quadp (p42, positive)

Percentage Accurate: 52.6% → 86.3%

Time: 12.5s

Alternatives: 7

Speedup: 2.4×

• could not determine a ground truth

  1. a = 2.019707714234225e-307
  2. b = 2.2130301631800477e+57
  3. c = 4.6273233374406555e-155

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: 52.6% 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: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+26}:\\ \;\;\;\;\frac{\left|b \cdot \sqrt{\mathsf{fma}\left(a \cdot -4, \frac{c}{b \cdot b}, 1\right)}\right| - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.04 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{c \cdot \mathsf{fma}\left(a, -4, \frac{b \cdot b}{c}\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.5e+26)
   (/ (- (fabs (* b (sqrt (fma (* a -4.0) (/ c (* b b)) 1.0)))) b) (* a 2.0))
   (if (<= b 1.04e-109)
     (/ (- (sqrt (* c (fma a -4.0 (/ (* b b) c)))) b) (* a 2.0))
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e+26) {
		tmp = (fabs((b * sqrt(fma((a * -4.0), (c / (b * b)), 1.0)))) - b) / (a * 2.0);
	} else if (b <= 1.04e-109) {
		tmp = (sqrt((c * fma(a, -4.0, ((b * b) / c)))) - b) / (a * 2.0);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.5e+26)
		tmp = Float64(Float64(abs(Float64(b * sqrt(fma(Float64(a * -4.0), Float64(c / Float64(b * b)), 1.0)))) - b) / Float64(a * 2.0));
	elseif (b <= 1.04e-109)
		tmp = Float64(Float64(sqrt(Float64(c * fma(a, -4.0, Float64(Float64(b * b) / c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.5e+26], N[(N[(N[Abs[N[(b * N[Sqrt[N[(N[(a * -4.0), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.04e-109], N[(N[(N[Sqrt[N[(c * N[(a * -4.0 + N[(N[(b * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.49999999999999999e26

   1. Initial program 63.0%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-lft-identity N/A

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

      16. +-commutative N/A

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

      17. distribute-rgt-out N/A

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

      18. mul0-lft N/A

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

      19. accelerator-lowering-fma.f64 63.0

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

   4. Applied egg-rr 63.0%

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

   5. Taylor expanded in b around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 63.0

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

   7. Simplified 63.0%

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

      1. rem-square-sqrt N/A

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

      2. rem-sqrt-square N/A

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

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

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

      4. sqrt-prod N/A

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

      5. sqrt-prod N/A

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

      6. rem-square-sqrt N/A

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

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

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

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

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

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

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

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

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

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

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

      12. *-lowering-*.f64 97.3

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

   9. Applied egg-rr 97.3%

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

   if -1.49999999999999999e26 < b < 1.03999999999999996e-109

   1. Initial program 85.2%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-lft-identity N/A

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

      16. +-commutative N/A

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

      17. distribute-rgt-out N/A

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

      18. mul0-lft N/A

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

      19. accelerator-lowering-fma.f64 85.2

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

   4. Applied egg-rr 85.2%

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

   5. Taylor expanded in c around inf

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

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

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

      2. *-commutative N/A

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

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

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 85.2

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

   7. Simplified 85.2%

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

   if 1.03999999999999996e-109 < b

   1. Initial program 18.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 88.2

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

   5. Simplified 88.2%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   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 88.2

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

   7. Applied egg-rr 88.2%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+26}:\\ \;\;\;\;\frac{\left|b \cdot \sqrt{\mathsf{fma}\left(a \cdot -4, \frac{c}{b \cdot b}, 1\right)}\right| - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.04 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{c \cdot \mathsf{fma}\left(a, -4, \frac{b \cdot b}{c}\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{b}{0 - a}\\ \mathbf{elif}\;b \leq 1.04 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.4e+154)
   (/ b (- 0.0 a))
   (if (<= b 1.04e-109)
     (/ (- (sqrt (fma c (* a -4.0) (* b b))) b) (* a 2.0))
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.4e+154) {
		tmp = b / (0.0 - a);
	} else if (b <= 1.04e-109) {
		tmp = (sqrt(fma(c, (a * -4.0), (b * b))) - b) / (a * 2.0);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.4e+154)
		tmp = Float64(b / Float64(0.0 - a));
	elseif (b <= 1.04e-109)
		tmp = Float64(Float64(sqrt(fma(c, Float64(a * -4.0), Float64(b * b))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.4e+154], N[(b / N[(0.0 - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.04e-109], N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.40000000000000015e154

   1. Initial program 41.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 97.8

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

   5. Simplified 97.8%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 97.8

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

   7. Applied egg-rr 97.8%

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

   if -2.40000000000000015e154 < b < 1.03999999999999996e-109

   1. Initial program 87.2%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-lft-identity N/A

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

      16. +-commutative N/A

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

      17. distribute-rgt-out N/A

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

      18. mul0-lft N/A

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

      19. accelerator-lowering-fma.f64 87.2

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

   4. Applied egg-rr 87.2%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 87.2

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

   7. Simplified 87.2%

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

   if 1.03999999999999996e-109 < b

   1. Initial program 18.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 88.2

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

   5. Simplified 88.2%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   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 88.2

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

   7. Applied egg-rr 88.2%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{b}{0 - a}\\ \mathbf{elif}\;b \leq 1.04 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, \frac{b}{0 - a}\right)\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-116}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.3e-26)
   (fma b (/ c (* b b)) (/ b (- 0.0 a)))
   (if (<= b 6e-116)
     (/ (- (sqrt (* -4.0 (* a c))) b) (* a 2.0))
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.3e-26) {
		tmp = fma(b, (c / (b * b)), (b / (0.0 - a)));
	} else if (b <= 6e-116) {
		tmp = (sqrt((-4.0 * (a * c))) - b) / (a * 2.0);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.3e-26)
		tmp = fma(b, Float64(c / Float64(b * b)), Float64(b / Float64(0.0 - a)));
	elseif (b <= 6e-116)
		tmp = Float64(Float64(sqrt(Float64(-4.0 * Float64(a * c))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.3e-26], N[(b * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(b / N[(0.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6e-116], N[(N[(N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.29999999999999988e-26

   1. Initial program 66.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 87.3

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

   5. Simplified 87.3%

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

   if -4.29999999999999988e-26 < b < 6.00000000000000053e-116

   1. Initial program 84.3%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-lft-identity N/A

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

      16. +-commutative N/A

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

      17. distribute-rgt-out N/A

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

      18. mul0-lft N/A

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

      19. accelerator-lowering-fma.f64 84.3

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

   4. Applied egg-rr 84.3%

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

   5. Taylor expanded in a around inf

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 75.6

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

   7. Simplified 75.6%

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

   if 6.00000000000000053e-116 < b

   1. Initial program 18.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 88.2

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

   5. Simplified 88.2%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   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 88.2

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

   7. Applied egg-rr 88.2%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, \frac{b}{0 - a}\right)\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-116}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, \frac{b}{0 - a}\right)\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-110}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{-4 \cdot \left(a \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.75e-26)
   (fma b (/ c (* b b)) (/ b (- 0.0 a)))
   (if (<= b 4.4e-110)
     (* (/ -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 <= -1.75e-26) {
		tmp = fma(b, (c / (b * b)), (b / (0.0 - a)));
	} else if (b <= 4.4e-110) {
		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 <= -1.75e-26)
		tmp = fma(b, Float64(c / Float64(b * b)), Float64(b / Float64(0.0 - a)));
	elseif (b <= 4.4e-110)
		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, -1.75e-26], N[(b * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(b / N[(0.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.4e-110], 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 < -1.74999999999999992e-26

   1. Initial program 66.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 87.3

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

   5. Simplified 87.3%

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

   if -1.74999999999999992e-26 < b < 4.3999999999999999e-110

   1. Initial program 84.3%

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

   4. Applied egg-rr 84.1%

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

   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. *-lowering-*.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 75.5

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

   7. Simplified 75.5%

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

   if 4.3999999999999999e-110 < b

   1. Initial program 18.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 88.2

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

   5. Simplified 88.2%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   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 88.2

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

   7. Applied egg-rr 88.2%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, \frac{b}{0 - a}\right)\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-110}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{-4 \cdot \left(a \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{-285}:\\ \;\;\;\;\frac{b}{0 - a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.8e-285) (/ b (- 0.0 a)) (- 0.0 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.8e-285) {
		tmp = b / (0.0 - 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 <= 5.8d-285) then
        tmp = b / (0.0d0 - 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 <= 5.8e-285) {
		tmp = b / (0.0 - a);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.8e-285:
		tmp = b / (0.0 - a)
	else:
		tmp = 0.0 - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.8e-285)
		tmp = Float64(b / Float64(0.0 - 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 <= 5.8e-285)
		tmp = b / (0.0 - a);
	else
		tmp = 0.0 - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.8e-285], N[(b / N[(0.0 - a), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.7999999999999999e-285

   1. Initial program 73.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 63.2

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

   5. Simplified 63.2%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 63.2

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

   7. Applied egg-rr 63.2%

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

   if 5.7999999999999999e-285 < b

   1. Initial program 34.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 69.9

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

   5. Simplified 69.9%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   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 69.9

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

   7. Applied egg-rr 69.9%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{-285}:\\ \;\;\;\;\frac{b}{0 - a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 34.4% 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 53.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
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 36.7

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

5. Simplified 36.7%

   \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
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 36.7

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

7. Applied egg-rr 36.7%

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

8. Final simplification 36.7%

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

Alternative 7: 11.2% 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 53.2%

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

4. Applied egg-rr 53.1%

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

   1. sub-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. clear-num N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. div-inv N/A

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

   7. frac-2neg N/A

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

   8. neg-mul-1 N/A

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

   9. *-commutative N/A

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

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

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

   11. metadata-eval N/A

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

   12. times-frac N/A

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

   13. metadata-eval N/A

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

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

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

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

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

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

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

6. Applied egg-rr 51.8%

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

7. Taylor expanded in c around 0

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

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

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

   3. mul0-rgt 10.3

      \[\leadsto \color{blue}{0} \]

9. Simplified 10.3%

   \[\leadsto \color{blue}{0} \]
10. 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 2024199 
(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)))