Complex division, imag part

Percentage Accurate: 62.1% → 86.3%

Time: 8.8s

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: 62.1% 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: 86.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-a, \frac{d}{\mathsf{fma}\left(c, c, d \cdot d\right)}, b \cdot \frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)\\ t_1 := \frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{if}\;d \leq -7 \cdot 10^{+115}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -2.4 \cdot 10^{-49}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-158}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d \cdot a, \frac{-1}{c}, b\right)}{c}\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{+145}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma (- a) (/ d (fma c c (* d d))) (* b (/ c (fma d d (* c c))))))
        (t_1 (/ (fma c (/ b d) (- a)) d)))
   (if (<= d -7e+115)
     t_1
     (if (<= d -2.4e-49)
       t_0
       (if (<= d 1.35e-158)
         (/ (fma (* d a) (/ -1.0 c) b) c)
         (if (<= d 3.8e+145) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(-a, (d / fma(c, c, (d * d))), (b * (c / fma(d, d, (c * c)))));
	double t_1 = fma(c, (b / d), -a) / d;
	double tmp;
	if (d <= -7e+115) {
		tmp = t_1;
	} else if (d <= -2.4e-49) {
		tmp = t_0;
	} else if (d <= 1.35e-158) {
		tmp = fma((d * a), (-1.0 / c), b) / c;
	} else if (d <= 3.8e+145) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(Float64(-a), Float64(d / fma(c, c, Float64(d * d))), Float64(b * Float64(c / fma(d, d, Float64(c * c)))))
	t_1 = Float64(fma(c, Float64(b / d), Float64(-a)) / d)
	tmp = 0.0
	if (d <= -7e+115)
		tmp = t_1;
	elseif (d <= -2.4e-49)
		tmp = t_0;
	elseif (d <= 1.35e-158)
		tmp = Float64(fma(Float64(d * a), Float64(-1.0 / c), b) / c);
	elseif (d <= 3.8e+145)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) * N[(d / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(c / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -7e+115], t$95$1, If[LessEqual[d, -2.4e-49], t$95$0, If[LessEqual[d, 1.35e-158], N[(N[(N[(d * a), $MachinePrecision] * N[(-1.0 / c), $MachinePrecision] + b), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 3.8e+145], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.00000000000000011e115 or 3.80000000000000012e145 < d

   1. Initial program 31.2%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      13. /-lowering-/.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. neg-lowering-neg.f64 88.4

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

   5. Simplified 88.4%

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

   if -7.00000000000000011e115 < d < -2.39999999999999992e-49 or 1.3499999999999999e-158 < d < 3.80000000000000012e145

   1. Initial program 81.1%

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-lowering-*.f64 85.2

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

   4. Applied egg-rr 85.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 87.4

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

   6. Applied egg-rr 87.4%

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

   if -2.39999999999999992e-49 < d < 1.3499999999999999e-158

   1. Initial program 69.4%

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-lowering-*.f64 63.9

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

   4. Applied egg-rr 63.9%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 68.7

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

   6. Applied egg-rr 68.7%

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

   7. Taylor expanded in d around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. /-lowering-/.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. associate-*r* N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. neg-lowering-neg.f64 N/A

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

      16. /-lowering-/.f64 89.7

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

   9. Simplified 89.7%

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

       1. distribute-lft-neg-out N/A

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

       2. associate-*r/ N/A

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

       3. distribute-frac-neg2 N/A

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

       4. div-inv N/A

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

       5. accelerator-lowering-fma.f64 N/A

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 N/A

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

       8. metadata-eval N/A

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

       9. frac-2neg N/A

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

       10. /-lowering-/.f64 90.6

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

   11. Applied egg-rr 90.6%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(d \cdot a, \frac{-1}{c}, b\right)}}{c} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 84.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{if}\;d \leq -1.8 \cdot 10^{+100}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -2.4 \cdot 10^{-49}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{-155}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d \cdot a, \frac{-1}{c}, b\right)}{c}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(c, -b, d \cdot a\right) \cdot \frac{-1}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma c (/ b d) (- a)) d)))
   (if (<= d -1.8e+100)
     t_0
     (if (<= d -2.4e-49)
       (/ (- (* c b) (* d a)) (+ (* d d) (* c c)))
       (if (<= d 3.6e-155)
         (/ (fma (* d a) (/ -1.0 c) b) c)
         (if (<= d 4.5e+102)
           (* (fma c (- b) (* d a)) (/ -1.0 (fma c c (* d d))))
           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 <= -1.8e+100) {
		tmp = t_0;
	} else if (d <= -2.4e-49) {
		tmp = ((c * b) - (d * a)) / ((d * d) + (c * c));
	} else if (d <= 3.6e-155) {
		tmp = fma((d * a), (-1.0 / c), b) / c;
	} else if (d <= 4.5e+102) {
		tmp = fma(c, -b, (d * a)) * (-1.0 / fma(c, c, (d * d)));
	} 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 <= -1.8e+100)
		tmp = t_0;
	elseif (d <= -2.4e-49)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(d * d) + Float64(c * c)));
	elseif (d <= 3.6e-155)
		tmp = Float64(fma(Float64(d * a), Float64(-1.0 / c), b) / c);
	elseif (d <= 4.5e+102)
		tmp = Float64(fma(c, Float64(-b), Float64(d * a)) * Float64(-1.0 / fma(c, c, Float64(d * d))));
	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, -1.8e+100], t$95$0, If[LessEqual[d, -2.4e-49], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.6e-155], N[(N[(N[(d * a), $MachinePrecision] * N[(-1.0 / c), $MachinePrecision] + b), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 4.5e+102], N[(N[(c * (-b) + N[(d * a), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.8e100 or 4.50000000000000021e102 < d

   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

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      13. /-lowering-/.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. neg-lowering-neg.f64 85.7

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

   5. Simplified 85.7%

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

   if -1.8e100 < d < -2.39999999999999992e-49

   1. Initial program 88.4%

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

   if -2.39999999999999992e-49 < d < 3.59999999999999989e-155

   1. Initial program 69.7%

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-lowering-*.f64 64.3

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

   4. Applied egg-rr 64.3%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 69.1

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

   6. Applied egg-rr 69.1%

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

   7. Taylor expanded in d around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. /-lowering-/.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. associate-*r* N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. neg-lowering-neg.f64 N/A

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

      16. /-lowering-/.f64 89.8

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

   9. Simplified 89.8%

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

       1. distribute-lft-neg-out N/A

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

       2. associate-*r/ N/A

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

       3. distribute-frac-neg2 N/A

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

       4. div-inv N/A

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

       5. accelerator-lowering-fma.f64 N/A

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 N/A

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

       8. metadata-eval N/A

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

       9. frac-2neg N/A

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

       10. /-lowering-/.f64 90.7

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

   11. Applied egg-rr 90.7%

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

   if 3.59999999999999989e-155 < d < 4.50000000000000021e102

   1. Initial program 83.4%

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

      1. frac-2neg N/A

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

      2. div-inv N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-neg-in N/A

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

      6. *-commutative N/A

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

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

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

      8. remove-double-neg N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 N/A

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

      17. accelerator-lowering-fma.f64 N/A

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

      18. *-lowering-*.f64 83.4

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

   4. Applied egg-rr 83.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, -b, d \cdot a\right) \cdot \frac{-1}{\mathsf{fma}\left(c, c, d \cdot d\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.8 \cdot 10^{+100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq -2.4 \cdot 10^{-49}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{-155}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d \cdot a, \frac{-1}{c}, b\right)}{c}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(c, -b, d \cdot a\right) \cdot \frac{-1}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot b - d \cdot a\\ t_1 := \frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{if}\;d \leq -1.8 \cdot 10^{+100}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -2.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{t\_0}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-155}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d \cdot a, \frac{-1}{c}, b\right)}{c}\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+105}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* c b) (* d a))) (t_1 (/ (fma c (/ b d) (- a)) d)))
   (if (<= d -1.8e+100)
     t_1
     (if (<= d -2.4e-48)
       (/ t_0 (+ (* d d) (* c c)))
       (if (<= d 2.4e-155)
         (/ (fma (* d a) (/ -1.0 c) b) c)
         (if (<= d 3e+105) (/ t_0 (fma d d (* c c))) t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c * b) - (d * a);
	double t_1 = fma(c, (b / d), -a) / d;
	double tmp;
	if (d <= -1.8e+100) {
		tmp = t_1;
	} else if (d <= -2.4e-48) {
		tmp = t_0 / ((d * d) + (c * c));
	} else if (d <= 2.4e-155) {
		tmp = fma((d * a), (-1.0 / c), b) / c;
	} else if (d <= 3e+105) {
		tmp = t_0 / fma(d, d, (c * c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c * b) - Float64(d * a))
	t_1 = Float64(fma(c, Float64(b / d), Float64(-a)) / d)
	tmp = 0.0
	if (d <= -1.8e+100)
		tmp = t_1;
	elseif (d <= -2.4e-48)
		tmp = Float64(t_0 / Float64(Float64(d * d) + Float64(c * c)));
	elseif (d <= 2.4e-155)
		tmp = Float64(fma(Float64(d * a), Float64(-1.0 / c), b) / c);
	elseif (d <= 3e+105)
		tmp = Float64(t_0 / fma(d, d, Float64(c * c)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -1.8e+100], t$95$1, If[LessEqual[d, -2.4e-48], N[(t$95$0 / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.4e-155], N[(N[(N[(d * a), $MachinePrecision] * N[(-1.0 / c), $MachinePrecision] + b), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 3e+105], N[(t$95$0 / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.8e100 or 3.0000000000000001e105 < d

   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

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      13. /-lowering-/.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. neg-lowering-neg.f64 85.7

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

   5. Simplified 85.7%

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

   if -1.8e100 < d < -2.4e-48

   1. Initial program 88.4%

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

   if -2.4e-48 < d < 2.4e-155

   1. Initial program 69.7%

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-lowering-*.f64 64.3

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

   4. Applied egg-rr 64.3%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 69.1

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

   6. Applied egg-rr 69.1%

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

   7. Taylor expanded in d around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. /-lowering-/.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. associate-*r* N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. neg-lowering-neg.f64 N/A

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

      16. /-lowering-/.f64 89.8

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

   9. Simplified 89.8%

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

       1. distribute-lft-neg-out N/A

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

       2. associate-*r/ N/A

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

       3. distribute-frac-neg2 N/A

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

       4. div-inv N/A

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

       5. accelerator-lowering-fma.f64 N/A

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 N/A

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

       8. metadata-eval N/A

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

       9. frac-2neg N/A

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

       10. /-lowering-/.f64 90.7

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

   11. Applied egg-rr 90.7%

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

   if 2.4e-155 < d < 3.0000000000000001e105

   1. Initial program 83.4%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 83.4

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

   4. Applied egg-rr 83.4%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.8 \cdot 10^{+100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq -2.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-155}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d \cdot a, \frac{-1}{c}, b\right)}{c}\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+105}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ t_1 := \frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{if}\;d \leq -7.2 \cdot 10^{+99}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{-49}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.95 \cdot 10^{-155}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d \cdot a, \frac{-1}{c}, b\right)}{c}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{+104}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (fma d d (* c c))))
        (t_1 (/ (fma c (/ b d) (- a)) d)))
   (if (<= d -7.2e+99)
     t_1
     (if (<= d -7.2e-49)
       t_0
       (if (<= d 2.95e-155)
         (/ (fma (* d a) (/ -1.0 c) b) c)
         (if (<= d 1.15e+104) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / fma(d, d, (c * c));
	double t_1 = fma(c, (b / d), -a) / d;
	double tmp;
	if (d <= -7.2e+99) {
		tmp = t_1;
	} else if (d <= -7.2e-49) {
		tmp = t_0;
	} else if (d <= 2.95e-155) {
		tmp = fma((d * a), (-1.0 / c), b) / c;
	} else if (d <= 1.15e+104) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(d * a)) / fma(d, d, Float64(c * c)))
	t_1 = Float64(fma(c, Float64(b / d), Float64(-a)) / d)
	tmp = 0.0
	if (d <= -7.2e+99)
		tmp = t_1;
	elseif (d <= -7.2e-49)
		tmp = t_0;
	elseif (d <= 2.95e-155)
		tmp = Float64(fma(Float64(d * a), Float64(-1.0 / c), b) / c);
	elseif (d <= 1.15e+104)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -7.2e+99], t$95$1, If[LessEqual[d, -7.2e-49], t$95$0, If[LessEqual[d, 2.95e-155], N[(N[(N[(d * a), $MachinePrecision] * N[(-1.0 / c), $MachinePrecision] + b), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.15e+104], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.2000000000000003e99 or 1.14999999999999992e104 < d

   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

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      13. /-lowering-/.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. neg-lowering-neg.f64 85.7

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

   5. Simplified 85.7%

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

   if -7.2000000000000003e99 < d < -7.19999999999999939e-49 or 2.95e-155 < d < 1.14999999999999992e104

   1. Initial program 85.0%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 85.0

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

   4. Applied egg-rr 85.0%

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

   if -7.19999999999999939e-49 < d < 2.95e-155

   1. Initial program 69.7%

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-lowering-*.f64 64.3

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

   4. Applied egg-rr 64.3%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 69.1

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

   6. Applied egg-rr 69.1%

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

   7. Taylor expanded in d around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. /-lowering-/.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. associate-*r* N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. neg-lowering-neg.f64 N/A

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

      16. /-lowering-/.f64 89.8

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

   9. Simplified 89.8%

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

       1. distribute-lft-neg-out N/A

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

       2. associate-*r/ N/A

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

       3. distribute-frac-neg2 N/A

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

       4. div-inv N/A

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

       5. accelerator-lowering-fma.f64 N/A

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 N/A

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

       8. metadata-eval N/A

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

       9. frac-2neg N/A

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

       10. /-lowering-/.f64 90.7

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

   11. Applied egg-rr 90.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(d \cdot a, \frac{-1}{c}, b\right)}}{c} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{+99}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{-49}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 2.95 \cdot 10^{-155}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d \cdot a, \frac{-1}{c}, b\right)}{c}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{+104}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ t_1 := \frac{c \cdot b - d \cdot a}{d \cdot d}\\ \mathbf{if}\;d \leq -9.2 \cdot 10^{+80}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -1.55 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-70}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 1.26 \cdot 10^{+137}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)) (t_1 (/ (- (* c b) (* d a)) (* d d))))
   (if (<= d -9.2e+80)
     t_0
     (if (<= d -1.55e-26)
       t_1
       (if (<= d 1.45e-70)
         (/ (- b (/ (* d a) c)) c)
         (if (<= d 1.26e+137) t_1 t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double t_1 = ((c * b) - (d * a)) / (d * d);
	double tmp;
	if (d <= -9.2e+80) {
		tmp = t_0;
	} else if (d <= -1.55e-26) {
		tmp = t_1;
	} else if (d <= 1.45e-70) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 1.26e+137) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = -a / d
    t_1 = ((c * b) - (d * a)) / (d * d)
    if (d <= (-9.2d+80)) then
        tmp = t_0
    else if (d <= (-1.55d-26)) then
        tmp = t_1
    else if (d <= 1.45d-70) then
        tmp = (b - ((d * a) / c)) / c
    else if (d <= 1.26d+137) then
        tmp = t_1
    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 t_1 = ((c * b) - (d * a)) / (d * d);
	double tmp;
	if (d <= -9.2e+80) {
		tmp = t_0;
	} else if (d <= -1.55e-26) {
		tmp = t_1;
	} else if (d <= 1.45e-70) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 1.26e+137) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = -a / d
	t_1 = ((c * b) - (d * a)) / (d * d)
	tmp = 0
	if d <= -9.2e+80:
		tmp = t_0
	elif d <= -1.55e-26:
		tmp = t_1
	elif d <= 1.45e-70:
		tmp = (b - ((d * a) / c)) / c
	elif d <= 1.26e+137:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	t_1 = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(d * d))
	tmp = 0.0
	if (d <= -9.2e+80)
		tmp = t_0;
	elseif (d <= -1.55e-26)
		tmp = t_1;
	elseif (d <= 1.45e-70)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (d <= 1.26e+137)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = -a / d;
	t_1 = ((c * b) - (d * a)) / (d * d);
	tmp = 0.0;
	if (d <= -9.2e+80)
		tmp = t_0;
	elseif (d <= -1.55e-26)
		tmp = t_1;
	elseif (d <= 1.45e-70)
		tmp = (b - ((d * a) / c)) / c;
	elseif (d <= 1.26e+137)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -9.2e+80], t$95$0, If[LessEqual[d, -1.55e-26], t$95$1, If[LessEqual[d, 1.45e-70], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.26e+137], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -9.20000000000000016e80 or 1.2599999999999999e137 < d

   1. Initial program 36.5%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. /-lowering-/.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. neg-lowering-neg.f64 79.3

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

   5. Simplified 79.3%

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

   if -9.20000000000000016e80 < d < -1.54999999999999992e-26 or 1.44999999999999986e-70 < d < 1.2599999999999999e137

   1. Initial program 88.4%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 66.8

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

   5. Simplified 66.8%

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

   if -1.54999999999999992e-26 < d < 1.44999999999999986e-70

   1. Initial program 71.1%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.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. --lowering--.f64 N/A

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

      5. /-lowering-/.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. *-lowering-*.f64 84.6

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

   5. Simplified 84.6%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.2 \cdot 10^{+80}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -1.55 \cdot 10^{-26}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{d \cdot d}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-70}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 1.26 \cdot 10^{+137}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ t_1 := \frac{c \cdot b - d \cdot a}{d \cdot d}\\ \mathbf{if}\;d \leq -9 \cdot 10^{+80}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -1.9 \cdot 10^{-49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-100}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{+137}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)) (t_1 (/ (- (* c b) (* d a)) (* d d))))
   (if (<= d -9e+80)
     t_0
     (if (<= d -1.9e-49)
       t_1
       (if (<= d 5.8e-100) (/ b c) (if (<= d 5.8e+137) t_1 t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double t_1 = ((c * b) - (d * a)) / (d * d);
	double tmp;
	if (d <= -9e+80) {
		tmp = t_0;
	} else if (d <= -1.9e-49) {
		tmp = t_1;
	} else if (d <= 5.8e-100) {
		tmp = b / c;
	} else if (d <= 5.8e+137) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = -a / d
    t_1 = ((c * b) - (d * a)) / (d * d)
    if (d <= (-9d+80)) then
        tmp = t_0
    else if (d <= (-1.9d-49)) then
        tmp = t_1
    else if (d <= 5.8d-100) then
        tmp = b / c
    else if (d <= 5.8d+137) then
        tmp = t_1
    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 t_1 = ((c * b) - (d * a)) / (d * d);
	double tmp;
	if (d <= -9e+80) {
		tmp = t_0;
	} else if (d <= -1.9e-49) {
		tmp = t_1;
	} else if (d <= 5.8e-100) {
		tmp = b / c;
	} else if (d <= 5.8e+137) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = -a / d
	t_1 = ((c * b) - (d * a)) / (d * d)
	tmp = 0
	if d <= -9e+80:
		tmp = t_0
	elif d <= -1.9e-49:
		tmp = t_1
	elif d <= 5.8e-100:
		tmp = b / c
	elif d <= 5.8e+137:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	t_1 = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(d * d))
	tmp = 0.0
	if (d <= -9e+80)
		tmp = t_0;
	elseif (d <= -1.9e-49)
		tmp = t_1;
	elseif (d <= 5.8e-100)
		tmp = Float64(b / c);
	elseif (d <= 5.8e+137)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = -a / d;
	t_1 = ((c * b) - (d * a)) / (d * d);
	tmp = 0.0;
	if (d <= -9e+80)
		tmp = t_0;
	elseif (d <= -1.9e-49)
		tmp = t_1;
	elseif (d <= 5.8e-100)
		tmp = b / c;
	elseif (d <= 5.8e+137)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -9e+80], t$95$0, If[LessEqual[d, -1.9e-49], t$95$1, If[LessEqual[d, 5.8e-100], N[(b / c), $MachinePrecision], If[LessEqual[d, 5.8e+137], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -9.00000000000000013e80 or 5.79999999999999969e137 < d

   1. Initial program 36.5%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. /-lowering-/.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. neg-lowering-neg.f64 79.3

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

   5. Simplified 79.3%

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

   if -9.00000000000000013e80 < d < -1.8999999999999999e-49 or 5.79999999999999951e-100 < d < 5.79999999999999969e137

   1. Initial program 85.3%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 63.7

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

   5. Simplified 63.7%

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

   if -1.8999999999999999e-49 < d < 5.79999999999999951e-100

   1. Initial program 70.5%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 75.4

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

   5. Simplified 75.4%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9 \cdot 10^{+80}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -1.9 \cdot 10^{-49}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{d \cdot d}\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-100}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{+137}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -8 \cdot 10^{+99}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -1.7 \cdot 10^{-49}:\\ \;\;\;\;\frac{d \cdot a}{-\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq 10^{-97}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{+147}:\\ \;\;\;\;-a \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -8e+99)
     t_0
     (if (<= d -1.7e-49)
       (/ (* d a) (- (fma c c (* d d))))
       (if (<= d 1e-97)
         (/ b c)
         (if (<= d 1.05e+147) (- (* a (/ d (fma d d (* c c))))) t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -8e+99) {
		tmp = t_0;
	} else if (d <= -1.7e-49) {
		tmp = (d * a) / -fma(c, c, (d * d));
	} else if (d <= 1e-97) {
		tmp = b / c;
	} else if (d <= 1.05e+147) {
		tmp = -(a * (d / fma(d, d, (c * c))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (d <= -8e+99)
		tmp = t_0;
	elseif (d <= -1.7e-49)
		tmp = Float64(Float64(d * a) / Float64(-fma(c, c, Float64(d * d))));
	elseif (d <= 1e-97)
		tmp = Float64(b / c);
	elseif (d <= 1.05e+147)
		tmp = Float64(-Float64(a * Float64(d / fma(d, d, Float64(c * c)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, If[LessEqual[d, -8e+99], t$95$0, If[LessEqual[d, -1.7e-49], N[(N[(d * a), $MachinePrecision] / (-N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[d, 1e-97], N[(b / c), $MachinePrecision], If[LessEqual[d, 1.05e+147], (-N[(a * N[(d / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -7.9999999999999997e99 or 1.05000000000000003e147 < d

   1. Initial program 32.8%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   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. /-lowering-/.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. neg-lowering-neg.f64 81.8

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

   5. Simplified 81.8%

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

   if -7.9999999999999997e99 < d < -1.70000000000000002e-49

   1. Initial program 88.8%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. *-lowering-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 57.6

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

   5. Simplified 57.6%

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

      1. accelerator-lowering-fma.f64 N/A

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

      2. *-lowering-*.f64 57.6

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

   7. Applied egg-rr 57.6%

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

   if -1.70000000000000002e-49 < d < 1.00000000000000004e-97

   1. Initial program 70.8%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 75.6

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

   5. Simplified 75.6%

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

   if 1.00000000000000004e-97 < d < 1.05000000000000003e147

   1. Initial program 80.0%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. *-lowering-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 58.4

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

   5. Simplified 58.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. neg-lowering-neg.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 60.5

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

   7. Applied egg-rr 60.5%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8 \cdot 10^{+99}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -1.7 \cdot 10^{-49}:\\ \;\;\;\;\frac{d \cdot a}{-\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq 10^{-97}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{+147}:\\ \;\;\;\;-a \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ t_1 := -a \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{if}\;d \leq -1.4 \cdot 10^{+91}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -1.1 \cdot 10^{-49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 8.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{+144}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)) (t_1 (- (* a (/ d (fma d d (* c c)))))))
   (if (<= d -1.4e+91)
     t_0
     (if (<= d -1.1e-49)
       t_1
       (if (<= d 8.5e-98) (/ b c) (if (<= d 4.2e+144) t_1 t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double t_1 = -(a * (d / fma(d, d, (c * c))));
	double tmp;
	if (d <= -1.4e+91) {
		tmp = t_0;
	} else if (d <= -1.1e-49) {
		tmp = t_1;
	} else if (d <= 8.5e-98) {
		tmp = b / c;
	} else if (d <= 4.2e+144) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	t_1 = Float64(-Float64(a * Float64(d / fma(d, d, Float64(c * c)))))
	tmp = 0.0
	if (d <= -1.4e+91)
		tmp = t_0;
	elseif (d <= -1.1e-49)
		tmp = t_1;
	elseif (d <= 8.5e-98)
		tmp = Float64(b / c);
	elseif (d <= 4.2e+144)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, Block[{t$95$1 = (-N[(a * N[(d / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[d, -1.4e+91], t$95$0, If[LessEqual[d, -1.1e-49], t$95$1, If[LessEqual[d, 8.5e-98], N[(b / c), $MachinePrecision], If[LessEqual[d, 4.2e+144], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.3999999999999999e91 or 4.19999999999999993e144 < d

   1. Initial program 33.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. /-lowering-/.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. neg-lowering-neg.f64 81.1

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

   5. Simplified 81.1%

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

   if -1.3999999999999999e91 < d < -1.09999999999999995e-49 or 8.4999999999999997e-98 < d < 4.19999999999999993e144

   1. Initial program 83.9%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. *-lowering-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 58.2

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

   5. Simplified 58.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. neg-lowering-neg.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 59.6

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

   7. Applied egg-rr 59.6%

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

   if -1.09999999999999995e-49 < d < 8.4999999999999997e-98

   1. Initial program 70.8%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 75.6

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

   5. Simplified 75.6%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.4 \cdot 10^{+91}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -1.1 \cdot 10^{-49}:\\ \;\;\;\;-a \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 8.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{+144}:\\ \;\;\;\;-a \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{if}\;d \leq -6.2 \cdot 10^{-27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{-72}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d \cdot a, \frac{-1}{c}, b\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma c (/ b d) (- a)) d)))
   (if (<= d -6.2e-27)
     t_0
     (if (<= d 4.6e-72) (/ (fma (* d a) (/ -1.0 c) b) 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 <= -6.2e-27) {
		tmp = t_0;
	} else if (d <= 4.6e-72) {
		tmp = fma((d * a), (-1.0 / c), b) / 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 <= -6.2e-27)
		tmp = t_0;
	elseif (d <= 4.6e-72)
		tmp = Float64(fma(Float64(d * a), Float64(-1.0 / c), b) / 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, -6.2e-27], t$95$0, If[LessEqual[d, 4.6e-72], N[(N[(N[(d * a), $MachinePrecision] * N[(-1.0 / c), $MachinePrecision] + b), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -6.1999999999999997e-27 or 4.59999999999999989e-72 < d

   1. Initial program 55.9%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      13. /-lowering-/.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. neg-lowering-neg.f64 78.2

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

   5. Simplified 78.2%

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

   if -6.1999999999999997e-27 < d < 4.59999999999999989e-72

   1. Initial program 71.1%

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-lowering-*.f64 66.9

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

   4. Applied egg-rr 66.9%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 72.4

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

   6. Applied egg-rr 72.4%

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

   7. Taylor expanded in d around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. /-lowering-/.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. associate-*r* N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. neg-lowering-neg.f64 N/A

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

      16. /-lowering-/.f64 84.3

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

   9. Simplified 84.3%

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

       1. distribute-lft-neg-out N/A

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

       2. associate-*r/ N/A

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

       3. distribute-frac-neg2 N/A

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

       4. div-inv N/A

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

       5. accelerator-lowering-fma.f64 N/A

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 N/A

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

       8. metadata-eval N/A

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

       9. frac-2neg N/A

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

       10. /-lowering-/.f64 84.7

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

   11. Applied egg-rr 84.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(d \cdot a, \frac{-1}{c}, b\right)}}{c} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 79.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{if}\;d \leq -1.75 \cdot 10^{-26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-70}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma c (/ b d) (- a)) d)))
   (if (<= d -1.75e-26)
     t_0
     (if (<= d 1.45e-70) (/ (- b (/ (* d a) 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 <= -1.75e-26) {
		tmp = t_0;
	} else if (d <= 1.45e-70) {
		tmp = (b - ((d * a) / 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 <= -1.75e-26)
		tmp = t_0;
	elseif (d <= 1.45e-70)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / 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, -1.75e-26], t$95$0, If[LessEqual[d, 1.45e-70], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.74999999999999992e-26 or 1.44999999999999986e-70 < d

   1. Initial program 55.9%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      13. /-lowering-/.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. neg-lowering-neg.f64 78.2

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

   5. Simplified 78.2%

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

   if -1.74999999999999992e-26 < d < 1.44999999999999986e-70

   1. Initial program 71.1%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.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. --lowering--.f64 N/A

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

      5. /-lowering-/.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. *-lowering-*.f64 84.6

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

   5. Simplified 84.6%

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

Alternative 11: 63.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -2.3 \cdot 10^{-49}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 10^{-97}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -2.3e-49) t_0 (if (<= d 1e-97) (/ b c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -2.3e-49) {
		tmp = t_0;
	} else if (d <= 1e-97) {
		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 <= (-2.3d-49)) then
        tmp = t_0
    else if (d <= 1d-97) 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 <= -2.3e-49) {
		tmp = t_0;
	} else if (d <= 1e-97) {
		tmp = b / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = -a / d
	tmp = 0
	if d <= -2.3e-49:
		tmp = t_0
	elif d <= 1e-97:
		tmp = b / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (d <= -2.3e-49)
		tmp = t_0;
	elseif (d <= 1e-97)
		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 <= -2.3e-49)
		tmp = t_0;
	elseif (d <= 1e-97)
		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, -2.3e-49], t$95$0, If[LessEqual[d, 1e-97], N[(b / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -2.2999999999999999e-49 or 1.00000000000000004e-97 < d

   1. Initial program 57.6%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.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. neg-lowering-neg.f64 62.8

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

   5. Simplified 62.8%

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

   if -2.2999999999999999e-49 < d < 1.00000000000000004e-97

   1. Initial program 70.8%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 75.6

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

   5. Simplified 75.6%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{-49}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq 10^{-97}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.7% 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 62.9%

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

3. Taylor expanded in c around inf

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

   1. /-lowering-/.f64 40.8

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

5. Simplified 40.8%

   \[\leadsto \color{blue}{\frac{b}{c}} \]
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 2024198 
(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))))