Complex division, imag part

Percentage Accurate: 61.6% → 83.8%

Time: 9.7s

Alternatives: 12

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: 61.6% 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: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \mathsf{fma}\left(c, c, d \cdot d\right)\\ \mathbf{if}\;c \leq -2.7 \cdot 10^{+73}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, \frac{a}{c}, b\right)}{c}\\ \mathbf{elif}\;c \leq -6.2 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-a}{t\_1}, d, \frac{c \cdot b}{t\_1}\right)\\ \mathbf{elif}\;c \leq 7.6 \cdot 10^{-97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{elif}\;c \leq 3.6 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{t\_0}, b, \frac{a \cdot \left(-d\right)}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -\frac{d}{c}, b\right)}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))) (t_1 (fma c c (* d d))))
   (if (<= c -2.7e+73)
     (/ (fma (- d) (/ a c) b) c)
     (if (<= c -6.2e-61)
       (fma (/ (- a) t_1) d (/ (* c b) t_1))
       (if (<= c 7.6e-97)
         (/ (fma b (/ c d) (- a)) d)
         (if (<= c 3.6e+132)
           (fma (/ c t_0) b (/ (* a (- d)) t_0))
           (/ (fma a (- (/ 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, c, (d * d));
	double tmp;
	if (c <= -2.7e+73) {
		tmp = fma(-d, (a / c), b) / c;
	} else if (c <= -6.2e-61) {
		tmp = fma((-a / t_1), d, ((c * b) / t_1));
	} else if (c <= 7.6e-97) {
		tmp = fma(b, (c / d), -a) / d;
	} else if (c <= 3.6e+132) {
		tmp = fma((c / t_0), b, ((a * -d) / t_0));
	} else {
		tmp = fma(a, -(d / c), b) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = fma(c, c, Float64(d * d))
	tmp = 0.0
	if (c <= -2.7e+73)
		tmp = Float64(fma(Float64(-d), Float64(a / c), b) / c);
	elseif (c <= -6.2e-61)
		tmp = fma(Float64(Float64(-a) / t_1), d, Float64(Float64(c * b) / t_1));
	elseif (c <= 7.6e-97)
		tmp = Float64(fma(b, Float64(c / d), Float64(-a)) / d);
	elseif (c <= 3.6e+132)
		tmp = fma(Float64(c / t_0), b, Float64(Float64(a * Float64(-d)) / t_0));
	else
		tmp = Float64(fma(a, Float64(-Float64(d / c)), 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[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.7e+73], N[(N[((-d) * N[(a / c), $MachinePrecision] + b), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -6.2e-61], N[(N[((-a) / t$95$1), $MachinePrecision] * d + N[(N[(c * b), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 7.6e-97], N[(N[(b * N[(c / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 3.6e+132], N[(N[(c / t$95$0), $MachinePrecision] * b + N[(N[(a * (-d)), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(a * (-N[(d / c), $MachinePrecision]) + b), $MachinePrecision] / c), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -2.6999999999999999e73

   1. Initial program 36.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 81.6

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

   5. Applied rewrites 81.6%

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

   7. Applied rewrites 89.0%

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

   if -2.6999999999999999e73 < c < -6.1999999999999999e-61

   1. Initial program 81.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. distribute-neg-frac2 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 81.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. associate-*l/ N/A

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

   6. Applied rewrites 92.2%

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

   if -6.1999999999999999e-61 < c < 7.6000000000000001e-97

   1. Initial program 74.5%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 75.6

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in d around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. associate-/l* N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 89.3

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

   8. Applied rewrites 89.3%

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

   if 7.6000000000000001e-97 < c < 3.60000000000000016e132

   1. Initial program 78.6%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. distribute-neg-frac2 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 84.7%

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

   if 3.60000000000000016e132 < c

   1. Initial program 30.5%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 16.8

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

   5. Applied rewrites 16.8%

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

   6. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 92.0

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

   8. Applied rewrites 92.0%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.7 \cdot 10^{+73}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, \frac{a}{c}, b\right)}{c}\\ \mathbf{elif}\;c \leq -6.2 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-a}{\mathsf{fma}\left(c, c, d \cdot d\right)}, d, \frac{c \cdot b}{\mathsf{fma}\left(c, c, d \cdot d\right)}\right)\\ \mathbf{elif}\;c \leq 7.6 \cdot 10^{-97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{elif}\;c \leq 3.6 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}, b, \frac{a \cdot \left(-d\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -\frac{d}{c}, b\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ \mathbf{if}\;c \leq -2.7 \cdot 10^{+73}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, \frac{a}{c}, b\right)}{c}\\ \mathbf{elif}\;c \leq -6.2 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(-d, \frac{a}{t\_0}, \frac{c \cdot b}{t\_0}\right)\\ \mathbf{elif}\;c \leq 7.6 \cdot 10^{-97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{elif}\;c \leq 3.6 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{t\_0}, b, \frac{a \cdot \left(-d\right)}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -\frac{d}{c}, b\right)}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))))
   (if (<= c -2.7e+73)
     (/ (fma (- d) (/ a c) b) c)
     (if (<= c -6.2e-61)
       (fma (- d) (/ a t_0) (/ (* c b) t_0))
       (if (<= c 7.6e-97)
         (/ (fma b (/ c d) (- a)) d)
         (if (<= c 3.6e+132)
           (fma (/ c t_0) b (/ (* a (- d)) t_0))
           (/ (fma a (- (/ d c)) b) c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double tmp;
	if (c <= -2.7e+73) {
		tmp = fma(-d, (a / c), b) / c;
	} else if (c <= -6.2e-61) {
		tmp = fma(-d, (a / t_0), ((c * b) / t_0));
	} else if (c <= 7.6e-97) {
		tmp = fma(b, (c / d), -a) / d;
	} else if (c <= 3.6e+132) {
		tmp = fma((c / t_0), b, ((a * -d) / t_0));
	} else {
		tmp = fma(a, -(d / c), b) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	tmp = 0.0
	if (c <= -2.7e+73)
		tmp = Float64(fma(Float64(-d), Float64(a / c), b) / c);
	elseif (c <= -6.2e-61)
		tmp = fma(Float64(-d), Float64(a / t_0), Float64(Float64(c * b) / t_0));
	elseif (c <= 7.6e-97)
		tmp = Float64(fma(b, Float64(c / d), Float64(-a)) / d);
	elseif (c <= 3.6e+132)
		tmp = fma(Float64(c / t_0), b, Float64(Float64(a * Float64(-d)) / t_0));
	else
		tmp = Float64(fma(a, Float64(-Float64(d / c)), b) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.7e+73], N[(N[((-d) * N[(a / c), $MachinePrecision] + b), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -6.2e-61], N[((-d) * N[(a / t$95$0), $MachinePrecision] + N[(N[(c * b), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 7.6e-97], N[(N[(b * N[(c / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 3.6e+132], N[(N[(c / t$95$0), $MachinePrecision] * b + N[(N[(a * (-d)), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(a * (-N[(d / c), $MachinePrecision]) + b), $MachinePrecision] / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -2.6999999999999999e73

   1. Initial program 36.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 81.6

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

   5. Applied rewrites 81.6%

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

   7. Applied rewrites 89.0%

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

   if -2.6999999999999999e73 < c < -6.1999999999999999e-61

   1. Initial program 81.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 92.1

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

   4. Applied rewrites 92.1%

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

   if -6.1999999999999999e-61 < c < 7.6000000000000001e-97

   1. Initial program 74.5%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 75.6

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in d around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. associate-/l* N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 89.3

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

   8. Applied rewrites 89.3%

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

   if 7.6000000000000001e-97 < c < 3.60000000000000016e132

   1. Initial program 78.6%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. distribute-neg-frac2 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 84.7%

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

   if 3.60000000000000016e132 < c

   1. Initial program 30.5%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 16.8

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

   5. Applied rewrites 16.8%

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

   6. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 92.0

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

   8. Applied rewrites 92.0%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.7 \cdot 10^{+73}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, \frac{a}{c}, b\right)}{c}\\ \mathbf{elif}\;c \leq -6.2 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(-d, \frac{a}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{c \cdot b}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)\\ \mathbf{elif}\;c \leq 7.6 \cdot 10^{-97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{elif}\;c \leq 3.6 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}, b, \frac{a \cdot \left(-d\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -\frac{d}{c}, b\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \mathsf{fma}\left(-d, \frac{a}{t\_0}, \frac{c \cdot b}{t\_0}\right)\\ \mathbf{if}\;c \leq -2.7 \cdot 10^{+73}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, \frac{a}{c}, b\right)}{c}\\ \mathbf{elif}\;c \leq -6.2 \cdot 10^{-61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 7.6 \cdot 10^{-97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{elif}\;c \leq 5.1 \cdot 10^{+105}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -\frac{d}{c}, b\right)}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))) (t_1 (fma (- d) (/ a t_0) (/ (* c b) t_0))))
   (if (<= c -2.7e+73)
     (/ (fma (- d) (/ a c) b) c)
     (if (<= c -6.2e-61)
       t_1
       (if (<= c 7.6e-97)
         (/ (fma b (/ c d) (- a)) d)
         (if (<= c 5.1e+105) t_1 (/ (fma a (- (/ 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(-d, (a / t_0), ((c * b) / t_0));
	double tmp;
	if (c <= -2.7e+73) {
		tmp = fma(-d, (a / c), b) / c;
	} else if (c <= -6.2e-61) {
		tmp = t_1;
	} else if (c <= 7.6e-97) {
		tmp = fma(b, (c / d), -a) / d;
	} else if (c <= 5.1e+105) {
		tmp = t_1;
	} else {
		tmp = fma(a, -(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(-d), Float64(a / t_0), Float64(Float64(c * b) / t_0))
	tmp = 0.0
	if (c <= -2.7e+73)
		tmp = Float64(fma(Float64(-d), Float64(a / c), b) / c);
	elseif (c <= -6.2e-61)
		tmp = t_1;
	elseif (c <= 7.6e-97)
		tmp = Float64(fma(b, Float64(c / d), Float64(-a)) / d);
	elseif (c <= 5.1e+105)
		tmp = t_1;
	else
		tmp = Float64(fma(a, Float64(-Float64(d / c)), 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[((-d) * N[(a / t$95$0), $MachinePrecision] + N[(N[(c * b), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.7e+73], N[(N[((-d) * N[(a / c), $MachinePrecision] + b), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -6.2e-61], t$95$1, If[LessEqual[c, 7.6e-97], N[(N[(b * N[(c / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 5.1e+105], t$95$1, N[(N[(a * (-N[(d / c), $MachinePrecision]) + b), $MachinePrecision] / c), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.6999999999999999e73

   1. Initial program 36.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 81.6

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

   5. Applied rewrites 81.6%

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

   7. Applied rewrites 89.0%

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

   if -2.6999999999999999e73 < c < -6.1999999999999999e-61 or 7.6000000000000001e-97 < c < 5.09999999999999991e105

   1. Initial program 79.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 85.5

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

   4. Applied rewrites 85.5%

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

   if -6.1999999999999999e-61 < c < 7.6000000000000001e-97

   1. Initial program 74.5%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 75.6

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in d around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. associate-/l* N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 89.3

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

   8. Applied rewrites 89.3%

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

   if 5.09999999999999991e105 < c

   1. Initial program 35.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 19.0

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

   5. Applied rewrites 19.0%

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

   6. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 90.6

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

   8. Applied rewrites 90.6%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.7 \cdot 10^{+73}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, \frac{a}{c}, b\right)}{c}\\ \mathbf{elif}\;c \leq -6.2 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(-d, \frac{a}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{c \cdot b}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)\\ \mathbf{elif}\;c \leq 7.6 \cdot 10^{-97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{elif}\;c \leq 5.1 \cdot 10^{+105}:\\ \;\;\;\;\mathsf{fma}\left(-d, \frac{a}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{c \cdot b}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -\frac{d}{c}, b\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - a \cdot d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{if}\;c \leq -2.7 \cdot 10^{+73}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, \frac{a}{c}, b\right)}{c}\\ \mathbf{elif}\;c \leq -3.6 \cdot 10^{-61}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 7.6 \cdot 10^{-97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{+104}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -\frac{d}{c}, b\right)}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* a d)) (fma c c (* d d)))))
   (if (<= c -2.7e+73)
     (/ (fma (- d) (/ a c) b) c)
     (if (<= c -3.6e-61)
       t_0
       (if (<= c 7.6e-97)
         (/ (fma b (/ c d) (- a)) d)
         (if (<= c 2.6e+104) t_0 (/ (fma a (- (/ d c)) b) c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (a * d)) / fma(c, c, (d * d));
	double tmp;
	if (c <= -2.7e+73) {
		tmp = fma(-d, (a / c), b) / c;
	} else if (c <= -3.6e-61) {
		tmp = t_0;
	} else if (c <= 7.6e-97) {
		tmp = fma(b, (c / d), -a) / d;
	} else if (c <= 2.6e+104) {
		tmp = t_0;
	} else {
		tmp = fma(a, -(d / c), b) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(a * d)) / fma(c, c, Float64(d * d)))
	tmp = 0.0
	if (c <= -2.7e+73)
		tmp = Float64(fma(Float64(-d), Float64(a / c), b) / c);
	elseif (c <= -3.6e-61)
		tmp = t_0;
	elseif (c <= 7.6e-97)
		tmp = Float64(fma(b, Float64(c / d), Float64(-a)) / d);
	elseif (c <= 2.6e+104)
		tmp = t_0;
	else
		tmp = Float64(fma(a, Float64(-Float64(d / c)), b) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * b), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.7e+73], N[(N[((-d) * N[(a / c), $MachinePrecision] + b), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -3.6e-61], t$95$0, If[LessEqual[c, 7.6e-97], N[(N[(b * N[(c / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 2.6e+104], t$95$0, N[(N[(a * (-N[(d / c), $MachinePrecision]) + b), $MachinePrecision] / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.6999999999999999e73

   1. Initial program 36.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 81.6

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

   5. Applied rewrites 81.6%

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

   7. Applied rewrites 89.0%

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

   if -2.6999999999999999e73 < c < -3.60000000000000014e-61 or 7.6000000000000001e-97 < c < 2.6e104

   1. Initial program 79.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 79.8

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

   4. Applied rewrites 79.8%

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

   if -3.60000000000000014e-61 < c < 7.6000000000000001e-97

   1. Initial program 74.5%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 75.6

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in d around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. associate-/l* N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 89.3

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

   8. Applied rewrites 89.3%

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

   if 2.6e104 < c

   1. Initial program 35.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 19.0

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

   5. Applied rewrites 19.0%

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

   6. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 90.6

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

   8. Applied rewrites 90.6%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.7 \cdot 10^{+73}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, \frac{a}{c}, b\right)}{c}\\ \mathbf{elif}\;c \leq -3.6 \cdot 10^{-61}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 7.6 \cdot 10^{-97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{+104}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -\frac{d}{c}, b\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{if}\;d \leq -7.8 \cdot 10^{+98}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+16}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma c (/ b d) (- a)) d)))
   (if (<= d -7.8e+98) t_0 (if (<= d 1.7e+16) (/ (- b (/ (* a d) c)) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(c, (b / d), -a) / d;
	double tmp;
	if (d <= -7.8e+98) {
		tmp = t_0;
	} else if (d <= 1.7e+16) {
		tmp = (b - ((a * d) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(c, Float64(b / d), Float64(-a)) / d)
	tmp = 0.0
	if (d <= -7.8e+98)
		tmp = t_0;
	elseif (d <= 1.7e+16)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -7.8e+98], t$95$0, If[LessEqual[d, 1.7e+16], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -7.7999999999999999e98 or 1.7e16 < d

   1. Initial program 56.6%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 87.0

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

   5. Applied rewrites 87.0%

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

   if -7.7999999999999999e98 < d < 1.7e16

   1. Initial program 65.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 79.6

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

   5. Applied rewrites 79.6%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.8 \cdot 10^{+98}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+16}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{if}\;d \leq -2.5 \cdot 10^{+100}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+16}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma b (/ c d) (- a)) d)))
   (if (<= d -2.5e+100)
     t_0
     (if (<= d 1.7e+16) (/ (- b (/ (* a d) c)) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(b, (c / d), -a) / d;
	double tmp;
	if (d <= -2.5e+100) {
		tmp = t_0;
	} else if (d <= 1.7e+16) {
		tmp = (b - ((a * d) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(b, Float64(c / d), Float64(-a)) / d)
	tmp = 0.0
	if (d <= -2.5e+100)
		tmp = t_0;
	elseif (d <= 1.7e+16)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * N[(c / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -2.5e+100], t$95$0, If[LessEqual[d, 1.7e+16], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -2.4999999999999999e100 or 1.7e16 < d

   1. Initial program 56.6%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 78.3

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

   5. Applied rewrites 78.3%

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

   6. Taylor expanded in d around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. associate-/l* N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 86.2

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

   8. Applied rewrites 86.2%

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

   if -2.4999999999999999e100 < d < 1.7e16

   1. Initial program 65.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 79.6

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

   5. Applied rewrites 79.6%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.5 \cdot 10^{+100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+16}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+16}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -2.5e+103)
   (/ a (- d))
   (if (<= d 1.7e+16) (/ (- b (/ (* a d) c)) c) (/ (- (/ (* c b) d) a) d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2.5e+103) {
		tmp = a / -d;
	} else if (d <= 1.7e+16) {
		tmp = (b - ((a * d) / c)) / c;
	} else {
		tmp = (((c * b) / d) - a) / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-2.5d+103)) then
        tmp = a / -d
    else if (d <= 1.7d+16) then
        tmp = (b - ((a * d) / c)) / c
    else
        tmp = (((c * b) / d) - a) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2.5e+103) {
		tmp = a / -d;
	} else if (d <= 1.7e+16) {
		tmp = (b - ((a * d) / c)) / c;
	} else {
		tmp = (((c * b) / d) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -2.5e+103:
		tmp = a / -d
	elif d <= 1.7e+16:
		tmp = (b - ((a * d) / c)) / c
	else:
		tmp = (((c * b) / d) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -2.5e+103)
		tmp = Float64(a / Float64(-d));
	elseif (d <= 1.7e+16)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	else
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -2.5e+103)
		tmp = a / -d;
	elseif (d <= 1.7e+16)
		tmp = (b - ((a * d) / c)) / c;
	else
		tmp = (((c * b) / d) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -2.5e+103], N[(a / (-d)), $MachinePrecision], If[LessEqual[d, 1.7e+16], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.5e103

   1. Initial program 49.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 80.1

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

   5. Applied rewrites 80.1%

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

   if -2.5e103 < d < 1.7e16

   1. Initial program 65.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 79.6

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

   5. Applied rewrites 79.6%

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

   if 1.7e16 < d

   1. Initial program 61.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 9.8

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

   5. Applied rewrites 9.8%

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

   6. Taylor expanded in d around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 85.2

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

   8. Applied rewrites 85.2%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+16}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{-d}\\ \mathbf{if}\;d \leq -2.5 \cdot 10^{+103}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{+16}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ a (- d))))
   (if (<= d -2.5e+103)
     t_0
     (if (<= d 2.9e+16) (/ (- b (/ (* a d) c)) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = a / -d;
	double tmp;
	if (d <= -2.5e+103) {
		tmp = t_0;
	} else if (d <= 2.9e+16) {
		tmp = (b - ((a * d) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / -d
    if (d <= (-2.5d+103)) then
        tmp = t_0
    else if (d <= 2.9d+16) then
        tmp = (b - ((a * d) / c)) / c
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = a / -d;
	double tmp;
	if (d <= -2.5e+103) {
		tmp = t_0;
	} else if (d <= 2.9e+16) {
		tmp = (b - ((a * d) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = a / -d
	tmp = 0
	if d <= -2.5e+103:
		tmp = t_0
	elif d <= 2.9e+16:
		tmp = (b - ((a * d) / c)) / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(a / Float64(-d))
	tmp = 0.0
	if (d <= -2.5e+103)
		tmp = t_0;
	elseif (d <= 2.9e+16)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = a / -d;
	tmp = 0.0;
	if (d <= -2.5e+103)
		tmp = t_0;
	elseif (d <= 2.9e+16)
		tmp = (b - ((a * d) / c)) / c;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(a / (-d)), $MachinePrecision]}, If[LessEqual[d, -2.5e+103], t$95$0, If[LessEqual[d, 2.9e+16], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -2.5e103 or 2.9e16 < d

   1. Initial program 56.6%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 78.3

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

   5. Applied rewrites 78.3%

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

   if -2.5e103 < d < 2.9e16

   1. Initial program 65.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 79.6

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

   5. Applied rewrites 79.6%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{+16}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{-d}\\ \mathbf{if}\;d \leq -2.4 \cdot 10^{+139}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -9500000000:\\ \;\;\;\;a \cdot \frac{-d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq 2.55 \cdot 10^{+16}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ a (- d))))
   (if (<= d -2.4e+139)
     t_0
     (if (<= d -9500000000.0)
       (* a (/ (- d) (fma c c (* d d))))
       (if (<= d 2.55e+16) (/ b c) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = a / -d;
	double tmp;
	if (d <= -2.4e+139) {
		tmp = t_0;
	} else if (d <= -9500000000.0) {
		tmp = a * (-d / fma(c, c, (d * d)));
	} else if (d <= 2.55e+16) {
		tmp = b / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(a / Float64(-d))
	tmp = 0.0
	if (d <= -2.4e+139)
		tmp = t_0;
	elseif (d <= -9500000000.0)
		tmp = Float64(a * Float64(Float64(-d) / fma(c, c, Float64(d * d))));
	elseif (d <= 2.55e+16)
		tmp = Float64(b / c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(a / (-d)), $MachinePrecision]}, If[LessEqual[d, -2.4e+139], t$95$0, If[LessEqual[d, -9500000000.0], N[(a * N[((-d) / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.55e+16], N[(b / c), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.40000000000000008e139 or 2.55e16 < d

   1. Initial program 54.6%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 78.9

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

   5. Applied rewrites 78.9%

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

   if -2.40000000000000008e139 < d < -9.5e9

   1. Initial program 68.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 57.1

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

   5. Applied rewrites 57.1%

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

   6. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 75.2

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

   8. Applied rewrites 75.2%

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

   if -9.5e9 < d < 2.55e16

   1. Initial program 65.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 67.0

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

   5. Applied rewrites 67.0%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.4 \cdot 10^{+139}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{elif}\;d \leq -9500000000:\\ \;\;\;\;a \cdot \frac{-d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq 2.55 \cdot 10^{+16}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -17000000000:\\ \;\;\;\;\frac{-1}{\frac{d}{a}}\\ \mathbf{elif}\;d \leq 2.55 \cdot 10^{+16}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -17000000000.0)
   (/ -1.0 (/ d a))
   (if (<= d 2.55e+16) (/ b c) (/ a (- d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -17000000000.0) {
		tmp = -1.0 / (d / a);
	} else if (d <= 2.55e+16) {
		tmp = b / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-17000000000.0d0)) then
        tmp = (-1.0d0) / (d / a)
    else if (d <= 2.55d+16) then
        tmp = b / c
    else
        tmp = a / -d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -17000000000.0) {
		tmp = -1.0 / (d / a);
	} else if (d <= 2.55e+16) {
		tmp = b / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -17000000000.0:
		tmp = -1.0 / (d / a)
	elif d <= 2.55e+16:
		tmp = b / c
	else:
		tmp = a / -d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -17000000000.0)
		tmp = Float64(-1.0 / Float64(d / a));
	elseif (d <= 2.55e+16)
		tmp = Float64(b / c);
	else
		tmp = Float64(a / Float64(-d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -17000000000.0)
		tmp = -1.0 / (d / a);
	elseif (d <= 2.55e+16)
		tmp = b / c;
	else
		tmp = a / -d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -17000000000.0], N[(-1.0 / N[(d / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.55e+16], N[(b / c), $MachinePrecision], N[(a / (-d)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.7e10

   1. Initial program 53.9%

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

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. clear-num N/A

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

      6. frac-2neg N/A

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

      7. metadata-eval N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 53.9%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 70.2

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

   7. Applied rewrites 70.2%

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

   if -1.7e10 < d < 2.55e16

   1. Initial program 65.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 67.0

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

   5. Applied rewrites 67.0%

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

   if 2.55e16 < d

   1. Initial program 61.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 77.3

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

   5. Applied rewrites 77.3%

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

Alternative 11: 63.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{-d}\\ \mathbf{if}\;d \leq -17000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.55 \cdot 10^{+16}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ a (- d))))
   (if (<= d -17000000000.0) t_0 (if (<= d 2.55e+16) (/ b c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = a / -d;
	double tmp;
	if (d <= -17000000000.0) {
		tmp = t_0;
	} else if (d <= 2.55e+16) {
		tmp = b / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / -d
    if (d <= (-17000000000.0d0)) then
        tmp = t_0
    else if (d <= 2.55d+16) then
        tmp = b / c
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = a / -d;
	double tmp;
	if (d <= -17000000000.0) {
		tmp = t_0;
	} else if (d <= 2.55e+16) {
		tmp = b / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = a / -d
	tmp = 0
	if d <= -17000000000.0:
		tmp = t_0
	elif d <= 2.55e+16:
		tmp = b / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(a / Float64(-d))
	tmp = 0.0
	if (d <= -17000000000.0)
		tmp = t_0;
	elseif (d <= 2.55e+16)
		tmp = Float64(b / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = a / -d;
	tmp = 0.0;
	if (d <= -17000000000.0)
		tmp = t_0;
	elseif (d <= 2.55e+16)
		tmp = b / c;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(a / (-d)), $MachinePrecision]}, If[LessEqual[d, -17000000000.0], t$95$0, If[LessEqual[d, 2.55e+16], N[(b / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.7e10 or 2.55e16 < d

   1. Initial program 57.7%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 73.8

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

   5. Applied rewrites 73.8%

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

   if -1.7e10 < d < 2.55e16

   1. Initial program 65.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 67.0

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

   5. Applied rewrites 67.0%

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

Alternative 12: 42.3% 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 61.9%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 43.1

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

5. Applied rewrites 43.1%

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

Developer Target 1: 99.4% 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 2024233 
(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))))