Complex division, imag part

Percentage Accurate: 60.8% → 78.9%

Time: 6.5s

Alternatives: 7

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: 60.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 78.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\frac{a}{{c}^{4}} \cdot d - \frac{b}{{c}^{3}}, d, \frac{\frac{-a}{c}}{c}\right), d, \frac{b}{c}\right)\\ \mathbf{if}\;c \leq -9 \cdot 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 2.05 \cdot 10^{-79}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \mathbf{elif}\;c \leq 9 \cdot 10^{+153}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{b}, -a, c\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0
         (fma
          (fma
           (- (* (/ a (pow c 4.0)) d) (/ b (pow c 3.0)))
           d
           (/ (/ (- a) c) c))
          d
          (/ b c))))
   (if (<= c -9e-10)
     t_0
     (if (<= c 2.05e-79)
       (/ (fma (/ c d) b (- a)) d)
       (if (<= c 9e+153)
         (* (/ (fma (/ d b) (- a) c) (fma d d (* c c))) b)
         t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(fma((((a / pow(c, 4.0)) * d) - (b / pow(c, 3.0))), d, ((-a / c) / c)), d, (b / c));
	double tmp;
	if (c <= -9e-10) {
		tmp = t_0;
	} else if (c <= 2.05e-79) {
		tmp = fma((c / d), b, -a) / d;
	} else if (c <= 9e+153) {
		tmp = (fma((d / b), -a, c) / fma(d, d, (c * c))) * b;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(fma(Float64(Float64(Float64(a / (c ^ 4.0)) * d) - Float64(b / (c ^ 3.0))), d, Float64(Float64(Float64(-a) / c) / c)), d, Float64(b / c))
	tmp = 0.0
	if (c <= -9e-10)
		tmp = t_0;
	elseif (c <= 2.05e-79)
		tmp = Float64(fma(Float64(c / d), b, Float64(-a)) / d);
	elseif (c <= 9e+153)
		tmp = Float64(Float64(fma(Float64(d / b), Float64(-a), c) / fma(d, d, Float64(c * c))) * b);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(N[(N[(a / N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] - N[(b / N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d + N[(N[((-a) / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * d + N[(b / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -9e-10], t$95$0, If[LessEqual[c, 2.05e-79], N[(N[(N[(c / d), $MachinePrecision] * b + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 9e+153], N[(N[(N[(N[(d / b), $MachinePrecision] * (-a) + c), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if c < -8.9999999999999999e-10 or 9.0000000000000002e153 < c

   1. Initial program 39.6%

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

   3. Taylor expanded in d around 0

      \[\leadsto \color{blue}{d \cdot \left(-1 \cdot \frac{a}{{c}^{2}} + d \cdot \left(\frac{a \cdot d}{{c}^{4}} - \frac{b}{{c}^{3}}\right)\right) + \frac{b}{c}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{a}{{c}^{2}} + d \cdot \left(\frac{a \cdot d}{{c}^{4}} - \frac{b}{{c}^{3}}\right)\right) \cdot d} + \frac{b}{c} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{a}{{c}^{2}} + d \cdot \left(\frac{a \cdot d}{{c}^{4}} - \frac{b}{{c}^{3}}\right), d, \frac{b}{c}\right)} \]

   5. Applied rewrites 83.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{a}{{c}^{4}} \cdot d - \frac{b}{{c}^{3}}, d, \frac{\frac{-a}{c}}{c}\right), d, \frac{b}{c}\right)} \]

   if -8.9999999999999999e-10 < c < 2.04999999999999997e-79

   1. Initial program 70.5%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. mul-1-neg N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

   5. Applied rewrites 60.2%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 88.1%

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

   if 2.04999999999999997e-79 < c < 9.0000000000000002e153

   1. Initial program 68.8%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. mul-1-neg N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

   5. Applied rewrites 80.1%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -9 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{a}{{c}^{4}} \cdot d - \frac{b}{{c}^{3}}, d, \frac{\frac{-a}{c}}{c}\right), d, \frac{b}{c}\right)\\ \mathbf{elif}\;c \leq 2.05 \cdot 10^{-79}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \mathbf{elif}\;c \leq 9 \cdot 10^{+153}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{b}, -a, c\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{a}{{c}^{4}} \cdot d - \frac{b}{{c}^{3}}, d, \frac{\frac{-a}{c}}{c}\right), d, \frac{b}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -9 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-d}{c}, a, b\right)}{c}\\ \mathbf{elif}\;c \leq 2.05 \cdot 10^{-79}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \mathbf{elif}\;c \leq 9 \cdot 10^{+153}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{b}, -a, c\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d}{c} \cdot a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -9e-10)
   (/ (fma (/ (- d) c) a b) c)
   (if (<= c 2.05e-79)
     (/ (fma (/ c d) b (- a)) d)
     (if (<= c 9e+153)
       (* (/ (fma (/ d b) (- a) c) (fma d d (* c c))) b)
       (/ (- b (* (/ d c) a)) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -9e-10) {
		tmp = fma((-d / c), a, b) / c;
	} else if (c <= 2.05e-79) {
		tmp = fma((c / d), b, -a) / d;
	} else if (c <= 9e+153) {
		tmp = (fma((d / b), -a, c) / fma(d, d, (c * c))) * b;
	} else {
		tmp = (b - ((d / c) * a)) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -9e-10)
		tmp = Float64(fma(Float64(Float64(-d) / c), a, b) / c);
	elseif (c <= 2.05e-79)
		tmp = Float64(fma(Float64(c / d), b, Float64(-a)) / d);
	elseif (c <= 9e+153)
		tmp = Float64(Float64(fma(Float64(d / b), Float64(-a), c) / fma(d, d, Float64(c * c))) * b);
	else
		tmp = Float64(Float64(b - Float64(Float64(d / c) * a)) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -9e-10], N[(N[(N[((-d) / c), $MachinePrecision] * a + b), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 2.05e-79], N[(N[(N[(c / d), $MachinePrecision] * b + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 9e+153], N[(N[(N[(N[(d / b), $MachinePrecision] * (-a) + c), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[(N[(b - N[(N[(d / c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -8.9999999999999999e-10

   1. Initial program 52.9%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 79.9

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

   5. Applied rewrites 79.9%

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

   7. Applied rewrites 80.1%

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

   9. Applied rewrites 80.1%

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

   if -8.9999999999999999e-10 < c < 2.04999999999999997e-79

   1. Initial program 70.5%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. mul-1-neg N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

   5. Applied rewrites 60.2%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 88.1%

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

   if 2.04999999999999997e-79 < c < 9.0000000000000002e153

   1. Initial program 68.8%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. mul-1-neg N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

   5. Applied rewrites 80.1%

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

   if 9.0000000000000002e153 < c

   1. Initial program 23.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 77.5

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

   5. Applied rewrites 77.5%

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

   7. Applied rewrites 87.9%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -9 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-d}{c}, a, b\right)}{c}\\ \mathbf{elif}\;c \leq 2.05 \cdot 10^{-79}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \mathbf{elif}\;c \leq 9 \cdot 10^{+153}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{b}, -a, c\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d}{c} \cdot a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -9 \cdot 10^{-10} \lor \neg \left(c \leq 9.2 \cdot 10^{+23}\right):\\ \;\;\;\;\frac{b - \frac{d}{c} \cdot a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -9e-10) (not (<= c 9.2e+23)))
   (/ (- b (* (/ d c) a)) c)
   (/ (fma (/ c d) b (- a)) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -9e-10) || !(c <= 9.2e+23)) {
		tmp = (b - ((d / c) * a)) / c;
	} else {
		tmp = fma((c / d), b, -a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -9e-10) || !(c <= 9.2e+23))
		tmp = Float64(Float64(b - Float64(Float64(d / c) * a)) / c);
	else
		tmp = Float64(fma(Float64(c / d), b, Float64(-a)) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -9e-10], N[Not[LessEqual[c, 9.2e+23]], $MachinePrecision]], N[(N[(b - N[(N[(d / c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(c / d), $MachinePrecision] * b + (-a)), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -8.9999999999999999e-10 or 9.2000000000000002e23 < c

   1. Initial program 45.2%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 77.9

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

   5. Applied rewrites 77.9%

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

   7. Applied rewrites 81.7%

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

   if -8.9999999999999999e-10 < c < 9.2000000000000002e23

   1. Initial program 71.3%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. mul-1-neg N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

   5. Applied rewrites 61.9%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 84.6%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{\color{blue}{d}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -9 \cdot 10^{-10} \lor \neg \left(c \leq 9.2 \cdot 10^{+23}\right):\\ \;\;\;\;\frac{b - \frac{d}{c} \cdot a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -9 \cdot 10^{-10} \lor \neg \left(c \leq 5.2 \cdot 10^{+29}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -9e-10) (not (<= c 5.2e+29)))
   (/ b c)
   (/ (fma (/ c d) b (- a)) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -9e-10) || !(c <= 5.2e+29)) {
		tmp = b / c;
	} else {
		tmp = fma((c / d), b, -a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -9e-10) || !(c <= 5.2e+29))
		tmp = Float64(b / c);
	else
		tmp = Float64(fma(Float64(c / d), b, Float64(-a)) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -9e-10], N[Not[LessEqual[c, 5.2e+29]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[(N[(N[(c / d), $MachinePrecision] * b + (-a)), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -8.9999999999999999e-10 or 5.2e29 < c

   1. Initial program 45.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 69.4

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

   5. Applied rewrites 69.4%

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

   if -8.9999999999999999e-10 < c < 5.2e29

   1. Initial program 71.3%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. mul-1-neg N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

   5. Applied rewrites 61.9%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 84.6%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{\color{blue}{d}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -9 \cdot 10^{-10} \lor \neg \left(c \leq 5.2 \cdot 10^{+29}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -9 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-d}{c}, a, b\right)}{c}\\ \mathbf{elif}\;c \leq 9.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d}{c} \cdot a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -9e-10)
   (/ (fma (/ (- d) c) a b) c)
   (if (<= c 9.2e+23) (/ (fma (/ c d) b (- a)) d) (/ (- b (* (/ d c) a)) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -9e-10) {
		tmp = fma((-d / c), a, b) / c;
	} else if (c <= 9.2e+23) {
		tmp = fma((c / d), b, -a) / d;
	} else {
		tmp = (b - ((d / c) * a)) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -9e-10)
		tmp = Float64(fma(Float64(Float64(-d) / c), a, b) / c);
	elseif (c <= 9.2e+23)
		tmp = Float64(fma(Float64(c / d), b, Float64(-a)) / d);
	else
		tmp = Float64(Float64(b - Float64(Float64(d / c) * a)) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -9e-10], N[(N[(N[((-d) / c), $MachinePrecision] * a + b), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 9.2e+23], N[(N[(N[(c / d), $MachinePrecision] * b + (-a)), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(N[(d / c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -8.9999999999999999e-10

   1. Initial program 52.9%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 79.9

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

   5. Applied rewrites 79.9%

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

   7. Applied rewrites 80.1%

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

   9. Applied rewrites 80.1%

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

   if -8.9999999999999999e-10 < c < 9.2000000000000002e23

   1. Initial program 71.3%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. mul-1-neg N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

   5. Applied rewrites 61.9%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 84.6%

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

   if 9.2000000000000002e23 < c

   1. Initial program 39.6%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

   7. Applied rewrites 82.8%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -9 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-d}{c}, a, b\right)}{c}\\ \mathbf{elif}\;c \leq 9.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d}{c} \cdot a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -6.2 \cdot 10^{+83} \lor \neg \left(d \leq 2.2 \cdot 10^{+23}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -6.2e+83) (not (<= d 2.2e+23))) (/ (- a) d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -6.2e+83) || !(d <= 2.2e+23)) {
		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 <= (-6.2d+83)) .or. (.not. (d <= 2.2d+23))) 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 <= -6.2e+83) || !(d <= 2.2e+23)) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -6.2e+83) or not (d <= 2.2e+23):
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -6.2e+83) || !(d <= 2.2e+23))
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -6.2e+83) || ~((d <= 2.2e+23)))
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -6.2e+83], N[Not[LessEqual[d, 2.2e+23]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -6.19999999999999984e83 or 2.20000000000000008e23 < d

   1. Initial program 45.5%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 72.7

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

   5. Applied rewrites 72.7%

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

   if -6.19999999999999984e83 < d < 2.20000000000000008e23

   1. Initial program 65.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 65.4

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

   5. Applied rewrites 65.4%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.2 \cdot 10^{+83} \lor \neg \left(d \leq 2.2 \cdot 10^{+23}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.5% 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 57.8%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 47.1

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

5. Applied rewrites 47.1%

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

Developer Target 1: 99.4% 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 2024344 
(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))))