quadp (p42, positive)

Percentage Accurate: 52.0% → 85.1%

Time: 10.3s

Alternatives: 6

Speedup: 2.4×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{+152}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.06 \cdot 10^{-13}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, c \cdot -4, 0\right)\right)} - b\right) \cdot 0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.5e+152)
   (- 0.0 (/ b a))
   (if (<= b 2.06e-13)
     (/ (* (- (sqrt (fma b b (fma a (* c -4.0) 0.0))) b) 0.5) a)
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.5e+152) {
		tmp = 0.0 - (b / a);
	} else if (b <= 2.06e-13) {
		tmp = ((sqrt(fma(b, b, fma(a, (c * -4.0), 0.0))) - b) * 0.5) / a;
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9.5e+152)
		tmp = Float64(0.0 - Float64(b / a));
	elseif (b <= 2.06e-13)
		tmp = Float64(Float64(Float64(sqrt(fma(b, b, fma(a, Float64(c * -4.0), 0.0))) - b) * 0.5) / a);
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9.5e+152], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.06e-13], N[(N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * 0.5), $MachinePrecision] / a), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.49999999999999916e152

   1. Initial program 28.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

         \[\leadsto \frac{b}{\color{blue}{0 - a}} \]

      7. --lowering--.f64 96.5

         \[\leadsto \frac{b}{\color{blue}{0 - a}} \]

   5. Simplified 96.5%

      \[\leadsto \color{blue}{\frac{b}{0 - a}} \]
   6. Step-by-step derivation

      1. neg-sub0 N/A

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

      2. neg-lowering-neg.f64 96.5

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

   7. Applied egg-rr 96.5%

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

   if -9.49999999999999916e152 < b < 2.06000000000000006e-13

   1. Initial program 82.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{b \cdot b + 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a} \]

      2. clear-num N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\frac{1}{\frac{b \cdot b + 4 \cdot \left(a \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}}}}{2 \cdot a} \]

      3. sqrt-div N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{b \cdot b + 4 \cdot \left(a \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}}}}{2 \cdot a} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \frac{\color{blue}{1}}{\sqrt{\frac{b \cdot b + 4 \cdot \left(a \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}}}{2 \cdot a} \]

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

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \color{blue}{\frac{1}{\sqrt{\frac{b \cdot b + 4 \cdot \left(a \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}}}}{2 \cdot a} \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \frac{1}{\color{blue}{\sqrt{\frac{b \cdot b + 4 \cdot \left(a \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}}}}{2 \cdot a} \]

      7. clear-num N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \frac{1}{\sqrt{\color{blue}{\frac{1}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{b \cdot b + 4 \cdot \left(a \cdot c\right)}}}}}}{2 \cdot a} \]

      8. flip-- N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \frac{1}{\sqrt{\frac{1}{\color{blue}{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}}{2 \cdot a} \]

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

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \frac{1}{\sqrt{\color{blue}{\frac{1}{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}}{2 \cdot a} \]

      10. sub-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \frac{1}{\sqrt{\frac{1}{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)\right)}}}}}{2 \cdot a} \]

   4. Applied egg-rr 81.8%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\frac{1}{\sqrt{\frac{1}{\mathsf{fma}\left(a, c \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)}}}}}{2 \cdot a} \]
   5. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\mathsf{neg}\left(b\right)\right) + \frac{1}{\sqrt{\frac{1}{a \cdot \left(c \cdot -4\right) + \left(b \cdot b + 0\right)}}}}{2}}{a}} \]

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

         \[\leadsto \color{blue}{\frac{\frac{\left(\mathsf{neg}\left(b\right)\right) + \frac{1}{\sqrt{\frac{1}{a \cdot \left(c \cdot -4\right) + \left(b \cdot b + 0\right)}}}}{2}}{a}} \]

   6. Applied egg-rr 82.8%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, c \cdot -4, 0\right)\right)} - b\right) \cdot 0.5}{a}} \]

   if 2.06000000000000006e-13 < b

   1. Initial program 16.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

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

         \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

      4. /-lowering-/.f64 85.7

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

   5. Simplified 85.7%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

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

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

      3. /-lowering-/.f64 85.7

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

   7. Applied egg-rr 85.7%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{+152}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.06 \cdot 10^{-13}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, c \cdot -4, 0\right)\right)} - b\right) \cdot 0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{-104}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, 0 - \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 2.25 \cdot 10^{-13}:\\ \;\;\;\;\frac{0.5 \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.85e-104)
   (fma b (/ c (* b b)) (- 0.0 (/ b a)))
   (if (<= b 2.25e-13)
     (/ (* 0.5 (- (sqrt (* a (* c -4.0))) b)) a)
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.85e-104) {
		tmp = fma(b, (c / (b * b)), (0.0 - (b / a)));
	} else if (b <= 2.25e-13) {
		tmp = (0.5 * (sqrt((a * (c * -4.0))) - b)) / a;
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.85e-104)
		tmp = fma(b, Float64(c / Float64(b * b)), Float64(0.0 - Float64(b / a)));
	elseif (b <= 2.25e-13)
		tmp = Float64(Float64(0.5 * Float64(sqrt(Float64(a * Float64(c * -4.0))) - b)) / a);
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.85e-104], N[(b * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.25e-13], N[(N[(0.5 * N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.85e-104

   1. Initial program 59.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. remove-double-neg N/A

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

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

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

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

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

      12. unpow2 N/A

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

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

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

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

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

      17. mul-1-neg N/A

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

      18. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{c}{b \cdot b}, \frac{b}{\color{blue}{0 - a}}\right) \]

      19. --lowering--.f64 87.1

          \[\leadsto \mathsf{fma}\left(b, \frac{c}{b \cdot b}, \frac{b}{\color{blue}{0 - a}}\right) \]

   5. Simplified 87.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{c}{b \cdot b}, \frac{b}{0 - a}\right)} \]

   if -1.85e-104 < b < 2.25e-13

   1. Initial program 74.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{b \cdot b + 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a} \]

      2. clear-num N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\frac{1}{\frac{b \cdot b + 4 \cdot \left(a \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}}}}{2 \cdot a} \]

      3. sqrt-div N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{b \cdot b + 4 \cdot \left(a \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}}}}{2 \cdot a} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \frac{\color{blue}{1}}{\sqrt{\frac{b \cdot b + 4 \cdot \left(a \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}}}{2 \cdot a} \]

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

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \color{blue}{\frac{1}{\sqrt{\frac{b \cdot b + 4 \cdot \left(a \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}}}}{2 \cdot a} \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \frac{1}{\color{blue}{\sqrt{\frac{b \cdot b + 4 \cdot \left(a \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}}}}{2 \cdot a} \]

      7. clear-num N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \frac{1}{\sqrt{\color{blue}{\frac{1}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{b \cdot b + 4 \cdot \left(a \cdot c\right)}}}}}}{2 \cdot a} \]

      8. flip-- N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \frac{1}{\sqrt{\frac{1}{\color{blue}{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}}{2 \cdot a} \]

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

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \frac{1}{\sqrt{\color{blue}{\frac{1}{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}}{2 \cdot a} \]

      10. sub-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \frac{1}{\sqrt{\frac{1}{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)\right)}}}}}{2 \cdot a} \]

   4. Applied egg-rr 74.5%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\frac{1}{\sqrt{\frac{1}{\mathsf{fma}\left(a, c \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)}}}}}{2 \cdot a} \]
   5. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\mathsf{neg}\left(b\right)\right) + \frac{1}{\sqrt{\frac{1}{a \cdot \left(c \cdot -4\right) + \left(b \cdot b + 0\right)}}}}{2}}{a}} \]

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

         \[\leadsto \color{blue}{\frac{\frac{\left(\mathsf{neg}\left(b\right)\right) + \frac{1}{\sqrt{\frac{1}{a \cdot \left(c \cdot -4\right) + \left(b \cdot b + 0\right)}}}}{2}}{a}} \]

   6. Applied egg-rr 76.0%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, c \cdot -4, 0\right)\right)} - b\right) \cdot 0.5}{a}} \]

   7. Taylor expanded in b around 0

      \[\leadsto \frac{\left(\sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}} - b\right) \cdot \frac{1}{2}}{a} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}} - b\right) \cdot \frac{1}{2}}{a} \]

      2. rem-square-sqrt N/A

         \[\leadsto \frac{\left(\sqrt{\left(a \cdot c\right) \cdot \color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)}} - b\right) \cdot \frac{1}{2}}{a} \]

      3. unpow2 N/A

         \[\leadsto \frac{\left(\sqrt{\left(a \cdot c\right) \cdot \color{blue}{{\left(\sqrt{-4}\right)}^{2}}} - b\right) \cdot \frac{1}{2}}{a} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(\sqrt{\color{blue}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}} - b\right) \cdot \frac{1}{2}}{a} \]

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

         \[\leadsto \frac{\left(\sqrt{\color{blue}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}} - b\right) \cdot \frac{1}{2}}{a} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(\sqrt{a \cdot \left(c \cdot \color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)}\right)} - b\right) \cdot \frac{1}{2}}{a} \]

      7. rem-square-sqrt N/A

         \[\leadsto \frac{\left(\sqrt{a \cdot \left(c \cdot \color{blue}{-4}\right)} - b\right) \cdot \frac{1}{2}}{a} \]

      8. *-lowering-*.f64 73.7

         \[\leadsto \frac{\left(\sqrt{a \cdot \color{blue}{\left(c \cdot -4\right)}} - b\right) \cdot 0.5}{a} \]

   9. Simplified 73.7%

      \[\leadsto \frac{\left(\sqrt{\color{blue}{a \cdot \left(c \cdot -4\right)}} - b\right) \cdot 0.5}{a} \]

   if 2.25e-13 < b

   1. Initial program 16.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

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

         \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

      4. /-lowering-/.f64 85.7

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

   5. Simplified 85.7%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

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

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

      3. /-lowering-/.f64 85.7

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

   7. Applied egg-rr 85.7%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{-104}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, 0 - \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 2.25 \cdot 10^{-13}:\\ \;\;\;\;\frac{0.5 \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-104}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, 0 - \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.8e-104)
   (fma b (/ c (* b b)) (- 0.0 (/ b a)))
   (if (<= b 4.2e-13)
     (* (/ 0.5 a) (+ b (sqrt (* -4.0 (* a c)))))
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.8e-104) {
		tmp = fma(b, (c / (b * b)), (0.0 - (b / a)));
	} else if (b <= 4.2e-13) {
		tmp = (0.5 / a) * (b + sqrt((-4.0 * (a * c))));
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.8e-104)
		tmp = fma(b, Float64(c / Float64(b * b)), Float64(0.0 - Float64(b / a)));
	elseif (b <= 4.2e-13)
		tmp = Float64(Float64(0.5 / a) * Float64(b + sqrt(Float64(-4.0 * Float64(a * c)))));
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.8e-104], N[(b * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.2e-13], N[(N[(0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.7999999999999999e-104

   1. Initial program 59.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. remove-double-neg N/A

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

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

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

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

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

      12. unpow2 N/A

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

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

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

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

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

      17. mul-1-neg N/A

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

      18. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{c}{b \cdot b}, \frac{b}{\color{blue}{0 - a}}\right) \]

      19. --lowering--.f64 87.1

          \[\leadsto \mathsf{fma}\left(b, \frac{c}{b \cdot b}, \frac{b}{\color{blue}{0 - a}}\right) \]

   5. Simplified 87.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{c}{b \cdot b}, \frac{b}{0 - a}\right)} \]

   if -1.7999999999999999e-104 < b < 4.19999999999999977e-13

   1. Initial program 74.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied egg-rr 73.5%

      \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)}\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \frac{\frac{1}{2}}{a} \cdot \left(b + \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{1}{2}}{a} \cdot \left(b + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}\right) \]

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

         \[\leadsto \frac{\frac{1}{2}}{a} \cdot \left(b + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}\right) \]

      3. *-lowering-*.f64 73.4

         \[\leadsto \frac{0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -4}\right) \]

   7. Simplified 73.4%

      \[\leadsto \frac{0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}\right) \]

   if 4.19999999999999977e-13 < b

   1. Initial program 16.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

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

         \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

      4. /-lowering-/.f64 85.7

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

   5. Simplified 85.7%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

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

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

      3. /-lowering-/.f64 85.7

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

   7. Applied egg-rr 85.7%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-104}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, 0 - \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (- 0.0 (/ b a)) (- 0.0 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = 0.0 - (b / a);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = 0.0d0 - (b / a)
    else
        tmp = 0.0d0 - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = 0.0 - (b / a);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = 0.0 - (b / a)
	else:
		tmp = 0.0 - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(0.0 - Float64(b / a));
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = 0.0 - (b / a);
	else
		tmp = 0.0 - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 65.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

         \[\leadsto \frac{b}{\color{blue}{0 - a}} \]

      7. --lowering--.f64 68.7

         \[\leadsto \frac{b}{\color{blue}{0 - a}} \]

   5. Simplified 68.7%

      \[\leadsto \color{blue}{\frac{b}{0 - a}} \]
   6. Step-by-step derivation

      1. neg-sub0 N/A

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

      2. neg-lowering-neg.f64 68.7

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

   7. Applied egg-rr 68.7%

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

   if -4.999999999999985e-310 < b

   1. Initial program 37.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

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

         \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

      4. /-lowering-/.f64 58.5

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

   5. Simplified 58.5%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

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

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

      3. /-lowering-/.f64 58.5

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

   7. Applied egg-rr 58.5%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 35.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ 0 - \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (- 0.0 (/ c b)))

C

double code(double a, double b, double c) {
	return 0.0 - (c / b);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0 - (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 - (c / b);
}

Python

def code(a, b, c):
	return 0.0 - (c / b)

Julia

function code(a, b, c)
	return Float64(0.0 - Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 - (c / b);
end

Wolfram

code[a_, b_, c_] := N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around inf

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

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

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   4. /-lowering-/.f64 29.2

      \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

5. Simplified 29.2%

   \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
6. Step-by-step derivation

   1. sub0-neg N/A

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

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

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

   3. /-lowering-/.f64 29.2

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

7. Applied egg-rr 29.2%

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

8. Final simplification 29.2%

   \[\leadsto 0 - \frac{c}{b} \]
9. Add Preprocessing

Alternative 6: 12.1% accurate, 50.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 52.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
2. Add Preprocessing

4. Applied egg-rr 35.9%

   \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)}\right)} \]

5. Taylor expanded in a around 0

   \[\leadsto \frac{\frac{1}{2}}{a} \cdot \left(b + \sqrt{\color{blue}{{b}^{2}}}\right) \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\frac{1}{2}}{a} \cdot \left(b + \sqrt{\color{blue}{b \cdot b}}\right) \]

   2. *-lowering-*.f64 8.6

      \[\leadsto \frac{0.5}{a} \cdot \left(b + \sqrt{\color{blue}{b \cdot b}}\right) \]

7. Simplified 8.6%

   \[\leadsto \frac{0.5}{a} \cdot \left(b + \sqrt{\color{blue}{b \cdot b}}\right) \]

8. Taylor expanded in b around -inf

   \[\leadsto \color{blue}{0} \]
9. Step-by-step derivation

10. Simplified 10.2%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t\_0 - t\_1} \cdot \sqrt{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t\_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{t\_2 - \frac{b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{\frac{b}{2} + t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ (- t_2 (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) t_2)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = (t_2 - (b / 2.0)) / a
	else:
		tmp_1 = -c / ((b / 2.0) + t_2)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(Float64(t_2 - Float64(b / 2.0)) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(Float64(b / 2.0) + t_2));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = (t_2 - (b / 2.0)) / a;
	else
		tmp_2 = -c / ((b / 2.0) + t_2);
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Abs[N[(b / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(b / 2.0), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024194 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2)) x)) (sqrt (+ (fabs (/ b 2)) x))) (hypot (/ b 2) x))))) (if (< b 0) (/ (- sqtD (/ b 2)) a) (/ (- c) (+ (/ b 2) sqtD)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))