Complex division, imag part

Percentage Accurate: 62.2% → 85.5%

Time: 10.2s

Alternatives: 13

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: 62.2% 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: 85.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \mathsf{fma}\left(\frac{d}{t\_0}, \frac{-1}{\frac{1}{a}}, \frac{c}{t\_0} \cdot b\right)\\ \mathbf{if}\;d \leq -2.05 \cdot 10^{+96}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq -3 \cdot 10^{-126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{-137}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{+129}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c)))
        (t_1 (fma (/ d t_0) (/ -1.0 (/ 1.0 a)) (* (/ c t_0) b))))
   (if (<= d -2.05e+96)
     (/ (fma c (/ b d) (- a)) d)
     (if (<= d -3e-126)
       t_1
       (if (<= d 1.65e-137)
         (/ (- b (/ (* a d) c)) c)
         (if (<= d 3.3e+129) t_1 (/ (fma (/ c d) b (- a)) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double t_1 = fma((d / t_0), (-1.0 / (1.0 / a)), ((c / t_0) * b));
	double tmp;
	if (d <= -2.05e+96) {
		tmp = fma(c, (b / d), -a) / d;
	} else if (d <= -3e-126) {
		tmp = t_1;
	} else if (d <= 1.65e-137) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 3.3e+129) {
		tmp = t_1;
	} else {
		tmp = fma((c / d), b, -a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = fma(Float64(d / t_0), Float64(-1.0 / Float64(1.0 / a)), Float64(Float64(c / t_0) * b))
	tmp = 0.0
	if (d <= -2.05e+96)
		tmp = Float64(fma(c, Float64(b / d), Float64(-a)) / d);
	elseif (d <= -3e-126)
		tmp = t_1;
	elseif (d <= 1.65e-137)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	elseif (d <= 3.3e+129)
		tmp = t_1;
	else
		tmp = Float64(fma(Float64(c / d), b, Float64(-a)) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(d / t$95$0), $MachinePrecision] * N[(-1.0 / N[(1.0 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / t$95$0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.05e+96], N[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -3e-126], t$95$1, If[LessEqual[d, 1.65e-137], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 3.3e+129], t$95$1, N[(N[(N[(c / d), $MachinePrecision] * b + (-a)), $MachinePrecision] / d), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.04999999999999999e96

   1. Initial program 30.7%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 81.8

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

   5. Applied rewrites 81.8%

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

   7. Applied rewrites 90.1%

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

   if -2.04999999999999999e96 < d < -3.0000000000000002e-126 or 1.6500000000000001e-137 < d < 3.2999999999999999e129

   1. Initial program 75.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

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

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 80.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lift-neg.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. *-commutative N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 84.8

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

   6. Applied rewrites 84.8%

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

   if -3.0000000000000002e-126 < d < 1.6500000000000001e-137

   1. Initial program 73.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 92.6

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

   5. Applied rewrites 92.6%

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

   if 3.2999999999999999e129 < d

   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

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

      1. lower-/.f64 13.3

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

   5. Applied rewrites 13.3%

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

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]
   7. 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. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 94.1

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

   8. Applied rewrites 94.1%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.05 \cdot 10^{+96}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq -3 \cdot 10^{-126}:\\ \;\;\;\;\mathsf{fma}\left(\frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{-1}{\frac{1}{a}}, \frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\right)\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{-137}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{+129}:\\ \;\;\;\;\mathsf{fma}\left(\frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{-1}{\frac{1}{a}}, \frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ \mathbf{if}\;d \leq -1.25 \cdot 10^{+57}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq -1.28 \cdot 10^{-79}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, a, c \cdot b\right)}{t\_0}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{-147}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{+128}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{t\_0}, b, \frac{a}{t\_0} \cdot \left(-d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))))
   (if (<= d -1.25e+57)
     (/ (fma c (/ b d) (- a)) d)
     (if (<= d -1.28e-79)
       (/ (fma (- d) a (* c b)) t_0)
       (if (<= d 3.6e-147)
         (/ (- b (/ (* a d) c)) c)
         (if (<= d 3.3e+128)
           (fma (/ c t_0) b (* (/ a t_0) (- d)))
           (/ (fma (/ c d) b (- a)) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double tmp;
	if (d <= -1.25e+57) {
		tmp = fma(c, (b / d), -a) / d;
	} else if (d <= -1.28e-79) {
		tmp = fma(-d, a, (c * b)) / t_0;
	} else if (d <= 3.6e-147) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 3.3e+128) {
		tmp = fma((c / t_0), b, ((a / t_0) * -d));
	} else {
		tmp = fma((c / d), b, -a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	tmp = 0.0
	if (d <= -1.25e+57)
		tmp = Float64(fma(c, Float64(b / d), Float64(-a)) / d);
	elseif (d <= -1.28e-79)
		tmp = Float64(fma(Float64(-d), a, Float64(c * b)) / t_0);
	elseif (d <= 3.6e-147)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	elseif (d <= 3.3e+128)
		tmp = fma(Float64(c / t_0), b, Float64(Float64(a / t_0) * Float64(-d)));
	else
		tmp = Float64(fma(Float64(c / d), b, Float64(-a)) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.25e+57], N[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -1.28e-79], N[(N[((-d) * a + N[(c * b), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[d, 3.6e-147], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 3.3e+128], N[(N[(c / t$95$0), $MachinePrecision] * b + N[(N[(a / t$95$0), $MachinePrecision] * (-d)), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c / d), $MachinePrecision] * b + (-a)), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.24999999999999993e57

   1. Initial program 33.0%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 81.3

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

   5. Applied rewrites 81.3%

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

   7. Applied rewrites 88.7%

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

   if -1.24999999999999993e57 < d < -1.28e-79

   1. Initial program 84.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 84.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 84.2

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

   4. Applied rewrites 84.2%

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

   if -1.28e-79 < d < 3.60000000000000012e-147

   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

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   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 88.6

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

   5. Applied rewrites 88.6%

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

   if 3.60000000000000012e-147 < d < 3.3000000000000001e128

   1. Initial program 78.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

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

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 85.8%

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

   if 3.3000000000000001e128 < d

   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

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

      1. lower-/.f64 13.3

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

   5. Applied rewrites 13.3%

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

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]
   7. 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. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 94.1

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

   8. Applied rewrites 94.1%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.25 \cdot 10^{+57}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq -1.28 \cdot 10^{-79}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, a, c \cdot b\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{-147}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{+128}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}, b, \frac{a}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot \left(-d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, c, d \cdot d\right)\\ t_1 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -2.4 \cdot 10^{+142}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -0.0175:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{d \cdot d}\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-274}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 4.7 \cdot 10^{-76}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, a, c \cdot b\right)}{c \cdot c}\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{+33}:\\ \;\;\;\;\frac{d}{t\_0} \cdot \left(-a\right)\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{+99}:\\ \;\;\;\;\frac{b}{t\_0} \cdot c\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma c c (* d d))) (t_1 (/ (- a) d)))
   (if (<= d -2.4e+142)
     t_1
     (if (<= d -0.0175)
       (/ (- (* c b) (* a d)) (* d d))
       (if (<= d 5.8e-274)
         (/ b c)
         (if (<= d 4.7e-76)
           (/ (fma (- d) a (* c b)) (* c c))
           (if (<= d 1.2e+33)
             (* (/ d t_0) (- a))
             (if (<= d 1.65e+99) (* (/ b t_0) c) t_1))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(c, c, (d * d));
	double t_1 = -a / d;
	double tmp;
	if (d <= -2.4e+142) {
		tmp = t_1;
	} else if (d <= -0.0175) {
		tmp = ((c * b) - (a * d)) / (d * d);
	} else if (d <= 5.8e-274) {
		tmp = b / c;
	} else if (d <= 4.7e-76) {
		tmp = fma(-d, a, (c * b)) / (c * c);
	} else if (d <= 1.2e+33) {
		tmp = (d / t_0) * -a;
	} else if (d <= 1.65e+99) {
		tmp = (b / t_0) * c;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(c, c, Float64(d * d))
	t_1 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (d <= -2.4e+142)
		tmp = t_1;
	elseif (d <= -0.0175)
		tmp = Float64(Float64(Float64(c * b) - Float64(a * d)) / Float64(d * d));
	elseif (d <= 5.8e-274)
		tmp = Float64(b / c);
	elseif (d <= 4.7e-76)
		tmp = Float64(fma(Float64(-d), a, Float64(c * b)) / Float64(c * c));
	elseif (d <= 1.2e+33)
		tmp = Float64(Float64(d / t_0) * Float64(-a));
	elseif (d <= 1.65e+99)
		tmp = Float64(Float64(b / t_0) * c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-a) / d), $MachinePrecision]}, If[LessEqual[d, -2.4e+142], t$95$1, If[LessEqual[d, -0.0175], N[(N[(N[(c * b), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.8e-274], N[(b / c), $MachinePrecision], If[LessEqual[d, 4.7e-76], N[(N[((-d) * a + N[(c * b), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.2e+33], N[(N[(d / t$95$0), $MachinePrecision] * (-a)), $MachinePrecision], If[LessEqual[d, 1.65e+99], N[(N[(b / t$95$0), $MachinePrecision] * c), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if d < -2.3999999999999999e142 or 1.65e99 < d

   1. Initial program 23.2%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 79.7

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

   5. Applied rewrites 79.7%

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

   if -2.3999999999999999e142 < d < -0.017500000000000002

   1. Initial program 81.0%

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

   3. Taylor expanded in c around 0

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{d}^{2}}} \]
   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 62.8

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

   5. Applied rewrites 62.8%

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

   if -0.017500000000000002 < d < 5.79999999999999952e-274

   1. Initial program 71.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 74.1

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

   5. Applied rewrites 74.1%

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

   if 5.79999999999999952e-274 < d < 4.7000000000000002e-76

   1. Initial program 80.9%

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

   3. Taylor expanded in 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 78.7

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

   5. Applied rewrites 78.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. lift-neg.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 78.7

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 78.7

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

   7. Applied rewrites 78.7%

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

   if 4.7000000000000002e-76 < d < 1.2e33

   1. Initial program 73.6%

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

   3. Taylor expanded in b around 0

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

      1. 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

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 56.9

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

   5. Applied rewrites 56.9%

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

   if 1.2e33 < d < 1.65e99

   1. Initial program 74.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

      \[\leadsto \color{blue}{\frac{b}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot c} \]
3. Recombined 6 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.4 \cdot 10^{+142}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -0.0175:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{d \cdot d}\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-274}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 4.7 \cdot 10^{-76}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, a, c \cdot b\right)}{c \cdot c}\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{+33}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot \left(-a\right)\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{+99}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot c\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, c, d \cdot d\right)\\ t_1 := \frac{-a}{d}\\ t_2 := c \cdot b - a \cdot d\\ \mathbf{if}\;d \leq -2.4 \cdot 10^{+142}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -0.0175:\\ \;\;\;\;\frac{t\_2}{d \cdot d}\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-274}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 4.7 \cdot 10^{-76}:\\ \;\;\;\;\frac{t\_2}{c \cdot c}\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{+33}:\\ \;\;\;\;\frac{d}{t\_0} \cdot \left(-a\right)\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{+99}:\\ \;\;\;\;\frac{b}{t\_0} \cdot c\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma c c (* d d))) (t_1 (/ (- a) d)) (t_2 (- (* c b) (* a d))))
   (if (<= d -2.4e+142)
     t_1
     (if (<= d -0.0175)
       (/ t_2 (* d d))
       (if (<= d 5.8e-274)
         (/ b c)
         (if (<= d 4.7e-76)
           (/ t_2 (* c c))
           (if (<= d 1.2e+33)
             (* (/ d t_0) (- a))
             (if (<= d 1.65e+99) (* (/ b t_0) c) t_1))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(c, c, (d * d));
	double t_1 = -a / d;
	double t_2 = (c * b) - (a * d);
	double tmp;
	if (d <= -2.4e+142) {
		tmp = t_1;
	} else if (d <= -0.0175) {
		tmp = t_2 / (d * d);
	} else if (d <= 5.8e-274) {
		tmp = b / c;
	} else if (d <= 4.7e-76) {
		tmp = t_2 / (c * c);
	} else if (d <= 1.2e+33) {
		tmp = (d / t_0) * -a;
	} else if (d <= 1.65e+99) {
		tmp = (b / t_0) * c;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(c, c, Float64(d * d))
	t_1 = Float64(Float64(-a) / d)
	t_2 = Float64(Float64(c * b) - Float64(a * d))
	tmp = 0.0
	if (d <= -2.4e+142)
		tmp = t_1;
	elseif (d <= -0.0175)
		tmp = Float64(t_2 / Float64(d * d));
	elseif (d <= 5.8e-274)
		tmp = Float64(b / c);
	elseif (d <= 4.7e-76)
		tmp = Float64(t_2 / Float64(c * c));
	elseif (d <= 1.2e+33)
		tmp = Float64(Float64(d / t_0) * Float64(-a));
	elseif (d <= 1.65e+99)
		tmp = Float64(Float64(b / t_0) * c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-a) / d), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * b), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.4e+142], t$95$1, If[LessEqual[d, -0.0175], N[(t$95$2 / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.8e-274], N[(b / c), $MachinePrecision], If[LessEqual[d, 4.7e-76], N[(t$95$2 / N[(c * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.2e+33], N[(N[(d / t$95$0), $MachinePrecision] * (-a)), $MachinePrecision], If[LessEqual[d, 1.65e+99], N[(N[(b / t$95$0), $MachinePrecision] * c), $MachinePrecision], t$95$1]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if d < -2.3999999999999999e142 or 1.65e99 < d

   1. Initial program 23.2%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 79.7

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

   5. Applied rewrites 79.7%

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

   if -2.3999999999999999e142 < d < -0.017500000000000002

   1. Initial program 81.0%

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

   3. Taylor expanded in c around 0

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{d}^{2}}} \]
   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 62.8

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

   5. Applied rewrites 62.8%

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

   if -0.017500000000000002 < d < 5.79999999999999952e-274

   1. Initial program 71.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 74.1

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

   5. Applied rewrites 74.1%

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

   if 5.79999999999999952e-274 < d < 4.7000000000000002e-76

   1. Initial program 80.9%

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

   3. Taylor expanded in 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 78.7

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

   5. Applied rewrites 78.7%

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

   if 4.7000000000000002e-76 < d < 1.2e33

   1. Initial program 73.6%

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

   3. Taylor expanded in b around 0

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

      1. 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

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 56.9

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

   5. Applied rewrites 56.9%

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

   if 1.2e33 < d < 1.65e99

   1. Initial program 74.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

      \[\leadsto \color{blue}{\frac{b}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot c} \]
3. Recombined 6 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.4 \cdot 10^{+142}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -0.0175:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{d \cdot d}\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-274}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 4.7 \cdot 10^{-76}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{c \cdot c}\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{+33}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot \left(-a\right)\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{+99}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot c\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, c, d \cdot d\right)\\ t_1 := \frac{-a}{d}\\ t_2 := \frac{d}{t\_0} \cdot \left(-a\right)\\ \mathbf{if}\;d \leq -3.4 \cdot 10^{+115}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -4.8 \cdot 10^{-18}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-274}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 4.7 \cdot 10^{-76}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{c \cdot c}\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{+33}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{+99}:\\ \;\;\;\;\frac{b}{t\_0} \cdot c\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma c c (* d d))) (t_1 (/ (- a) d)) (t_2 (* (/ d t_0) (- a))))
   (if (<= d -3.4e+115)
     t_1
     (if (<= d -4.8e-18)
       t_2
       (if (<= d 5.8e-274)
         (/ b c)
         (if (<= d 4.7e-76)
           (/ (- (* c b) (* a d)) (* c c))
           (if (<= d 1.2e+33)
             t_2
             (if (<= d 1.65e+99) (* (/ b t_0) c) t_1))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(c, c, (d * d));
	double t_1 = -a / d;
	double t_2 = (d / t_0) * -a;
	double tmp;
	if (d <= -3.4e+115) {
		tmp = t_1;
	} else if (d <= -4.8e-18) {
		tmp = t_2;
	} else if (d <= 5.8e-274) {
		tmp = b / c;
	} else if (d <= 4.7e-76) {
		tmp = ((c * b) - (a * d)) / (c * c);
	} else if (d <= 1.2e+33) {
		tmp = t_2;
	} else if (d <= 1.65e+99) {
		tmp = (b / t_0) * c;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(c, c, Float64(d * d))
	t_1 = Float64(Float64(-a) / d)
	t_2 = Float64(Float64(d / t_0) * Float64(-a))
	tmp = 0.0
	if (d <= -3.4e+115)
		tmp = t_1;
	elseif (d <= -4.8e-18)
		tmp = t_2;
	elseif (d <= 5.8e-274)
		tmp = Float64(b / c);
	elseif (d <= 4.7e-76)
		tmp = Float64(Float64(Float64(c * b) - Float64(a * d)) / Float64(c * c));
	elseif (d <= 1.2e+33)
		tmp = t_2;
	elseif (d <= 1.65e+99)
		tmp = Float64(Float64(b / t_0) * c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-a) / d), $MachinePrecision]}, Block[{t$95$2 = N[(N[(d / t$95$0), $MachinePrecision] * (-a)), $MachinePrecision]}, If[LessEqual[d, -3.4e+115], t$95$1, If[LessEqual[d, -4.8e-18], t$95$2, If[LessEqual[d, 5.8e-274], N[(b / c), $MachinePrecision], If[LessEqual[d, 4.7e-76], N[(N[(N[(c * b), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.2e+33], t$95$2, If[LessEqual[d, 1.65e+99], N[(N[(b / t$95$0), $MachinePrecision] * c), $MachinePrecision], t$95$1]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -3.4000000000000001e115 or 1.65e99 < d

   1. Initial program 25.1%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 79.1

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

   5. Applied rewrites 79.1%

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

   if -3.4000000000000001e115 < d < -4.79999999999999988e-18 or 4.7000000000000002e-76 < d < 1.2e33

   1. Initial program 78.5%

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

   3. Taylor expanded in b around 0

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

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 57.9

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

   5. Applied rewrites 57.9%

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

   if -4.79999999999999988e-18 < d < 5.79999999999999952e-274

   1. Initial program 69.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 75.4

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

   5. Applied rewrites 75.4%

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

   if 5.79999999999999952e-274 < d < 4.7000000000000002e-76

   1. Initial program 80.9%

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

   3. Taylor expanded in 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 78.7

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

   5. Applied rewrites 78.7%

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

   if 1.2e33 < d < 1.65e99

   1. Initial program 74.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.4 \cdot 10^{+115}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -4.8 \cdot 10^{-18}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot \left(-a\right)\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-274}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 4.7 \cdot 10^{-76}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{c \cdot c}\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{+33}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot \left(-a\right)\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{+99}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot c\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(-d, a, c \cdot b\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{if}\;d \leq -1.25 \cdot 10^{+57}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq -1.28 \cdot 10^{-79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-73}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 1.12 \cdot 10^{+123}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (- d) a (* c b)) (fma d d (* c c)))))
   (if (<= d -1.25e+57)
     (/ (fma c (/ b d) (- a)) d)
     (if (<= d -1.28e-79)
       t_0
       (if (<= d 4e-73)
         (/ (- b (/ (* a d) c)) c)
         (if (<= d 1.12e+123) t_0 (/ (fma (/ c d) b (- a)) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(-d, a, (c * b)) / fma(d, d, (c * c));
	double tmp;
	if (d <= -1.25e+57) {
		tmp = fma(c, (b / d), -a) / d;
	} else if (d <= -1.28e-79) {
		tmp = t_0;
	} else if (d <= 4e-73) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 1.12e+123) {
		tmp = t_0;
	} else {
		tmp = fma((c / d), b, -a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(-d), a, Float64(c * b)) / fma(d, d, Float64(c * c)))
	tmp = 0.0
	if (d <= -1.25e+57)
		tmp = Float64(fma(c, Float64(b / d), Float64(-a)) / d);
	elseif (d <= -1.28e-79)
		tmp = t_0;
	elseif (d <= 4e-73)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	elseif (d <= 1.12e+123)
		tmp = t_0;
	else
		tmp = Float64(fma(Float64(c / d), b, Float64(-a)) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[((-d) * a + N[(c * b), $MachinePrecision]), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.25e+57], N[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -1.28e-79], t$95$0, If[LessEqual[d, 4e-73], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.12e+123], t$95$0, N[(N[(N[(c / d), $MachinePrecision] * b + (-a)), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.24999999999999993e57

   1. Initial program 33.0%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 81.3

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

   5. Applied rewrites 81.3%

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

   7. Applied rewrites 88.7%

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

   if -1.24999999999999993e57 < d < -1.28e-79 or 3.99999999999999999e-73 < d < 1.12e123

   1. Initial program 79.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 79.8

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 79.8

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

   4. Applied rewrites 79.8%

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

   if -1.28e-79 < d < 3.99999999999999999e-73

   1. Initial program 72.9%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 87.8

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

   5. Applied rewrites 87.8%

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

   if 1.12e123 < d

   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

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

      1. lower-/.f64 13.3

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

   5. Applied rewrites 13.3%

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

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]
   7. 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. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 94.1

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

   8. Applied rewrites 94.1%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.25 \cdot 10^{+57}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq -1.28 \cdot 10^{-79}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, a, c \cdot b\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-73}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 1.12 \cdot 10^{+123}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, a, c \cdot b\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -2.4 \cdot 10^{+142}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -3.9:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{d \cdot d}\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{+99}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -2.4e+142)
     t_0
     (if (<= d -3.9)
       (/ (- (* c b) (* a d)) (* d d))
       (if (<= d 1.65e+99) (/ (- b (/ (* a d) c)) c) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -2.4e+142) {
		tmp = t_0;
	} else if (d <= -3.9) {
		tmp = ((c * b) - (a * d)) / (d * d);
	} else if (d <= 1.65e+99) {
		tmp = (b - ((a * d) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -a / d
    if (d <= (-2.4d+142)) then
        tmp = t_0
    else if (d <= (-3.9d0)) then
        tmp = ((c * b) - (a * d)) / (d * d)
    else if (d <= 1.65d+99) then
        tmp = (b - ((a * d) / c)) / c
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -2.4e+142) {
		tmp = t_0;
	} else if (d <= -3.9) {
		tmp = ((c * b) - (a * d)) / (d * d);
	} else if (d <= 1.65e+99) {
		tmp = (b - ((a * d) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = -a / d
	tmp = 0
	if d <= -2.4e+142:
		tmp = t_0
	elif d <= -3.9:
		tmp = ((c * b) - (a * d)) / (d * d)
	elif d <= 1.65e+99:
		tmp = (b - ((a * d) / c)) / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (d <= -2.4e+142)
		tmp = t_0;
	elseif (d <= -3.9)
		tmp = Float64(Float64(Float64(c * b) - Float64(a * d)) / Float64(d * d));
	elseif (d <= 1.65e+99)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = -a / d;
	tmp = 0.0;
	if (d <= -2.4e+142)
		tmp = t_0;
	elseif (d <= -3.9)
		tmp = ((c * b) - (a * d)) / (d * d);
	elseif (d <= 1.65e+99)
		tmp = (b - ((a * d) / c)) / c;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, If[LessEqual[d, -2.4e+142], t$95$0, If[LessEqual[d, -3.9], N[(N[(N[(c * b), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.65e+99], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.3999999999999999e142 or 1.65e99 < d

   1. Initial program 23.2%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 79.7

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

   5. Applied rewrites 79.7%

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

   if -2.3999999999999999e142 < d < -3.89999999999999991

   1. Initial program 81.0%

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

   3. Taylor expanded in c around 0

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{d}^{2}}} \]
   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 62.8

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

   5. Applied rewrites 62.8%

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

   if -3.89999999999999991 < d < 1.65e99

   1. Initial program 74.6%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. 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 79.0

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

   5. Applied rewrites 79.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.4 \cdot 10^{+142}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -3.9:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{d \cdot d}\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{+99}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.8 \cdot 10^{+30}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.92 \cdot 10^{-140}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{+92}:\\ \;\;\;\;\frac{c \cdot b}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.8e+30)
   (/ b c)
   (if (<= c 1.92e-140)
     (/ (- a) d)
     (if (<= c 2.6e+92) (/ (* c b) (fma d d (* c c))) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.8e+30) {
		tmp = b / c;
	} else if (c <= 1.92e-140) {
		tmp = -a / d;
	} else if (c <= 2.6e+92) {
		tmp = (c * b) / fma(d, d, (c * c));
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.8e+30)
		tmp = Float64(b / c);
	elseif (c <= 1.92e-140)
		tmp = Float64(Float64(-a) / d);
	elseif (c <= 2.6e+92)
		tmp = Float64(Float64(c * b) / fma(d, d, Float64(c * c)));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.8e+30], N[(b / c), $MachinePrecision], If[LessEqual[c, 1.92e-140], N[((-a) / d), $MachinePrecision], If[LessEqual[c, 2.6e+92], N[(N[(c * b), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.8000000000000001e30 or 2.5999999999999999e92 < c

   1. Initial program 44.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 70.9

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

   5. Applied rewrites 70.9%

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

   if -1.8000000000000001e30 < c < 1.9200000000000001e-140

   1. Initial program 64.3%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 67.9

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

   5. Applied rewrites 67.9%

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

   if 1.9200000000000001e-140 < c < 2.5999999999999999e92

   1. Initial program 82.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 65.5

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

   5. Applied rewrites 65.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 65.5

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

   7. Applied rewrites 65.5%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.8 \cdot 10^{+30}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.92 \cdot 10^{-140}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{+92}:\\ \;\;\;\;\frac{c \cdot b}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.9:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+120}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -3.9)
   (/ (fma c (/ b d) (- a)) d)
   (if (<= d 7.5e+120) (/ (- b (/ (* a d) c)) c) (/ (fma (/ c d) b (- a)) d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -3.9) {
		tmp = fma(c, (b / d), -a) / d;
	} else if (d <= 7.5e+120) {
		tmp = (b - ((a * d) / c)) / c;
	} else {
		tmp = fma((c / d), b, -a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -3.9)
		tmp = Float64(fma(c, Float64(b / d), Float64(-a)) / d);
	elseif (d <= 7.5e+120)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	else
		tmp = Float64(fma(Float64(c / d), b, Float64(-a)) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -3.9], N[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 7.5e+120], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(c / d), $MachinePrecision] * b + (-a)), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.89999999999999991

   1. Initial program 43.8%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 76.6

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

   5. Applied rewrites 76.6%

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

   7. Applied rewrites 82.5%

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

   if -3.89999999999999991 < d < 7.5000000000000006e120

   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 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 79.2

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

   5. Applied rewrites 79.2%

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

   if 7.5000000000000006e120 < d

   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

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

      1. lower-/.f64 13.3

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

   5. Applied rewrites 13.3%

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

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]
   7. 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. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 94.1

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

   8. Applied rewrites 94.1%

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

Alternative 10: 77.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{if}\;d \leq -3.9:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{+98}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma c (/ b d) (- a)) d)))
   (if (<= d -3.9) t_0 (if (<= d 7.2e+98) (/ (- b (/ (* a d) c)) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(c, (b / d), -a) / d;
	double tmp;
	if (d <= -3.9) {
		tmp = t_0;
	} else if (d <= 7.2e+98) {
		tmp = (b - ((a * d) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(c, Float64(b / d), Float64(-a)) / d)
	tmp = 0.0
	if (d <= -3.9)
		tmp = t_0;
	elseif (d <= 7.2e+98)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -3.9], t$95$0, If[LessEqual[d, 7.2e+98], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -3.89999999999999991 or 7.19999999999999962e98 < 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

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 80.5

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

   5. Applied rewrites 80.5%

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

   7. Applied rewrites 86.4%

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

   if -3.89999999999999991 < d < 7.19999999999999962e98

   1. Initial program 74.6%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. 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 79.0

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

   5. Applied rewrites 79.0%

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

Alternative 11: 74.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{if}\;d \leq -3.9:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+120}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (/ (* c b) d) a) d)))
   (if (<= d -3.9) t_0 (if (<= d 7.5e+120) (/ (- b (/ (* a d) c)) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (((c * b) / d) - a) / d;
	double tmp;
	if (d <= -3.9) {
		tmp = t_0;
	} else if (d <= 7.5e+120) {
		tmp = (b - ((a * d) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((c * b) / d) - a) / d
    if (d <= (-3.9d0)) then
        tmp = t_0
    else if (d <= 7.5d+120) then
        tmp = (b - ((a * d) / c)) / c
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (((c * b) / d) - a) / d;
	double tmp;
	if (d <= -3.9) {
		tmp = t_0;
	} else if (d <= 7.5e+120) {
		tmp = (b - ((a * d) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (((c * b) / d) - a) / d
	tmp = 0
	if d <= -3.9:
		tmp = t_0
	elif d <= 7.5e+120:
		tmp = (b - ((a * d) / c)) / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(Float64(c * b) / d) - a) / d)
	tmp = 0.0
	if (d <= -3.9)
		tmp = t_0;
	elseif (d <= 7.5e+120)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (((c * b) / d) - a) / d;
	tmp = 0.0;
	if (d <= -3.9)
		tmp = t_0;
	elseif (d <= 7.5e+120)
		tmp = (b - ((a * d) / c)) / c;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -3.9], t$95$0, If[LessEqual[d, 7.5e+120], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -3.89999999999999991 or 7.5000000000000006e120 < d

   1. Initial program 37.3%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 80.4

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

   5. Applied rewrites 80.4%

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

   if -3.89999999999999991 < d < 7.5000000000000006e120

   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 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 79.2

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

   5. Applied rewrites 79.2%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.9:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+120}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -3.1 \cdot 10^{+21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{+99}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -3.1e+21) t_0 (if (<= d 1.65e+99) (/ b c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -3.1e+21) {
		tmp = t_0;
	} else if (d <= 1.65e+99) {
		tmp = b / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -a / d
    if (d <= (-3.1d+21)) then
        tmp = t_0
    else if (d <= 1.65d+99) then
        tmp = b / c
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -3.1e+21) {
		tmp = t_0;
	} else if (d <= 1.65e+99) {
		tmp = b / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = -a / d
	tmp = 0
	if d <= -3.1e+21:
		tmp = t_0
	elif d <= 1.65e+99:
		tmp = b / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (d <= -3.1e+21)
		tmp = t_0;
	elseif (d <= 1.65e+99)
		tmp = Float64(b / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = -a / d;
	tmp = 0.0;
	if (d <= -3.1e+21)
		tmp = t_0;
	elseif (d <= 1.65e+99)
		tmp = b / c;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, If[LessEqual[d, -3.1e+21], t$95$0, If[LessEqual[d, 1.65e+99], N[(b / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -3.1e21 or 1.65e99 < d

   1. Initial program 35.7%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 71.9

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

   5. Applied rewrites 71.9%

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

   if -3.1e21 < d < 1.65e99

   1. Initial program 74.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 63.6

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

   5. Applied rewrites 63.6%

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

Alternative 13: 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 60.0%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 45.0

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

5. Applied rewrites 45.0%

   \[\leadsto \color{blue}{\frac{b}{c}} \]
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 2024240 
(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))))