:
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1850000000000.0)
		tmp = Float64(b / c);
	elseif (c <= 3.9e+29)
		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 (c <= -1850000000000.0)
		tmp = b / c;
	elseif (c <= 3.9e+29)
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1850000000000.0], N[(b / c), $MachinePrecision], If[LessEqual[c, 3.9e+29], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -1.85e12 or 3.89999999999999968e29 < c

   1. Initial program 51.0%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b}{c}} \]
   4. Step-by-step derivation

      1. lower-/.f64 68.2

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

   5. Applied rewrites 68.2%

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

   if -1.85e12 < c < 3.89999999999999968e29

   1. Initial program 80.1%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{a}{d}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{a}{\mathsf{neg}\left(d\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{a}{\color{blue}{\mathsf{neg}\left(d\right)}} \]

      6. lower-neg.f64 58.4

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

   5. Applied rewrites 58.4%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1850000000000:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 3.9 \cdot 10^{+29}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.7% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.9%

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

3. Taylor expanded in c around inf

   \[\leadsto \color{blue}{\frac{b}{c}} \]
4. Step-by-step derivation

   1. lower-/.f64 43.1

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

5. Applied rewrites 43.1%

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

Developer Target 1: 99.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (Math.abs(d) < Math.abs(c)) {
		tmp = (b - (a * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (-a + (b * (c / d))) / (d + (c * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if math.fabs(d) < math.fabs(c):
		tmp = (b - (a * (d / c))) / (c + (d * (d / c)))
	else:
		tmp = (-a + (b * (c / d))) / (d + (c * (c / d)))
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024238 
(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))))