Complex division, imag part

Percentage Accurate: 61.4% → 83.2%

Time: 8.1s

Alternatives: 16

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \mathsf{fma}\left(\frac{c}{t\_0}, b, \frac{a}{t\_0} \cdot \left(-d\right)\right)\\ \mathbf{if}\;c \leq -3.1 \cdot 10^{+125}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{elif}\;c \leq -1.1 \cdot 10^{+108}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, \frac{a}{d} \cdot \frac{c \cdot c}{d}\right) - a}{d}\\ \mathbf{elif}\;c \leq -1.05 \cdot 10^{-104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 1.85 \cdot 10^{-148}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 4.4 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-a}{c}, \frac{d}{c}, \frac{b}{c}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))) (t_1 (fma (/ c t_0) b (* (/ a t_0) (- d)))))
   (if (<= c -3.1e+125)
     (/ (- b (* (/ a c) d)) c)
     (if (<= c -1.1e+108)
       (/ (- (fma (/ c d) b (* (/ a d) (/ (* c c) d))) a) d)
       (if (<= c -1.05e-104)
         t_1
         (if (<= c 1.85e-148)
           (/ (- (/ (* b c) d) a) d)
           (if (<= c 4.4e+91) t_1 (fma (/ (- a) c) (/ d c) (/ b c)))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double t_1 = fma((c / t_0), b, ((a / t_0) * -d));
	double tmp;
	if (c <= -3.1e+125) {
		tmp = (b - ((a / c) * d)) / c;
	} else if (c <= -1.1e+108) {
		tmp = (fma((c / d), b, ((a / d) * ((c * c) / d))) - a) / d;
	} else if (c <= -1.05e-104) {
		tmp = t_1;
	} else if (c <= 1.85e-148) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 4.4e+91) {
		tmp = t_1;
	} else {
		tmp = fma((-a / c), (d / c), (b / c));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = fma(Float64(c / t_0), b, Float64(Float64(a / t_0) * Float64(-d)))
	tmp = 0.0
	if (c <= -3.1e+125)
		tmp = Float64(Float64(b - Float64(Float64(a / c) * d)) / c);
	elseif (c <= -1.1e+108)
		tmp = Float64(Float64(fma(Float64(c / d), b, Float64(Float64(a / d) * Float64(Float64(c * c) / d))) - a) / d);
	elseif (c <= -1.05e-104)
		tmp = t_1;
	elseif (c <= 1.85e-148)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 4.4e+91)
		tmp = t_1;
	else
		tmp = fma(Float64(Float64(-a) / c), Float64(d / c), Float64(b / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c / t$95$0), $MachinePrecision] * b + N[(N[(a / t$95$0), $MachinePrecision] * (-d)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -3.1e+125], N[(N[(b - N[(N[(a / c), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -1.1e+108], N[(N[(N[(N[(c / d), $MachinePrecision] * b + N[(N[(a / d), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, -1.05e-104], t$95$1, If[LessEqual[c, 1.85e-148], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 4.4e+91], t$95$1, N[(N[((-a) / c), $MachinePrecision] * N[(d / c), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -3.1e125

   1. Initial program 47.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. remove-double-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 47.0

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 47.0

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

   4. Applied rewrites 47.0%

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

   5. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 95.1

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

   7. Applied rewrites 95.1%

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

   8. Taylor expanded in c around inf

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

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 95.1

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

   10. Applied rewrites 95.1%

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

   if -3.1e125 < c < -1.1000000000000001e108

   1. Initial program 12.9%

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

      1. lift-/.f64 N/A

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

      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. frac-sub N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 0.3%

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

   5. Taylor expanded in d around -inf

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

   7. Applied rewrites 87.4%

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

   if -1.1000000000000001e108 < c < -1.04999999999999999e-104 or 1.85000000000000017e-148 < c < 4.39999999999999999e91

   1. Initial program 77.0%

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

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

   4. Applied rewrites 83.7%

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

   if -1.04999999999999999e-104 < c < 1.85000000000000017e-148

   1. Initial program 66.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 94.3

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

   5. Applied rewrites 94.3%

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

   if 4.39999999999999999e91 < c

   1. Initial program 37.1%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. remove-double-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 37.2

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 37.2

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

   4. Applied rewrites 37.2%

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

   5. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 91.0

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

   7. Applied rewrites 91.0%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.1 \cdot 10^{+125}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{elif}\;c \leq -1.1 \cdot 10^{+108}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, \frac{a}{d} \cdot \frac{c \cdot c}{d}\right) - a}{d}\\ \mathbf{elif}\;c \leq -1.05 \cdot 10^{-104}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}, b, \frac{a}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot \left(-d\right)\right)\\ \mathbf{elif}\;c \leq 1.85 \cdot 10^{-148}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 4.4 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}, b, \frac{a}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot \left(-d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-a}{c}, \frac{d}{c}, \frac{b}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \mathsf{fma}\left(\frac{c}{t\_0}, b, \frac{a}{t\_0} \cdot \left(-d\right)\right)\\ \mathbf{if}\;c \leq -3.1 \cdot 10^{+125}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{elif}\;c \leq -1.1 \cdot 10^{+108}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \mathbf{elif}\;c \leq -1.05 \cdot 10^{-104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 1.85 \cdot 10^{-148}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 4.4 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-a}{c}, \frac{d}{c}, \frac{b}{c}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))) (t_1 (fma (/ c t_0) b (* (/ a t_0) (- d)))))
   (if (<= c -3.1e+125)
     (/ (- b (* (/ a c) d)) c)
     (if (<= c -1.1e+108)
       (/ (fma (/ b d) c (- a)) d)
       (if (<= c -1.05e-104)
         t_1
         (if (<= c 1.85e-148)
           (/ (- (/ (* b c) d) a) d)
           (if (<= c 4.4e+91) t_1 (fma (/ (- a) c) (/ d c) (/ b c)))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double t_1 = fma((c / t_0), b, ((a / t_0) * -d));
	double tmp;
	if (c <= -3.1e+125) {
		tmp = (b - ((a / c) * d)) / c;
	} else if (c <= -1.1e+108) {
		tmp = fma((b / d), c, -a) / d;
	} else if (c <= -1.05e-104) {
		tmp = t_1;
	} else if (c <= 1.85e-148) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 4.4e+91) {
		tmp = t_1;
	} else {
		tmp = fma((-a / c), (d / c), (b / c));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = fma(Float64(c / t_0), b, Float64(Float64(a / t_0) * Float64(-d)))
	tmp = 0.0
	if (c <= -3.1e+125)
		tmp = Float64(Float64(b - Float64(Float64(a / c) * d)) / c);
	elseif (c <= -1.1e+108)
		tmp = Float64(fma(Float64(b / d), c, Float64(-a)) / d);
	elseif (c <= -1.05e-104)
		tmp = t_1;
	elseif (c <= 1.85e-148)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 4.4e+91)
		tmp = t_1;
	else
		tmp = fma(Float64(Float64(-a) / c), Float64(d / c), Float64(b / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c / t$95$0), $MachinePrecision] * b + N[(N[(a / t$95$0), $MachinePrecision] * (-d)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -3.1e+125], N[(N[(b - N[(N[(a / c), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -1.1e+108], N[(N[(N[(b / d), $MachinePrecision] * c + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, -1.05e-104], t$95$1, If[LessEqual[c, 1.85e-148], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 4.4e+91], t$95$1, N[(N[((-a) / c), $MachinePrecision] * N[(d / c), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -3.1e125

   1. Initial program 47.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. remove-double-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 47.0

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 47.0

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

   4. Applied rewrites 47.0%

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

   5. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 95.1

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

   7. Applied rewrites 95.1%

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

   8. Taylor expanded in c around inf

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

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 95.1

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

   10. Applied rewrites 95.1%

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

   if -3.1e125 < c < -1.1000000000000001e108

   1. Initial program 12.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. remove-double-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 13.0

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 13.0

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

   4. Applied rewrites 13.0%

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

   5. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 17.0

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

   7. Applied rewrites 17.0%

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

   8. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. associate-*l/ N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 87.4

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

   10. Applied rewrites 87.4%

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

   if -1.1000000000000001e108 < c < -1.04999999999999999e-104 or 1.85000000000000017e-148 < c < 4.39999999999999999e91

   1. Initial program 77.0%

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

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

   4. Applied rewrites 83.7%

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

   if -1.04999999999999999e-104 < c < 1.85000000000000017e-148

   1. Initial program 66.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 94.3

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

   5. Applied rewrites 94.3%

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

   if 4.39999999999999999e91 < c

   1. Initial program 37.1%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. remove-double-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 37.2

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 37.2

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

   4. Applied rewrites 37.2%

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

   5. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 91.0

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

   7. Applied rewrites 91.0%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.1 \cdot 10^{+125}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{elif}\;c \leq -1.1 \cdot 10^{+108}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \mathbf{elif}\;c \leq -1.05 \cdot 10^{-104}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}, b, \frac{a}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot \left(-d\right)\right)\\ \mathbf{elif}\;c \leq 1.85 \cdot 10^{-148}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 4.4 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}, b, \frac{a}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot \left(-d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-a}{c}, \frac{d}{c}, \frac{b}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.1 \cdot 10^{+125}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{elif}\;c \leq -1.1 \cdot 10^{+108}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \mathbf{elif}\;c \leq -6.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot \mathsf{fma}\left(-b, c, d \cdot a\right)\\ \mathbf{elif}\;c \leq 1.35 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{+85}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-a}{c}, \frac{d}{c}, \frac{b}{c}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3.1e+125)
   (/ (- b (* (/ a c) d)) c)
   (if (<= c -1.1e+108)
     (/ (fma (/ b d) c (- a)) d)
     (if (<= c -6.5e-80)
       (* (/ -1.0 (fma d d (* c c))) (fma (- b) c (* d a)))
       (if (<= c 1.35e-71)
         (/ (- (/ (* b c) d) a) d)
         (if (<= c 2.5e+85)
           (/ (- (* b c) (* d a)) (+ (* d d) (* c c)))
           (fma (/ (- a) c) (/ d c) (/ b c))))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.1e+125) {
		tmp = (b - ((a / c) * d)) / c;
	} else if (c <= -1.1e+108) {
		tmp = fma((b / d), c, -a) / d;
	} else if (c <= -6.5e-80) {
		tmp = (-1.0 / fma(d, d, (c * c))) * fma(-b, c, (d * a));
	} else if (c <= 1.35e-71) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 2.5e+85) {
		tmp = ((b * c) - (d * a)) / ((d * d) + (c * c));
	} else {
		tmp = fma((-a / c), (d / c), (b / c));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3.1e+125)
		tmp = Float64(Float64(b - Float64(Float64(a / c) * d)) / c);
	elseif (c <= -1.1e+108)
		tmp = Float64(fma(Float64(b / d), c, Float64(-a)) / d);
	elseif (c <= -6.5e-80)
		tmp = Float64(Float64(-1.0 / fma(d, d, Float64(c * c))) * fma(Float64(-b), c, Float64(d * a)));
	elseif (c <= 1.35e-71)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 2.5e+85)
		tmp = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(Float64(d * d) + Float64(c * c)));
	else
		tmp = fma(Float64(Float64(-a) / c), Float64(d / c), Float64(b / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3.1e+125], N[(N[(b - N[(N[(a / c), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -1.1e+108], N[(N[(N[(b / d), $MachinePrecision] * c + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, -6.5e-80], N[(N[(-1.0 / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[((-b) * c + N[(d * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.35e-71], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 2.5e+85], N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-a) / c), $MachinePrecision] * N[(d / c), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if c < -3.1e125

   1. Initial program 47.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. remove-double-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 47.0

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 47.0

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

   4. Applied rewrites 47.0%

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

   5. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 95.1

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

   7. Applied rewrites 95.1%

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

   8. Taylor expanded in c around inf

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

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 95.1

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

   10. Applied rewrites 95.1%

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

   if -3.1e125 < c < -1.1000000000000001e108

   1. Initial program 12.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. remove-double-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 13.0

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 13.0

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

   4. Applied rewrites 13.0%

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

   5. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 17.0

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

   7. Applied rewrites 17.0%

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

   8. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. associate-*l/ N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 87.4

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

   10. Applied rewrites 87.4%

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

   if -1.1000000000000001e108 < c < -6.49999999999999984e-80

   1. Initial program 75.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. remove-double-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 75.0

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 75.0

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

   4. Applied rewrites 75.0%

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

   if -6.49999999999999984e-80 < c < 1.3500000000000001e-71

   1. Initial program 66.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 90.8

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

   5. Applied rewrites 90.8%

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

   if 1.3500000000000001e-71 < c < 2.5e85

   1. Initial program 84.8%

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

   if 2.5e85 < c

   1. Initial program 36.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. remove-double-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 36.4

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 36.4

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

   4. Applied rewrites 36.4%

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

   5. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 89.0

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

   7. Applied rewrites 89.0%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.1 \cdot 10^{+125}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{elif}\;c \leq -1.1 \cdot 10^{+108}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \mathbf{elif}\;c \leq -6.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot \mathsf{fma}\left(-b, c, d \cdot a\right)\\ \mathbf{elif}\;c \leq 1.35 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{+85}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-a}{c}, \frac{d}{c}, \frac{b}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.1 \cdot 10^{+125}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{elif}\;c \leq -1.1 \cdot 10^{+108}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \mathbf{elif}\;c \leq -6.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot \mathsf{fma}\left(-b, c, d \cdot a\right)\\ \mathbf{elif}\;c \leq 1.35 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{+85}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{\frac{d}{c}}{c}, \frac{b}{c}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3.1e+125)
   (/ (- b (* (/ a c) d)) c)
   (if (<= c -1.1e+108)
     (/ (fma (/ b d) c (- a)) d)
     (if (<= c -6.5e-80)
       (* (/ -1.0 (fma d d (* c c))) (fma (- b) c (* d a)))
       (if (<= c 1.35e-71)
         (/ (- (/ (* b c) d) a) d)
         (if (<= c 2.5e+85)
           (/ (- (* b c) (* d a)) (+ (* d d) (* c c)))
           (fma (- a) (/ (/ d c) c) (/ b c))))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.1e+125) {
		tmp = (b - ((a / c) * d)) / c;
	} else if (c <= -1.1e+108) {
		tmp = fma((b / d), c, -a) / d;
	} else if (c <= -6.5e-80) {
		tmp = (-1.0 / fma(d, d, (c * c))) * fma(-b, c, (d * a));
	} else if (c <= 1.35e-71) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 2.5e+85) {
		tmp = ((b * c) - (d * a)) / ((d * d) + (c * c));
	} else {
		tmp = fma(-a, ((d / c) / c), (b / c));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3.1e+125)
		tmp = Float64(Float64(b - Float64(Float64(a / c) * d)) / c);
	elseif (c <= -1.1e+108)
		tmp = Float64(fma(Float64(b / d), c, Float64(-a)) / d);
	elseif (c <= -6.5e-80)
		tmp = Float64(Float64(-1.0 / fma(d, d, Float64(c * c))) * fma(Float64(-b), c, Float64(d * a)));
	elseif (c <= 1.35e-71)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 2.5e+85)
		tmp = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(Float64(d * d) + Float64(c * c)));
	else
		tmp = fma(Float64(-a), Float64(Float64(d / c) / c), Float64(b / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3.1e+125], N[(N[(b - N[(N[(a / c), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -1.1e+108], N[(N[(N[(b / d), $MachinePrecision] * c + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, -6.5e-80], N[(N[(-1.0 / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[((-b) * c + N[(d * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.35e-71], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 2.5e+85], N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-a) * N[(N[(d / c), $MachinePrecision] / c), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if c < -3.1e125

   1. Initial program 47.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. remove-double-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 47.0

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 47.0

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

   4. Applied rewrites 47.0%

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

   5. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 95.1

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

   7. Applied rewrites 95.1%

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

   8. Taylor expanded in c around inf

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

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 95.1

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

   10. Applied rewrites 95.1%

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

   if -3.1e125 < c < -1.1000000000000001e108

   1. Initial program 12.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. remove-double-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 13.0

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 13.0

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

   4. Applied rewrites 13.0%

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

   5. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 17.0

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

   7. Applied rewrites 17.0%

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

   8. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. associate-*l/ N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 87.4

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

   10. Applied rewrites 87.4%

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

   if -1.1000000000000001e108 < c < -6.49999999999999984e-80

   1. Initial program 75.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. remove-double-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 75.0

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 75.0

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

   4. Applied rewrites 75.0%

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

   if -6.49999999999999984e-80 < c < 1.3500000000000001e-71

   1. Initial program 66.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 90.8

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

   5. Applied rewrites 90.8%

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

   if 1.3500000000000001e-71 < c < 2.5e85

   1. Initial program 84.8%

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

   if 2.5e85 < c

   1. Initial program 36.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. remove-double-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 36.4

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 36.4

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

   4. Applied rewrites 36.4%

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

   5. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 89.0

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

   7. Applied rewrites 89.0%

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

   9. Applied rewrites 89.0%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.1 \cdot 10^{+125}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{elif}\;c \leq -1.1 \cdot 10^{+108}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \mathbf{elif}\;c \leq -6.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot \mathsf{fma}\left(-b, c, d \cdot a\right)\\ \mathbf{elif}\;c \leq 1.35 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{+85}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{\frac{d}{c}}{c}, \frac{b}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{if}\;c \leq -3.1 \cdot 10^{+125}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -1.1 \cdot 10^{+108}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \mathbf{elif}\;c \leq -6.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot \mathsf{fma}\left(-b, c, d \cdot a\right)\\ \mathbf{elif}\;c \leq 1.35 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{+85}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (* (/ a c) d)) c)))
   (if (<= c -3.1e+125)
     t_0
     (if (<= c -1.1e+108)
       (/ (fma (/ b d) c (- a)) d)
       (if (<= c -6.5e-80)
         (* (/ -1.0 (fma d d (* c c))) (fma (- b) c (* d a)))
         (if (<= c 1.35e-71)
           (/ (- (/ (* b c) d) a) d)
           (if (<= c 2.5e+85)
             (/ (- (* b c) (* d a)) (+ (* d d) (* c c)))
             t_0)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - ((a / c) * d)) / c;
	double tmp;
	if (c <= -3.1e+125) {
		tmp = t_0;
	} else if (c <= -1.1e+108) {
		tmp = fma((b / d), c, -a) / d;
	} else if (c <= -6.5e-80) {
		tmp = (-1.0 / fma(d, d, (c * c))) * fma(-b, c, (d * a));
	} else if (c <= 1.35e-71) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 2.5e+85) {
		tmp = ((b * c) - (d * a)) / ((d * d) + (c * c));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(Float64(a / c) * d)) / c)
	tmp = 0.0
	if (c <= -3.1e+125)
		tmp = t_0;
	elseif (c <= -1.1e+108)
		tmp = Float64(fma(Float64(b / d), c, Float64(-a)) / d);
	elseif (c <= -6.5e-80)
		tmp = Float64(Float64(-1.0 / fma(d, d, Float64(c * c))) * fma(Float64(-b), c, Float64(d * a)));
	elseif (c <= 1.35e-71)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 2.5e+85)
		tmp = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(Float64(d * d) + Float64(c * c)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(N[(a / c), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -3.1e+125], t$95$0, If[LessEqual[c, -1.1e+108], N[(N[(N[(b / d), $MachinePrecision] * c + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, -6.5e-80], N[(N[(-1.0 / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[((-b) * c + N[(d * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.35e-71], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 2.5e+85], N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -3.1e125 or 2.5e85 < c

   1. Initial program 41.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. remove-double-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 41.5

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 41.5

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

   4. Applied rewrites 41.5%

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

   5. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 91.9

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

   7. Applied rewrites 91.9%

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

   8. Taylor expanded in c around inf

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

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 91.9

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

   10. Applied rewrites 91.9%

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

   if -3.1e125 < c < -1.1000000000000001e108

   1. Initial program 12.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. remove-double-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 13.0

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 13.0

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

   4. Applied rewrites 13.0%

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

   5. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 17.0

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

   7. Applied rewrites 17.0%

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

   8. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. associate-*l/ N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 87.4

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

   10. Applied rewrites 87.4%

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

   if -1.1000000000000001e108 < c < -6.49999999999999984e-80

   1. Initial program 75.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. remove-double-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 75.0

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 75.0

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

   4. Applied rewrites 75.0%

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

   if -6.49999999999999984e-80 < c < 1.3500000000000001e-71

   1. Initial program 66.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 90.8

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

   5. Applied rewrites 90.8%

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

   if 1.3500000000000001e-71 < c < 2.5e85

   1. Initial program 84.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
3. Recombined 5 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.1 \cdot 10^{+125}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{elif}\;c \leq -1.1 \cdot 10^{+108}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \mathbf{elif}\;c \leq -6.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot \mathsf{fma}\left(-b, c, d \cdot a\right)\\ \mathbf{elif}\;c \leq 1.35 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{+85}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - \frac{a}{c} \cdot d}{c}\\ t_1 := \frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{if}\;c \leq -3.1 \cdot 10^{+125}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -1.1 \cdot 10^{+108}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \mathbf{elif}\;c \leq -6.5 \cdot 10^{-80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 1.35 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{+85}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (* (/ a c) d)) c))
        (t_1 (/ (- (* b c) (* d a)) (+ (* d d) (* c c)))))
   (if (<= c -3.1e+125)
     t_0
     (if (<= c -1.1e+108)
       (/ (fma (/ b d) c (- a)) d)
       (if (<= c -6.5e-80)
         t_1
         (if (<= c 1.35e-71)
           (/ (- (/ (* b c) d) a) d)
           (if (<= c 2.5e+85) t_1 t_0)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - ((a / c) * d)) / c;
	double t_1 = ((b * c) - (d * a)) / ((d * d) + (c * c));
	double tmp;
	if (c <= -3.1e+125) {
		tmp = t_0;
	} else if (c <= -1.1e+108) {
		tmp = fma((b / d), c, -a) / d;
	} else if (c <= -6.5e-80) {
		tmp = t_1;
	} else if (c <= 1.35e-71) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 2.5e+85) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(Float64(a / c) * d)) / c)
	t_1 = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(Float64(d * d) + Float64(c * c)))
	tmp = 0.0
	if (c <= -3.1e+125)
		tmp = t_0;
	elseif (c <= -1.1e+108)
		tmp = Float64(fma(Float64(b / d), c, Float64(-a)) / d);
	elseif (c <= -6.5e-80)
		tmp = t_1;
	elseif (c <= 1.35e-71)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 2.5e+85)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(N[(a / c), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -3.1e+125], t$95$0, If[LessEqual[c, -1.1e+108], N[(N[(N[(b / d), $MachinePrecision] * c + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, -6.5e-80], t$95$1, If[LessEqual[c, 1.35e-71], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 2.5e+85], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -3.1e125 or 2.5e85 < c

   1. Initial program 41.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. remove-double-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 41.5

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 41.5

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

   4. Applied rewrites 41.5%

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

   5. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 91.9

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

   7. Applied rewrites 91.9%

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

   8. Taylor expanded in c around inf

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

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 91.9

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

   10. Applied rewrites 91.9%

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

   if -3.1e125 < c < -1.1000000000000001e108

   1. Initial program 12.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. remove-double-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 13.0

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 13.0

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

   4. Applied rewrites 13.0%

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

   5. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 17.0

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

   7. Applied rewrites 17.0%

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

   8. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. associate-*l/ N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 87.4

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

   10. Applied rewrites 87.4%

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

   if -1.1000000000000001e108 < c < -6.49999999999999984e-80 or 1.3500000000000001e-71 < c < 2.5e85

   1. Initial program 79.8%

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

   if -6.49999999999999984e-80 < c < 1.3500000000000001e-71

   1. Initial program 66.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 90.8

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

   5. Applied rewrites 90.8%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.1 \cdot 10^{+125}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{elif}\;c \leq -1.1 \cdot 10^{+108}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \mathbf{elif}\;c \leq -6.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;c \leq 1.35 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{+85}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -8.5 \cdot 10^{+170}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -1.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{d}{t\_0} \cdot \left(-a\right)\\ \mathbf{elif}\;d \leq 1.52 \cdot 10^{-64}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{+73}:\\ \;\;\;\;\frac{\left(-a\right) \cdot d}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))) (t_1 (/ (- a) d)))
   (if (<= d -8.5e+170)
     t_1
     (if (<= d -1.2e-80)
       (* (/ d t_0) (- a))
       (if (<= d 1.52e-64)
         (/ b c)
         (if (<= d 3.6e+73) (/ (* (- a) d) t_0) t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double t_1 = -a / d;
	double tmp;
	if (d <= -8.5e+170) {
		tmp = t_1;
	} else if (d <= -1.2e-80) {
		tmp = (d / t_0) * -a;
	} else if (d <= 1.52e-64) {
		tmp = b / c;
	} else if (d <= 3.6e+73) {
		tmp = (-a * d) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (d <= -8.5e+170)
		tmp = t_1;
	elseif (d <= -1.2e-80)
		tmp = Float64(Float64(d / t_0) * Float64(-a));
	elseif (d <= 1.52e-64)
		tmp = Float64(b / c);
	elseif (d <= 3.6e+73)
		tmp = Float64(Float64(Float64(-a) * d) / t_0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-a) / d), $MachinePrecision]}, If[LessEqual[d, -8.5e+170], t$95$1, If[LessEqual[d, -1.2e-80], N[(N[(d / t$95$0), $MachinePrecision] * (-a)), $MachinePrecision], If[LessEqual[d, 1.52e-64], N[(b / c), $MachinePrecision], If[LessEqual[d, 3.6e+73], N[(N[((-a) * d), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -8.5000000000000004e170 or 3.5999999999999999e73 < d

   1. Initial program 33.9%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 68.8

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

   5. Applied rewrites 68.8%

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

   if -8.5000000000000004e170 < d < -1.2e-80

   1. Initial program 65.9%

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

   3. Taylor expanded in a around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 59.9

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

   5. Applied rewrites 59.9%

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

   if -1.2e-80 < d < 1.5200000000000001e-64

   1. Initial program 75.7%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 78.6

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

   5. Applied rewrites 78.6%

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

   if 1.5200000000000001e-64 < d < 3.5999999999999999e73

   1. Initial program 85.0%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 63.9

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

   5. Applied rewrites 63.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 63.9

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

   7. Applied rewrites 63.9%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.5 \cdot 10^{+170}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -1.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot \left(-a\right)\\ \mathbf{elif}\;d \leq 1.52 \cdot 10^{-64}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{+73}:\\ \;\;\;\;\frac{\left(-a\right) \cdot d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ t_1 := \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot \left(-a\right)\\ \mathbf{if}\;d \leq -8.5 \cdot 10^{+170}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -1.2 \cdot 10^{-80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 1.52 \cdot 10^{-64}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{+77}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)) (t_1 (* (/ d (fma d d (* c c))) (- a))))
   (if (<= d -8.5e+170)
     t_0
     (if (<= d -1.2e-80)
       t_1
       (if (<= d 1.52e-64) (/ b c) (if (<= d 3.8e+77) t_1 t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double t_1 = (d / fma(d, d, (c * c))) * -a;
	double tmp;
	if (d <= -8.5e+170) {
		tmp = t_0;
	} else if (d <= -1.2e-80) {
		tmp = t_1;
	} else if (d <= 1.52e-64) {
		tmp = b / c;
	} else if (d <= 3.8e+77) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	t_1 = Float64(Float64(d / fma(d, d, Float64(c * c))) * Float64(-a))
	tmp = 0.0
	if (d <= -8.5e+170)
		tmp = t_0;
	elseif (d <= -1.2e-80)
		tmp = t_1;
	elseif (d <= 1.52e-64)
		tmp = Float64(b / c);
	elseif (d <= 3.8e+77)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(d / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision]}, If[LessEqual[d, -8.5e+170], t$95$0, If[LessEqual[d, -1.2e-80], t$95$1, If[LessEqual[d, 1.52e-64], N[(b / c), $MachinePrecision], If[LessEqual[d, 3.8e+77], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -8.5000000000000004e170 or 3.8000000000000001e77 < d

   1. Initial program 34.7%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 69.2

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

   5. Applied rewrites 69.2%

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

   if -8.5000000000000004e170 < d < -1.2e-80 or 1.5200000000000001e-64 < d < 3.8000000000000001e77

   1. Initial program 71.1%

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

   3. Taylor expanded in a around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 61.0

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

   5. Applied rewrites 61.0%

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

   if -1.2e-80 < d < 1.5200000000000001e-64

   1. Initial program 75.7%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 78.6

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

   5. Applied rewrites 78.6%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.5 \cdot 10^{+170}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -1.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot \left(-a\right)\\ \mathbf{elif}\;d \leq 1.52 \cdot 10^{-64}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot \left(-a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.1 \cdot 10^{+46}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.75 \cdot 10^{-70}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 4 \cdot 10^{+83}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.1e+46)
   (/ b c)
   (if (<= c 1.75e-70)
     (/ (- a) d)
     (if (<= c 4e+83) (/ (- (* b c) (* d a)) (* c c)) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.1e+46) {
		tmp = b / c;
	} else if (c <= 1.75e-70) {
		tmp = -a / d;
	} else if (c <= 4e+83) {
		tmp = ((b * c) - (d * a)) / (c * c);
	} else {
		tmp = b / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-1.1d+46)) then
        tmp = b / c
    else if (c <= 1.75d-70) then
        tmp = -a / d
    else if (c <= 4d+83) then
        tmp = ((b * c) - (d * a)) / (c * c)
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.1e+46) {
		tmp = b / c;
	} else if (c <= 1.75e-70) {
		tmp = -a / d;
	} else if (c <= 4e+83) {
		tmp = ((b * c) - (d * a)) / (c * c);
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.1e+46:
		tmp = b / c
	elif c <= 1.75e-70:
		tmp = -a / d
	elif c <= 4e+83:
		tmp = ((b * c) - (d * a)) / (c * c)
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.1e+46)
		tmp = Float64(b / c);
	elseif (c <= 1.75e-70)
		tmp = Float64(Float64(-a) / d);
	elseif (c <= 4e+83)
		tmp = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(c * c));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.1e+46)
		tmp = b / c;
	elseif (c <= 1.75e-70)
		tmp = -a / d;
	elseif (c <= 4e+83)
		tmp = ((b * c) - (d * a)) / (c * c);
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.1e+46], N[(b / c), $MachinePrecision], If[LessEqual[c, 1.75e-70], N[((-a) / d), $MachinePrecision], If[LessEqual[c, 4e+83], N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(b / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.1e46 or 4.00000000000000012e83 < c

   1. Initial program 45.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 75.1

         \[\leadsto \color{blue}{\frac{b}{c}} \]

   5. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\frac{b}{c}} \]

   if -1.1e46 < c < 1.74999999999999987e-70

   1. Initial program 69.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{a}{d}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{a}{\mathsf{neg}\left(d\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{a}{\color{blue}{-1 \cdot d}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{-1 \cdot d}} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{a}{\color{blue}{\mathsf{neg}\left(d\right)}} \]

      6. lower-neg.f64 66.9

         \[\leadsto \frac{a}{\color{blue}{-d}} \]

   5. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\frac{a}{-d}} \]

   if 1.74999999999999987e-70 < c < 4.00000000000000012e83

   1. Initial program 83.6%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{c}^{2}}} \]
   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 68.6

         \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{c \cdot c}} \]

   5. Applied rewrites 68.6%

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{c \cdot c}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.1 \cdot 10^{+46}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.75 \cdot 10^{-70}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 4 \cdot 10^{+83}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.1 \cdot 10^{+46}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.12 \cdot 10^{-68}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.1e+46)
   (/ b c)
   (if (<= c 1.12e-68)
     (/ (- a) d)
     (if (<= c 1.5e+86) (* (/ c (fma d d (* c c))) b) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.1e+46) {
		tmp = b / c;
	} else if (c <= 1.12e-68) {
		tmp = -a / d;
	} else if (c <= 1.5e+86) {
		tmp = (c / fma(d, d, (c * c))) * b;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.1e+46)
		tmp = Float64(b / c);
	elseif (c <= 1.12e-68)
		tmp = Float64(Float64(-a) / d);
	elseif (c <= 1.5e+86)
		tmp = Float64(Float64(c / fma(d, d, Float64(c * c))) * b);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.1e+46], N[(b / c), $MachinePrecision], If[LessEqual[c, 1.12e-68], N[((-a) / d), $MachinePrecision], If[LessEqual[c, 1.5e+86], N[(N[(c / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[(b / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.1e46 or 1.49999999999999988e86 < c

   1. Initial program 43.5%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b}{c}} \]
   4. Step-by-step derivation

      1. lower-/.f64 75.2

         \[\leadsto \color{blue}{\frac{b}{c}} \]

   5. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\frac{b}{c}} \]

   if -1.1e46 < c < 1.11999999999999992e-68

   1. Initial program 69.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{a}{d}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{a}{\mathsf{neg}\left(d\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{a}{\color{blue}{-1 \cdot d}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{-1 \cdot d}} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{a}{\color{blue}{\mathsf{neg}\left(d\right)}} \]

      6. lower-neg.f64 67.2

         \[\leadsto \frac{a}{\color{blue}{-d}} \]

   5. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\frac{a}{-d}} \]

   if 1.11999999999999992e-68 < c < 1.49999999999999988e86

   1. Initial program 84.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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

          \[\leadsto \mathsf{fma}\left(\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}, b, \mathsf{neg}\left(\color{blue}{d \cdot \frac{a}{c \cdot c + d \cdot d}}\right)\right) \]

   4. Applied rewrites 86.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}, b, \left(-d\right) \cdot \frac{a}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{b \cdot c}{{c}^{2} + {d}^{2}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot b}}{{c}^{2} + {d}^{2}} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{c}{{c}^{2} + {d}^{2}} \cdot b} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c}{{c}^{2} + {d}^{2}} \cdot b} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c}{{c}^{2} + {d}^{2}}} \cdot b \]

      5. +-commutative N/A

         \[\leadsto \frac{c}{\color{blue}{{d}^{2} + {c}^{2}}} \cdot b \]

      6. unpow2 N/A

         \[\leadsto \frac{c}{\color{blue}{d \cdot d} + {c}^{2}} \cdot b \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{c}{\color{blue}{\mathsf{fma}\left(d, d, {c}^{2}\right)}} \cdot b \]

      8. unpow2 N/A

         \[\leadsto \frac{c}{\mathsf{fma}\left(d, d, \color{blue}{c \cdot c}\right)} \cdot b \]

      9. lower-*.f64 54.6

         \[\leadsto \frac{c}{\mathsf{fma}\left(d, d, \color{blue}{c \cdot c}\right)} \cdot b \]

   7. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.1 \cdot 10^{+46}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.12 \cdot 10^{-68}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{if}\;c \leq -2.2 \cdot 10^{+40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 7.8 \cdot 10^{-53}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (* (/ a c) d)) c)))
   (if (<= c -2.2e+40) t_0 (if (<= c 7.8e-53) (/ (- (/ (* b c) d) a) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - ((a / c) * d)) / c;
	double tmp;
	if (c <= -2.2e+40) {
		tmp = t_0;
	} else if (c <= 7.8e-53) {
		tmp = (((b * c) / d) - a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b - ((a / c) * d)) / c
    if (c <= (-2.2d+40)) then
        tmp = t_0
    else if (c <= 7.8d-53) then
        tmp = (((b * c) / d) - a) / d
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b - ((a / c) * d)) / c;
	double tmp;
	if (c <= -2.2e+40) {
		tmp = t_0;
	} else if (c <= 7.8e-53) {
		tmp = (((b * c) / d) - a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b - ((a / c) * d)) / c
	tmp = 0
	if c <= -2.2e+40:
		tmp = t_0
	elif c <= 7.8e-53:
		tmp = (((b * c) / d) - a) / d
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(Float64(a / c) * d)) / c)
	tmp = 0.0
	if (c <= -2.2e+40)
		tmp = t_0;
	elseif (c <= 7.8e-53)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b - ((a / c) * d)) / c;
	tmp = 0.0;
	if (c <= -2.2e+40)
		tmp = t_0;
	elseif (c <= 7.8e-53)
		tmp = (((b * c) / d) - a) / d;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(N[(a / c), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -2.2e+40], t$95$0, If[LessEqual[c, 7.8e-53], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -2.1999999999999999e40 or 7.8000000000000004e-53 < c

   1. Initial program 53.5%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(b \cdot c - a \cdot d\right)\right)}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)}} \]

      3. div-inv N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(b \cdot c - a \cdot d\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(b \cdot c - a \cdot d\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)}} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(b \cdot c - a \cdot d\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      6. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(b \cdot c + \left(\mathsf{neg}\left(a \cdot d\right)\right)\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      7. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(b \cdot c\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot d\right)\right)\right)\right)\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{b \cdot c}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot d\right)\right)\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot c} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot d\right)\right)\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      10. remove-double-neg N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(b\right)\right) \cdot c + \color{blue}{a \cdot d}\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(b\right), c, a \cdot d\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-b}, c, a \cdot d\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      13. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \frac{1}{\color{blue}{-1 \cdot \left(c \cdot c + d \cdot d\right)}} \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \color{blue}{\frac{\frac{1}{-1}}{c \cdot c + d \cdot d}} \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \frac{\color{blue}{-1}}{c \cdot c + d \cdot d} \]

      16. lower-/.f64 53.5

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \color{blue}{\frac{-1}{c \cdot c + d \cdot d}} \]

      17. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \frac{-1}{\color{blue}{c \cdot c + d \cdot d}} \]

      18. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \frac{-1}{\color{blue}{d \cdot d + c \cdot c}} \]

      19. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \frac{-1}{\color{blue}{d \cdot d} + c \cdot c} \]

      20. lower-fma.f64 53.5

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \frac{-1}{\color{blue}{\mathsf{fma}\left(d, d, c \cdot c\right)}} \]

   4. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \frac{-1}{\mathsf{fma}\left(d, d, c \cdot c\right)}} \]

   5. Taylor expanded in d around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(a \cdot d\right)}{{c}^{2}}} + \frac{b}{c} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot a\right) \cdot d}}{{c}^{2}} + \frac{b}{c} \]

      3. unpow2 N/A

         \[\leadsto \frac{\left(-1 \cdot a\right) \cdot d}{\color{blue}{c \cdot c}} + \frac{b}{c} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot a}{c} \cdot \frac{d}{c}} + \frac{b}{c} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1 \cdot a}{c}, \frac{d}{c}, \frac{b}{c}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1 \cdot a}{c}}, \frac{d}{c}, \frac{b}{c}\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\mathsf{neg}\left(a\right)}}{c}, \frac{d}{c}, \frac{b}{c}\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-a}}{c}, \frac{d}{c}, \frac{b}{c}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-a}{c}, \color{blue}{\frac{d}{c}}, \frac{b}{c}\right) \]

      10. lower-/.f64 81.0

          \[\leadsto \mathsf{fma}\left(\frac{-a}{c}, \frac{d}{c}, \color{blue}{\frac{b}{c}}\right) \]

   7. Applied rewrites 81.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-a}{c}, \frac{d}{c}, \frac{b}{c}\right)} \]

   8. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   9. 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. *-commutative N/A

         \[\leadsto \frac{b - \frac{\color{blue}{d \cdot a}}{c}}{c} \]

      6. associate-/l* N/A

         \[\leadsto \frac{b - \color{blue}{d \cdot \frac{a}{c}}}{c} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{b - \color{blue}{d \cdot \frac{a}{c}}}{c} \]

      8. lower-/.f64 80.9

         \[\leadsto \frac{b - d \cdot \color{blue}{\frac{a}{c}}}{c} \]

   10. Applied rewrites 80.9%

       \[\leadsto \color{blue}{\frac{b - d \cdot \frac{a}{c}}{c}} \]

   if -2.1999999999999999e40 < c < 7.8000000000000004e-53

   1. Initial program 70.1%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{b \cdot c}{{d}^{2}} + -1 \cdot \frac{a}{d}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{b \cdot c}{{d}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{a}{d}\right)\right)} \]

      3. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{b \cdot c}{{d}^{2}} - \frac{a}{d}} \]

      4. unpow2 N/A

         \[\leadsto \frac{b \cdot c}{\color{blue}{d \cdot d}} - \frac{a}{d} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b \cdot c}{d}}{d}} - \frac{a}{d} \]

      6. div-sub N/A

         \[\leadsto \color{blue}{\frac{\frac{b \cdot c}{d} - a}{d}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{b \cdot c}{d} - a}{d}} \]

      8. 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 84.3

          \[\leadsto \frac{\frac{\color{blue}{b \cdot c}}{d} - a}{d} \]

   5. Applied rewrites 84.3%

      \[\leadsto \color{blue}{\frac{\frac{b \cdot c}{d} - a}{d}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.2 \cdot 10^{+40}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{elif}\;c \leq 7.8 \cdot 10^{-53}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{if}\;c \leq -2.2 \cdot 10^{+40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 7.8 \cdot 10^{-53}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (* (/ a c) d)) c)))
   (if (<= c -2.2e+40)
     t_0
     (if (<= c 7.8e-53) (/ (fma (/ b d) c (- a)) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - ((a / c) * d)) / c;
	double tmp;
	if (c <= -2.2e+40) {
		tmp = t_0;
	} else if (c <= 7.8e-53) {
		tmp = fma((b / d), c, -a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(Float64(a / c) * d)) / c)
	tmp = 0.0
	if (c <= -2.2e+40)
		tmp = t_0;
	elseif (c <= 7.8e-53)
		tmp = Float64(fma(Float64(b / d), c, Float64(-a)) / d);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(N[(a / c), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -2.2e+40], t$95$0, If[LessEqual[c, 7.8e-53], N[(N[(N[(b / d), $MachinePrecision] * c + (-a)), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -2.1999999999999999e40 or 7.8000000000000004e-53 < c

   1. Initial program 53.5%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(b \cdot c - a \cdot d\right)\right)}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)}} \]

      3. div-inv N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(b \cdot c - a \cdot d\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(b \cdot c - a \cdot d\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)}} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(b \cdot c - a \cdot d\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      6. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(b \cdot c + \left(\mathsf{neg}\left(a \cdot d\right)\right)\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      7. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(b \cdot c\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot d\right)\right)\right)\right)\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{b \cdot c}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot d\right)\right)\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot c} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot d\right)\right)\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      10. remove-double-neg N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(b\right)\right) \cdot c + \color{blue}{a \cdot d}\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(b\right), c, a \cdot d\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-b}, c, a \cdot d\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      13. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \frac{1}{\color{blue}{-1 \cdot \left(c \cdot c + d \cdot d\right)}} \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \color{blue}{\frac{\frac{1}{-1}}{c \cdot c + d \cdot d}} \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \frac{\color{blue}{-1}}{c \cdot c + d \cdot d} \]

      16. lower-/.f64 53.5

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \color{blue}{\frac{-1}{c \cdot c + d \cdot d}} \]

      17. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \frac{-1}{\color{blue}{c \cdot c + d \cdot d}} \]

      18. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \frac{-1}{\color{blue}{d \cdot d + c \cdot c}} \]

      19. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \frac{-1}{\color{blue}{d \cdot d} + c \cdot c} \]

      20. lower-fma.f64 53.5

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \frac{-1}{\color{blue}{\mathsf{fma}\left(d, d, c \cdot c\right)}} \]

   4. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \frac{-1}{\mathsf{fma}\left(d, d, c \cdot c\right)}} \]

   5. Taylor expanded in d around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(a \cdot d\right)}{{c}^{2}}} + \frac{b}{c} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot a\right) \cdot d}}{{c}^{2}} + \frac{b}{c} \]

      3. unpow2 N/A

         \[\leadsto \frac{\left(-1 \cdot a\right) \cdot d}{\color{blue}{c \cdot c}} + \frac{b}{c} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot a}{c} \cdot \frac{d}{c}} + \frac{b}{c} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1 \cdot a}{c}, \frac{d}{c}, \frac{b}{c}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1 \cdot a}{c}}, \frac{d}{c}, \frac{b}{c}\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\mathsf{neg}\left(a\right)}}{c}, \frac{d}{c}, \frac{b}{c}\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-a}}{c}, \frac{d}{c}, \frac{b}{c}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-a}{c}, \color{blue}{\frac{d}{c}}, \frac{b}{c}\right) \]

      10. lower-/.f64 81.0

          \[\leadsto \mathsf{fma}\left(\frac{-a}{c}, \frac{d}{c}, \color{blue}{\frac{b}{c}}\right) \]

   7. Applied rewrites 81.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-a}{c}, \frac{d}{c}, \frac{b}{c}\right)} \]

   8. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   9. 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. *-commutative N/A

         \[\leadsto \frac{b - \frac{\color{blue}{d \cdot a}}{c}}{c} \]

      6. associate-/l* N/A

         \[\leadsto \frac{b - \color{blue}{d \cdot \frac{a}{c}}}{c} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{b - \color{blue}{d \cdot \frac{a}{c}}}{c} \]

      8. lower-/.f64 80.9

         \[\leadsto \frac{b - d \cdot \color{blue}{\frac{a}{c}}}{c} \]

   10. Applied rewrites 80.9%

       \[\leadsto \color{blue}{\frac{b - d \cdot \frac{a}{c}}{c}} \]

   if -2.1999999999999999e40 < c < 7.8000000000000004e-53

   1. Initial program 70.1%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(b \cdot c - a \cdot d\right)\right)}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)}} \]

      3. div-inv N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(b \cdot c - a \cdot d\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(b \cdot c - a \cdot d\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)}} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(b \cdot c - a \cdot d\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      6. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(b \cdot c + \left(\mathsf{neg}\left(a \cdot d\right)\right)\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      7. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(b \cdot c\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot d\right)\right)\right)\right)\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{b \cdot c}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot d\right)\right)\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot c} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot d\right)\right)\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      10. remove-double-neg N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(b\right)\right) \cdot c + \color{blue}{a \cdot d}\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(b\right), c, a \cdot d\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-b}, c, a \cdot d\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      13. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \frac{1}{\color{blue}{-1 \cdot \left(c \cdot c + d \cdot d\right)}} \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \color{blue}{\frac{\frac{1}{-1}}{c \cdot c + d \cdot d}} \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \frac{\color{blue}{-1}}{c \cdot c + d \cdot d} \]

      16. lower-/.f64 70.0

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \color{blue}{\frac{-1}{c \cdot c + d \cdot d}} \]

      17. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \frac{-1}{\color{blue}{c \cdot c + d \cdot d}} \]

      18. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \frac{-1}{\color{blue}{d \cdot d + c \cdot c}} \]

      19. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \frac{-1}{\color{blue}{d \cdot d} + c \cdot c} \]

      20. lower-fma.f64 70.0

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \frac{-1}{\color{blue}{\mathsf{fma}\left(d, d, c \cdot c\right)}} \]

   4. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \frac{-1}{\mathsf{fma}\left(d, d, c \cdot c\right)}} \]

   5. Taylor expanded in d around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(a \cdot d\right)}{{c}^{2}}} + \frac{b}{c} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot a\right) \cdot d}}{{c}^{2}} + \frac{b}{c} \]

      3. unpow2 N/A

         \[\leadsto \frac{\left(-1 \cdot a\right) \cdot d}{\color{blue}{c \cdot c}} + \frac{b}{c} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot a}{c} \cdot \frac{d}{c}} + \frac{b}{c} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1 \cdot a}{c}, \frac{d}{c}, \frac{b}{c}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1 \cdot a}{c}}, \frac{d}{c}, \frac{b}{c}\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\mathsf{neg}\left(a\right)}}{c}, \frac{d}{c}, \frac{b}{c}\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-a}}{c}, \frac{d}{c}, \frac{b}{c}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-a}{c}, \color{blue}{\frac{d}{c}}, \frac{b}{c}\right) \]

      10. lower-/.f64 24.5

          \[\leadsto \mathsf{fma}\left(\frac{-a}{c}, \frac{d}{c}, \color{blue}{\frac{b}{c}}\right) \]

   7. Applied rewrites 24.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-a}{c}, \frac{d}{c}, \frac{b}{c}\right)} \]

   8. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{b \cdot c}{{d}^{2}} + -1 \cdot \frac{a}{d}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{b \cdot c}{{d}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{a}{d}\right)\right)} \]

      3. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{b \cdot c}{{d}^{2}} - \frac{a}{d}} \]

      4. unpow2 N/A

         \[\leadsto \frac{b \cdot c}{\color{blue}{d \cdot d}} - \frac{a}{d} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b \cdot c}{d}}{d}} - \frac{a}{d} \]

      6. div-sub N/A

         \[\leadsto \color{blue}{\frac{\frac{b \cdot c}{d} - a}{d}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{b \cdot c}{d} - a}{d}} \]

      8. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\frac{b \cdot c}{d} + \left(\mathsf{neg}\left(a\right)\right)}}{d} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{b}{d} \cdot c} + \left(\mathsf{neg}\left(a\right)\right)}{d} \]

      10. mul-1-neg N/A

          \[\leadsto \frac{\frac{b}{d} \cdot c + \color{blue}{-1 \cdot a}}{d} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{b}{d}, c, -1 \cdot a\right)}}{d} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{b}{d}}, c, -1 \cdot a\right)}{d} \]

      13. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{b}{d}, c, \color{blue}{\mathsf{neg}\left(a\right)}\right)}{d} \]

      14. lower-neg.f64 82.1

          \[\leadsto \frac{\mathsf{fma}\left(\frac{b}{d}, c, \color{blue}{-a}\right)}{d} \]

   10. Applied rewrites 82.1%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.2 \cdot 10^{+40}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{elif}\;c \leq 7.8 \cdot 10^{-53}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 72.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -4.3 \cdot 10^{+52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+84}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -4.3e+52)
     t_0
     (if (<= d 1.85e+84) (/ (- b (* (/ a c) d)) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -4.3e+52) {
		tmp = t_0;
	} else if (d <= 1.85e+84) {
		tmp = (b - ((a / c) * d)) / 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 <= (-4.3d+52)) then
        tmp = t_0
    else if (d <= 1.85d+84) then
        tmp = (b - ((a / c) * d)) / 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 <= -4.3e+52) {
		tmp = t_0;
	} else if (d <= 1.85e+84) {
		tmp = (b - ((a / c) * d)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = -a / d
	tmp = 0
	if d <= -4.3e+52:
		tmp = t_0
	elif d <= 1.85e+84:
		tmp = (b - ((a / c) * d)) / 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 <= -4.3e+52)
		tmp = t_0;
	elseif (d <= 1.85e+84)
		tmp = Float64(Float64(b - Float64(Float64(a / c) * d)) / 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 <= -4.3e+52)
		tmp = t_0;
	elseif (d <= 1.85e+84)
		tmp = (b - ((a / c) * d)) / 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, -4.3e+52], t$95$0, If[LessEqual[d, 1.85e+84], N[(N[(b - N[(N[(a / c), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -4.3e52 or 1.85e84 < d

   1. Initial program 38.7%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{a}{d}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{a}{\mathsf{neg}\left(d\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{a}{\color{blue}{-1 \cdot d}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{-1 \cdot d}} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{a}{\color{blue}{\mathsf{neg}\left(d\right)}} \]

      6. lower-neg.f64 70.1

         \[\leadsto \frac{a}{\color{blue}{-d}} \]

   5. Applied rewrites 70.1%

      \[\leadsto \color{blue}{\frac{a}{-d}} \]

   if -4.3e52 < d < 1.85e84

   1. Initial program 74.9%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(b \cdot c - a \cdot d\right)\right)}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)}} \]

      3. div-inv N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(b \cdot c - a \cdot d\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(b \cdot c - a \cdot d\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)}} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(b \cdot c - a \cdot d\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      6. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(b \cdot c + \left(\mathsf{neg}\left(a \cdot d\right)\right)\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      7. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(b \cdot c\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot d\right)\right)\right)\right)\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{b \cdot c}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot d\right)\right)\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot c} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot d\right)\right)\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      10. remove-double-neg N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(b\right)\right) \cdot c + \color{blue}{a \cdot d}\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(b\right), c, a \cdot d\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-b}, c, a \cdot d\right) \cdot \frac{1}{\mathsf{neg}\left(\left(c \cdot c + d \cdot d\right)\right)} \]

      13. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \frac{1}{\color{blue}{-1 \cdot \left(c \cdot c + d \cdot d\right)}} \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \color{blue}{\frac{\frac{1}{-1}}{c \cdot c + d \cdot d}} \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \frac{\color{blue}{-1}}{c \cdot c + d \cdot d} \]

      16. lower-/.f64 74.9

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \color{blue}{\frac{-1}{c \cdot c + d \cdot d}} \]

      17. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \frac{-1}{\color{blue}{c \cdot c + d \cdot d}} \]

      18. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \frac{-1}{\color{blue}{d \cdot d + c \cdot c}} \]

      19. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \frac{-1}{\color{blue}{d \cdot d} + c \cdot c} \]

      20. lower-fma.f64 74.9

          \[\leadsto \mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \frac{-1}{\color{blue}{\mathsf{fma}\left(d, d, c \cdot c\right)}} \]

   4. Applied rewrites 74.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, c, a \cdot d\right) \cdot \frac{-1}{\mathsf{fma}\left(d, d, c \cdot c\right)}} \]

   5. Taylor expanded in d around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(a \cdot d\right)}{{c}^{2}}} + \frac{b}{c} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot a\right) \cdot d}}{{c}^{2}} + \frac{b}{c} \]

      3. unpow2 N/A

         \[\leadsto \frac{\left(-1 \cdot a\right) \cdot d}{\color{blue}{c \cdot c}} + \frac{b}{c} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot a}{c} \cdot \frac{d}{c}} + \frac{b}{c} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1 \cdot a}{c}, \frac{d}{c}, \frac{b}{c}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1 \cdot a}{c}}, \frac{d}{c}, \frac{b}{c}\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\mathsf{neg}\left(a\right)}}{c}, \frac{d}{c}, \frac{b}{c}\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-a}}{c}, \frac{d}{c}, \frac{b}{c}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-a}{c}, \color{blue}{\frac{d}{c}}, \frac{b}{c}\right) \]

      10. lower-/.f64 75.8

          \[\leadsto \mathsf{fma}\left(\frac{-a}{c}, \frac{d}{c}, \color{blue}{\frac{b}{c}}\right) \]

   7. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-a}{c}, \frac{d}{c}, \frac{b}{c}\right)} \]

   8. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   9. 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. *-commutative N/A

         \[\leadsto \frac{b - \frac{\color{blue}{d \cdot a}}{c}}{c} \]

      6. associate-/l* N/A

         \[\leadsto \frac{b - \color{blue}{d \cdot \frac{a}{c}}}{c} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{b - \color{blue}{d \cdot \frac{a}{c}}}{c} \]

      8. lower-/.f64 77.1

         \[\leadsto \frac{b - d \cdot \color{blue}{\frac{a}{c}}}{c} \]

   10. Applied rewrites 77.1%

       \[\leadsto \color{blue}{\frac{b - d \cdot \frac{a}{c}}{c}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.3 \cdot 10^{+52}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+84}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{+51}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{-63}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{d}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.6e+51) (/ (- a) d) (if (<= d 3.6e-63) (/ b c) (/ -1.0 (/ d a)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.6e+51) {
		tmp = -a / d;
	} else if (d <= 3.6e-63) {
		tmp = b / c;
	} else {
		tmp = -1.0 / (d / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-1.6d+51)) then
        tmp = -a / d
    else if (d <= 3.6d-63) then
        tmp = b / c
    else
        tmp = (-1.0d0) / (d / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.6e+51) {
		tmp = -a / d;
	} else if (d <= 3.6e-63) {
		tmp = b / c;
	} else {
		tmp = -1.0 / (d / a);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -1.6e+51:
		tmp = -a / d
	elif d <= 3.6e-63:
		tmp = b / c
	else:
		tmp = -1.0 / (d / a)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.6e+51)
		tmp = Float64(Float64(-a) / d);
	elseif (d <= 3.6e-63)
		tmp = Float64(b / c);
	else
		tmp = Float64(-1.0 / Float64(d / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -1.6e+51)
		tmp = -a / d;
	elseif (d <= 3.6e-63)
		tmp = b / c;
	else
		tmp = -1.0 / (d / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.6e+51], N[((-a) / d), $MachinePrecision], If[LessEqual[d, 3.6e-63], N[(b / c), $MachinePrecision], N[(-1.0 / N[(d / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.6000000000000001e51

   1. Initial program 31.1%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{a}{d}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{a}{\mathsf{neg}\left(d\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{a}{\color{blue}{-1 \cdot d}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{-1 \cdot d}} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{a}{\color{blue}{\mathsf{neg}\left(d\right)}} \]

      6. lower-neg.f64 70.7

         \[\leadsto \frac{a}{\color{blue}{-d}} \]

   5. Applied rewrites 70.7%

      \[\leadsto \color{blue}{\frac{a}{-d}} \]

   if -1.6000000000000001e51 < d < 3.60000000000000008e-63

   1. Initial program 74.1%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b}{c}} \]
   4. Step-by-step derivation

      1. lower-/.f64 70.5

         \[\leadsto \color{blue}{\frac{b}{c}} \]

   5. Applied rewrites 70.5%

      \[\leadsto \color{blue}{\frac{b}{c}} \]

   if 3.60000000000000008e-63 < d

   1. Initial program 57.6%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{a}{d}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{a}{\mathsf{neg}\left(d\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{a}{\color{blue}{-1 \cdot d}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{-1 \cdot d}} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{a}{\color{blue}{\mathsf{neg}\left(d\right)}} \]

      6. lower-neg.f64 58.8

         \[\leadsto \frac{a}{\color{blue}{-d}} \]

   5. Applied rewrites 58.8%

      \[\leadsto \color{blue}{\frac{a}{-d}} \]
   6. Step-by-step derivation

   7. Applied rewrites 59.5%

      \[\leadsto \frac{-1}{\color{blue}{\frac{d}{a}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{+51}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{-63}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{d}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.1 \cdot 10^{+46}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.95 \cdot 10^{-53}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.1e+46) (/ b c) (if (<= c 1.95e-53) (/ (- a) d) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.1e+46) {
		tmp = b / c;
	} else if (c <= 1.95e-53) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-1.1d+46)) then
        tmp = b / c
    else if (c <= 1.95d-53) then
        tmp = -a / d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.1e+46) {
		tmp = b / c;
	} else if (c <= 1.95e-53) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.1e+46:
		tmp = b / c
	elif c <= 1.95e-53:
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.1e+46)
		tmp = Float64(b / c);
	elseif (c <= 1.95e-53)
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.1e+46)
		tmp = b / c;
	elseif (c <= 1.95e-53)
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.1e+46], N[(b / c), $MachinePrecision], If[LessEqual[c, 1.95e-53], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -1.1e46 or 1.9500000000000001e-53 < c

   1. Initial program 53.5%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b}{c}} \]
   4. Step-by-step derivation

      1. lower-/.f64 67.1

         \[\leadsto \color{blue}{\frac{b}{c}} \]

   5. Applied rewrites 67.1%

      \[\leadsto \color{blue}{\frac{b}{c}} \]

   if -1.1e46 < c < 1.9500000000000001e-53

   1. Initial program 70.1%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{a}{d}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{a}{\mathsf{neg}\left(d\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{a}{\color{blue}{-1 \cdot d}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{-1 \cdot d}} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{a}{\color{blue}{\mathsf{neg}\left(d\right)}} \]

      6. lower-neg.f64 66.4

         \[\leadsto \frac{a}{\color{blue}{-d}} \]

   5. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\frac{a}{-d}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.1 \cdot 10^{+46}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.95 \cdot 10^{-53}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 43.3% 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 61.1%

   \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
2. Add Preprocessing

3. Taylor expanded in c around inf

   \[\leadsto \color{blue}{\frac{b}{c}} \]
4. Step-by-step derivation

   1. lower-/.f64 45.7

      \[\leadsto \color{blue}{\frac{b}{c}} \]

5. Applied rewrites 45.7%

   \[\leadsto \color{blue}{\frac{b}{c}} \]
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 2024304 
(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))))