Complex division, imag part

Percentage Accurate: 61.2% → 79.5%

Time: 6.8s

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: 61.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\mathsf{fma}\left(d, {a}^{-1}, c \cdot \frac{c}{\mathsf{fma}\left(b, -c, a \cdot d\right)}\right)}\\ \mathbf{if}\;c \leq -1.7 \cdot 10^{+138}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -6.5 \cdot 10^{+28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -1.05 \cdot 10^{-145}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2.2 \cdot 10^{-77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{elif}\;c \leq 8 \cdot 10^{+136}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ -1.0 (fma d (pow a -1.0) (* c (/ c (fma b (- c) (* a d))))))))
   (if (<= c -1.7e+138)
     (/ b c)
     (if (<= c -6.5e+28)
       t_0
       (if (<= c -1.05e-145)
         (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))
         (if (<= c 2.2e-77)
           (/ (fma b (/ c d) (- a)) d)
           (if (<= c 8e+136) t_0 (/ (- b (/ (* a d) c)) c))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -1.0 / fma(d, pow(a, -1.0), (c * (c / fma(b, -c, (a * d)))));
	double tmp;
	if (c <= -1.7e+138) {
		tmp = b / c;
	} else if (c <= -6.5e+28) {
		tmp = t_0;
	} else if (c <= -1.05e-145) {
		tmp = ((b * c) - (a * d)) / ((c * c) + (d * d));
	} else if (c <= 2.2e-77) {
		tmp = fma(b, (c / d), -a) / d;
	} else if (c <= 8e+136) {
		tmp = t_0;
	} else {
		tmp = (b - ((a * d) / c)) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(-1.0 / fma(d, (a ^ -1.0), Float64(c * Float64(c / fma(b, Float64(-c), Float64(a * d))))))
	tmp = 0.0
	if (c <= -1.7e+138)
		tmp = Float64(b / c);
	elseif (c <= -6.5e+28)
		tmp = t_0;
	elseif (c <= -1.05e-145)
		tmp = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 2.2e-77)
		tmp = Float64(fma(b, Float64(c / d), Float64(-a)) / d);
	elseif (c <= 8e+136)
		tmp = t_0;
	else
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(-1.0 / N[(d * N[Power[a, -1.0], $MachinePrecision] + N[(c * N[(c / N[(b * (-c) + N[(a * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.7e+138], N[(b / c), $MachinePrecision], If[LessEqual[c, -6.5e+28], t$95$0, If[LessEqual[c, -1.05e-145], N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.2e-77], N[(N[(b * N[(c / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 8e+136], t$95$0, N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -1.70000000000000006e138

   1. Initial program 43.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 87.5

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

   5. Applied rewrites 87.5%

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

   if -1.70000000000000006e138 < c < -6.5000000000000001e28 or 2.20000000000000007e-77 < c < 8.00000000000000047e136

   1. Initial program 64.9%

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

   4. Applied rewrites 64.7%

      \[\leadsto \color{blue}{\frac{-1}{\frac{\mathsf{fma}\left(d, d, c \cdot c\right)}{\mathsf{fma}\left(-c, b, a \cdot d\right)}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 72.4

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

      16. lift-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 72.4

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

   6. Applied rewrites 72.4%

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

   7. Taylor expanded in a around inf

      \[\leadsto \frac{-1}{\mathsf{fma}\left(d, \color{blue}{\frac{1}{a}}, c \cdot \frac{c}{\mathsf{fma}\left(b, -c, a \cdot d\right)}\right)} \]
   8. Step-by-step derivation

      1. lower-/.f64 88.0

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

   9. Applied rewrites 88.0%

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

   if -6.5000000000000001e28 < c < -1.04999999999999996e-145

   1. Initial program 90.4%

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

   if -1.04999999999999996e-145 < c < 2.20000000000000007e-77

   1. Initial program 69.3%

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

   4. Applied rewrites 69.3%

      \[\leadsto \color{blue}{\frac{-1}{\frac{\mathsf{fma}\left(d, d, c \cdot c\right)}{\mathsf{fma}\left(-c, b, a \cdot d\right)}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 83.7

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

      16. lift-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 83.7

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

   6. Applied rewrites 83.7%

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

   7. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 91.0

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

   9. Applied rewrites 91.0%

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

   if 8.00000000000000047e136 < c

   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 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. lower-*.f64 86.1

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

   5. Applied rewrites 86.1%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.7 \cdot 10^{+138}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -6.5 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(d, {a}^{-1}, c \cdot \frac{c}{\mathsf{fma}\left(b, -c, a \cdot d\right)}\right)}\\ \mathbf{elif}\;c \leq -1.05 \cdot 10^{-145}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2.2 \cdot 10^{-77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{elif}\;c \leq 8 \cdot 10^{+136}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(d, {a}^{-1}, c \cdot \frac{c}{\mathsf{fma}\left(b, -c, a \cdot d\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, -c, a \cdot d\right)\\ \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq \infty:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(d, \frac{d}{t\_0}, c \cdot \frac{c}{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma b (- c) (* a d))))
   (if (<= (/ (- (* b c) (* a d)) (+ (* c c) (* d d))) INFINITY)
     (/ -1.0 (fma d (/ d t_0) (* c (/ c t_0))))
     (/ (fma b (/ c d) (- a)) d))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(b, -c, (a * d));
	double tmp;
	if ((((b * c) - (a * d)) / ((c * c) + (d * d))) <= ((double) INFINITY)) {
		tmp = -1.0 / fma(d, (d / t_0), (c * (c / t_0)));
	} else {
		tmp = fma(b, (c / d), -a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(b, Float64(-c), Float64(a * d))
	tmp = 0.0
	if (Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d))) <= Inf)
		tmp = Float64(-1.0 / fma(d, Float64(d / t_0), Float64(c * Float64(c / t_0))));
	else
		tmp = Float64(fma(b, Float64(c / d), Float64(-a)) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(b * (-c) + N[(a * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(-1.0 / N[(d * N[(d / t$95$0), $MachinePrecision] + N[(c * N[(c / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * N[(c / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < +inf.0

   1. Initial program 78.2%

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

   4. Applied rewrites 78.0%

      \[\leadsto \color{blue}{\frac{-1}{\frac{\mathsf{fma}\left(d, d, c \cdot c\right)}{\mathsf{fma}\left(-c, b, a \cdot d\right)}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 94.3

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

      16. lift-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 94.3

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

   6. Applied rewrites 94.3%

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

   if +inf.0 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 0.0%

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

   4. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{-1}{\frac{\mathsf{fma}\left(d, d, c \cdot c\right)}{\mathsf{fma}\left(-c, b, a \cdot d\right)}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 2.8

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

      16. lift-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 2.8

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

   6. Applied rewrites 2.8%

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

   7. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 58.5

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

   9. Applied rewrites 58.5%

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

Alternative 3: 65.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ t_1 := b \cdot c - a \cdot d\\ \mathbf{if}\;d \leq -4.3 \cdot 10^{+121}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -1.1 \cdot 10^{-15}:\\ \;\;\;\;\frac{t\_1}{d \cdot d}\\ \mathbf{elif}\;d \leq 3.65 \cdot 10^{-186}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{-88}:\\ \;\;\;\;\frac{t\_1}{c \cdot c}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{+139}:\\ \;\;\;\;\left(-a\right) \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)) (t_1 (- (* b c) (* a d))))
   (if (<= d -4.3e+121)
     t_0
     (if (<= d -1.1e-15)
       (/ t_1 (* d d))
       (if (<= d 3.65e-186)
         (/ b c)
         (if (<= d 1.85e-88)
           (/ t_1 (* c c))
           (if (<= d 4.5e+139) (* (- a) (/ d (fma d d (* c c)))) t_0)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double t_1 = (b * c) - (a * d);
	double tmp;
	if (d <= -4.3e+121) {
		tmp = t_0;
	} else if (d <= -1.1e-15) {
		tmp = t_1 / (d * d);
	} else if (d <= 3.65e-186) {
		tmp = b / c;
	} else if (d <= 1.85e-88) {
		tmp = t_1 / (c * c);
	} else if (d <= 4.5e+139) {
		tmp = -a * (d / fma(d, d, (c * c)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	t_1 = Float64(Float64(b * c) - Float64(a * d))
	tmp = 0.0
	if (d <= -4.3e+121)
		tmp = t_0;
	elseif (d <= -1.1e-15)
		tmp = Float64(t_1 / Float64(d * d));
	elseif (d <= 3.65e-186)
		tmp = Float64(b / c);
	elseif (d <= 1.85e-88)
		tmp = Float64(t_1 / Float64(c * c));
	elseif (d <= 4.5e+139)
		tmp = Float64(Float64(-a) * Float64(d / fma(d, d, Float64(c * c))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -4.3e+121], t$95$0, If[LessEqual[d, -1.1e-15], N[(t$95$1 / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.65e-186], N[(b / c), $MachinePrecision], If[LessEqual[d, 1.85e-88], N[(t$95$1 / N[(c * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.5e+139], N[((-a) * N[(d / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -4.2999999999999997e121 or 4.4999999999999999e139 < d

   1. Initial program 37.9%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   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 81.0

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

   5. Applied rewrites 81.0%

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

   if -4.2999999999999997e121 < d < -1.09999999999999993e-15

   1. Initial program 96.5%

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

   3. Taylor expanded in c around 0

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{d}^{2}}} \]
   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 78.0

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

   5. Applied rewrites 78.0%

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

   if -1.09999999999999993e-15 < d < 3.65e-186

   1. Initial program 62.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 69.1

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

   5. Applied rewrites 69.1%

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

   if 3.65e-186 < d < 1.8499999999999999e-88

   1. Initial program 87.9%

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

   3. Taylor expanded in c around inf

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{c}^{2}}} \]
   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 73.2

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

   5. Applied rewrites 73.2%

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

   if 1.8499999999999999e-88 < d < 4.4999999999999999e139

   1. Initial program 68.7%

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

   3. Taylor expanded in a around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 58.6

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

   5. Applied rewrites 58.6%

      \[\leadsto \color{blue}{\left(-a\right) \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.3 \cdot 10^{+121}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -1.1 \cdot 10^{-15}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{d \cdot d}\\ \mathbf{elif}\;d \leq 3.65 \cdot 10^{-186}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{-88}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{+139}:\\ \;\;\;\;\left(-a\right) \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ t_1 := \left(-a\right) \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{if}\;d \leq -2.5 \cdot 10^{+78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-90}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 3.65 \cdot 10^{-186}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{-88}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{+139}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)) (t_1 (* (- a) (/ d (fma d d (* c c))))))
   (if (<= d -2.5e+78)
     t_0
     (if (<= d -1e-90)
       t_1
       (if (<= d 3.65e-186)
         (/ b c)
         (if (<= d 1.85e-88)
           (/ (- (* b c) (* a d)) (* c c))
           (if (<= d 4.5e+139) t_1 t_0)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double t_1 = -a * (d / fma(d, d, (c * c)));
	double tmp;
	if (d <= -2.5e+78) {
		tmp = t_0;
	} else if (d <= -1e-90) {
		tmp = t_1;
	} else if (d <= 3.65e-186) {
		tmp = b / c;
	} else if (d <= 1.85e-88) {
		tmp = ((b * c) - (a * d)) / (c * c);
	} else if (d <= 4.5e+139) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	t_1 = Float64(Float64(-a) * Float64(d / fma(d, d, Float64(c * c))))
	tmp = 0.0
	if (d <= -2.5e+78)
		tmp = t_0;
	elseif (d <= -1e-90)
		tmp = t_1;
	elseif (d <= 3.65e-186)
		tmp = Float64(b / c);
	elseif (d <= 1.85e-88)
		tmp = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(c * c));
	elseif (d <= 4.5e+139)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, Block[{t$95$1 = N[((-a) * N[(d / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.5e+78], t$95$0, If[LessEqual[d, -1e-90], t$95$1, If[LessEqual[d, 3.65e-186], N[(b / c), $MachinePrecision], If[LessEqual[d, 1.85e-88], N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.5e+139], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.49999999999999992e78 or 4.4999999999999999e139 < 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 80.9

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

   5. Applied rewrites 80.9%

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

   if -2.49999999999999992e78 < d < -9.99999999999999995e-91 or 1.8499999999999999e-88 < d < 4.4999999999999999e139

   1. Initial program 74.7%

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

   3. Taylor expanded in a around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 60.1

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

   5. Applied rewrites 60.1%

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

   if -9.99999999999999995e-91 < d < 3.65e-186

   1. Initial program 60.7%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 73.0

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

   5. Applied rewrites 73.0%

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

   if 3.65e-186 < d < 1.8499999999999999e-88

   1. Initial program 87.9%

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

   3. Taylor expanded in c around inf

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{c}^{2}}} \]
   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 73.2

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

   5. Applied rewrites 73.2%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.5 \cdot 10^{+78}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-90}:\\ \;\;\;\;\left(-a\right) \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 3.65 \cdot 10^{-186}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{-88}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{+139}:\\ \;\;\;\;\left(-a\right) \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ t_1 := \frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{if}\;c \leq -4.3 \cdot 10^{+125}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -8.6 \cdot 10^{+46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -1.05 \cdot 10^{-145}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma b (/ c d) (- a)) d)) (t_1 (/ (- b (/ (* a d) c)) c)))
   (if (<= c -4.3e+125)
     t_1
     (if (<= c -8.6e+46)
       t_0
       (if (<= c -1.05e-145)
         (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))
         (if (<= c 1.7e+16) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(b, (c / d), -a) / d;
	double t_1 = (b - ((a * d) / c)) / c;
	double tmp;
	if (c <= -4.3e+125) {
		tmp = t_1;
	} else if (c <= -8.6e+46) {
		tmp = t_0;
	} else if (c <= -1.05e-145) {
		tmp = ((b * c) - (a * d)) / ((c * c) + (d * d));
	} else if (c <= 1.7e+16) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(b, Float64(c / d), Float64(-a)) / d)
	t_1 = Float64(Float64(b - Float64(Float64(a * d) / c)) / c)
	tmp = 0.0
	if (c <= -4.3e+125)
		tmp = t_1;
	elseif (c <= -8.6e+46)
		tmp = t_0;
	elseif (c <= -1.05e-145)
		tmp = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 1.7e+16)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * N[(c / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -4.3e+125], t$95$1, If[LessEqual[c, -8.6e+46], t$95$0, If[LessEqual[c, -1.05e-145], N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.7e+16], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -4.30000000000000035e125 or 1.7e16 < c

   1. Initial program 48.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. lower-*.f64 83.4

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

   5. Applied rewrites 83.4%

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

   if -4.30000000000000035e125 < c < -8.60000000000000009e46 or -1.04999999999999996e-145 < c < 1.7e16

   1. Initial program 65.0%

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

   4. Applied rewrites 65.0%

      \[\leadsto \color{blue}{\frac{-1}{\frac{\mathsf{fma}\left(d, d, c \cdot c\right)}{\mathsf{fma}\left(-c, b, a \cdot d\right)}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 77.6

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

      16. lift-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 77.6

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

   6. Applied rewrites 77.6%

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

   7. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 87.1

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

   9. Applied rewrites 87.1%

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

   if -8.60000000000000009e46 < c < -1.04999999999999996e-145

   1. Initial program 90.6%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.3 \cdot 10^{+125}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;c \leq -8.6 \cdot 10^{+46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{elif}\;c \leq -1.05 \cdot 10^{-145}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ t_1 := \left(-a\right) \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{if}\;d \leq -2.5 \cdot 10^{+78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-90}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{+139}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)) (t_1 (* (- a) (/ d (fma d d (* c c))))))
   (if (<= d -2.5e+78)
     t_0
     (if (<= d -1e-90)
       t_1
       (if (<= d 4.2e-145) (/ b c) (if (<= d 4.5e+139) t_1 t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double t_1 = -a * (d / fma(d, d, (c * c)));
	double tmp;
	if (d <= -2.5e+78) {
		tmp = t_0;
	} else if (d <= -1e-90) {
		tmp = t_1;
	} else if (d <= 4.2e-145) {
		tmp = b / c;
	} else if (d <= 4.5e+139) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	t_1 = Float64(Float64(-a) * Float64(d / fma(d, d, Float64(c * c))))
	tmp = 0.0
	if (d <= -2.5e+78)
		tmp = t_0;
	elseif (d <= -1e-90)
		tmp = t_1;
	elseif (d <= 4.2e-145)
		tmp = Float64(b / c);
	elseif (d <= 4.5e+139)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, Block[{t$95$1 = N[((-a) * N[(d / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.5e+78], t$95$0, If[LessEqual[d, -1e-90], t$95$1, If[LessEqual[d, 4.2e-145], N[(b / c), $MachinePrecision], If[LessEqual[d, 4.5e+139], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.49999999999999992e78 or 4.4999999999999999e139 < 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 80.9

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

   5. Applied rewrites 80.9%

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

   if -2.49999999999999992e78 < d < -9.99999999999999995e-91 or 4.19999999999999982e-145 < d < 4.4999999999999999e139

   1. Initial program 78.0%

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

   3. Taylor expanded in a around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 59.4

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

   5. Applied rewrites 59.4%

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

   if -9.99999999999999995e-91 < d < 4.19999999999999982e-145

   1. Initial program 63.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 70.3

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

   5. Applied rewrites 70.3%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.5 \cdot 10^{+78}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-90}:\\ \;\;\;\;\left(-a\right) \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{+139}:\\ \;\;\;\;\left(-a\right) \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.1 \cdot 10^{+131}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{elif}\;c \leq 7.6 \cdot 10^{+136}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.1e+131)
   (/ b c)
   (if (<= c 1.7e+16)
     (/ (fma b (/ c d) (- a)) d)
     (if (<= c 7.6e+136) (/ (- (* b c) (* a d)) (* c c)) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.1e+131) {
		tmp = b / c;
	} else if (c <= 1.7e+16) {
		tmp = fma(b, (c / d), -a) / d;
	} else if (c <= 7.6e+136) {
		tmp = ((b * c) - (a * d)) / (c * c);
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.1e+131)
		tmp = Float64(b / c);
	elseif (c <= 1.7e+16)
		tmp = Float64(fma(b, Float64(c / d), Float64(-a)) / d);
	elseif (c <= 7.6e+136)
		tmp = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(c * c));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.1e+131], N[(b / c), $MachinePrecision], If[LessEqual[c, 1.7e+16], N[(N[(b * N[(c / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 7.6e+136], N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(b / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.09999999999999985e131 or 7.60000000000000029e136 < c

   1. Initial program 39.7%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 83.0

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

   5. Applied rewrites 83.0%

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

   if -2.09999999999999985e131 < c < 1.7e16

   1. Initial program 72.0%

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

   4. Applied rewrites 71.9%

      \[\leadsto \color{blue}{\frac{-1}{\frac{\mathsf{fma}\left(d, d, c \cdot c\right)}{\mathsf{fma}\left(-c, b, a \cdot d\right)}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 82.3

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

      16. lift-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 82.3

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

   6. Applied rewrites 82.3%

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

   7. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 79.2

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

   9. Applied rewrites 79.2%

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

   if 1.7e16 < c < 7.60000000000000029e136

   1. Initial program 79.3%

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

   3. Taylor expanded in c around inf

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{c}^{2}}} \]
   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 69.4

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

   5. Applied rewrites 69.4%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.1 \cdot 10^{+131}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{elif}\;c \leq 7.6 \cdot 10^{+136}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.3 \cdot 10^{+125} \lor \neg \left(c \leq 1.7 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -4.3e+125) (not (<= c 1.7e+16)))
   (/ (- b (/ (* a d) c)) c)
   (/ (fma b (/ c d) (- a)) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4.3e+125) || !(c <= 1.7e+16)) {
		tmp = (b - ((a * d) / c)) / c;
	} else {
		tmp = fma(b, (c / d), -a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -4.3e+125) || !(c <= 1.7e+16))
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	else
		tmp = Float64(fma(b, Float64(c / d), Float64(-a)) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -4.3e+125], N[Not[LessEqual[c, 1.7e+16]], $MachinePrecision]], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(b * N[(c / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -4.30000000000000035e125 or 1.7e16 < c

   1. Initial program 48.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. lower-*.f64 83.4

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

   5. Applied rewrites 83.4%

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

   if -4.30000000000000035e125 < c < 1.7e16

   1. Initial program 71.8%

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

   4. Applied rewrites 71.7%

      \[\leadsto \color{blue}{\frac{-1}{\frac{\mathsf{fma}\left(d, d, c \cdot c\right)}{\mathsf{fma}\left(-c, b, a \cdot d\right)}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 82.2

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

      16. lift-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 82.2

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

   6. Applied rewrites 82.2%

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

   7. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 79.7

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

   9. Applied rewrites 79.7%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.3 \cdot 10^{+125} \lor \neg \left(c \leq 1.7 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.6 \cdot 10^{+123} \lor \neg \left(c \leq 7.1 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -3.6e+123) (not (<= c 7.1e+16))) (/ b c) (/ (- a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3.6e+123) || !(c <= 7.1e+16)) {
		tmp = b / c;
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-3.6d+123)) .or. (.not. (c <= 7.1d+16))) then
        tmp = b / c
    else
        tmp = -a / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3.6e+123) || !(c <= 7.1e+16)) {
		tmp = b / c;
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -3.6e+123) or not (c <= 7.1e+16):
		tmp = b / c
	else:
		tmp = -a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -3.6e+123) || !(c <= 7.1e+16))
		tmp = Float64(b / c);
	else
		tmp = Float64(Float64(-a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -3.6e+123) || ~((c <= 7.1e+16)))
		tmp = b / c;
	else
		tmp = -a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -3.6e+123], N[Not[LessEqual[c, 7.1e+16]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[((-a) / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -3.59999999999999998e123 or 7.1e16 < c

   1. Initial program 47.9%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 70.7

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

   5. Applied rewrites 70.7%

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

   if -3.59999999999999998e123 < c < 7.1e16

   1. Initial program 72.2%

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

   3. Taylor expanded in 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 60.1

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

   5. Applied rewrites 60.1%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.6 \cdot 10^{+123} \lor \neg \left(c \leq 7.1 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 44.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.6 \cdot 10^{+82} \lor \neg \left(d \leq 6.4 \cdot 10^{+173}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -2.6e+82) (not (<= d 6.4e+173))) (/ a d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.6e+82) || !(d <= 6.4e+173)) {
		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 <= (-2.6d+82)) .or. (.not. (d <= 6.4d+173))) 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 <= -2.6e+82) || !(d <= 6.4e+173)) {
		tmp = a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -2.6e+82) or not (d <= 6.4e+173):
		tmp = a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -2.6e+82) || !(d <= 6.4e+173))
		tmp = Float64(a / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -2.6e+82) || ~((d <= 6.4e+173)))
		tmp = a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -2.6e+82], N[Not[LessEqual[d, 6.4e+173]], $MachinePrecision]], N[(a / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.5999999999999998e82 or 6.4000000000000005e173 < d

   1. Initial program 45.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 82.7

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

   5. Applied rewrites 82.7%

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

   7. Applied rewrites 36.7%

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

   if -2.5999999999999998e82 < d < 6.4000000000000005e173

   1. Initial program 70.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 47.5

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

   5. Applied rewrites 47.5%

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

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.6 \cdot 10^{+82} \lor \neg \left(d \leq 6.4 \cdot 10^{+173}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 10.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.2%

   \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
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 43.2

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

5. Applied rewrites 43.2%

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

7. Applied rewrites 12.9%

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

8. Final simplification 12.9%

   \[\leadsto \frac{a}{d} \]
9. Add Preprocessing

Developer Target 1: 99.2% 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 2024305 
(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))))